Publications

Monographs

  • V. John, Finite Element Methods for Incompressible Flow Problems, 51 of Springer Series in Computational Mathematics, Springer International Publishing AG, Cham, 2016, xiii+812 pages, (Monograph Published).

  • S. Canann, S. Owen, H. Si, eds., 25th International Meshing Roundtable, 163 of Procedia Engineering, Elsevier, Amsterdam, 2016, 366 pages, (Collection Published).

Articles in Refereed Journals

  • F. Anker, Ch. Bayer, M. Eigel, M. Ladkau, J. Neumann, J.G.M. Schoenmakers, SDE based regression for random PDEs, SIAM Journal on Scientific Computing, 39 (2017) pp. A1168--A1200.
    Abstract
    A simulation based method for the numerical solution of PDE with random coefficients is presented. By the Feynman-Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behaviour.

  • F. Anker, Ch. Bayer, M. Eigel, J. Neumann, J.G.M. Schoenmakers, A fully adaptive interpolated stochastic sampling method for linear random PDEs, International Journal for Uncertainty Quantification, 7 (2017) pp. 189--205, DOI 10.1615/Int.J.UncertaintyQuantification.2017019428 .
    Abstract
    A numerical method for the fully adaptive sampling and interpolation of PDE with random data is presented. It is based on the idea that the solution of the PDE with stochastic data can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE). The physical domain is decomposed subject to a non-uniform grid and a classical Euler scheme is employed to approximately solve the SDE at grid vertices. Interpolation with a conforming finite element basis is employed to reconstruct a global solution of the problem. An a posteriori error estimator is introduced which provides a measure of the different error contributions. This facilitates the formulation of an adaptive algorithm to control the overall error by either reducing the stochastic error by locally evaluating more samples, or the approximation error by locally refining the underlying mesh. Numerical examples illustrate the performance of the presented novel method.

  • F. Dassi, H. Si, S. Perotto, T. Streckenbach, A priori anisotropic mesh adaptation driven by a higher dimensional embedding, Computer-Aided Design, 85 (2017) pp. 111--122, DOI https://doi.org/10.1016/j.cad.2016.07.012 .
    Abstract
    In this paper we provide a novel anisotropic mesh adaptation technique for adaptive finite element analysis. It is based on the concept of higher dimensional embedding, which was exploited in [1-4] to obtain an anisotropic curvature adapted mesh that fits a complex surface in R^3. In the context of adaptive finite element simulation, the solution (which is an unknown function f : Ω ⊂ R^d → R) is sought by iteratively modifying a finite element mesh according to a mesh sizing field described via a (discrete) metric tensor field that is typically obtained through an error estimator. We proposed to use a higher dimensional embedding, Φ_f(x) := (x_1, …, x_d, s f (x_1, …, x_d), s ∇ f (x_1, …, x_d))^t, instead of the mesh sizing field for the mesh adaption. This embedding contains both informations of the function f itself and its gradient. An isotropic mesh in this embedded space will correspond to an anisotropic mesh in the actual space, where the mesh elements are stretched and aligned according to the features of the function f. To better capture the anisotropy and gradation of the mesh, it is necessary to balance the contribution of the components in this embedding. We have properly adjusted Φ_f(x) for adaptive finite element analysis. To better understand and validate the proposed mesh adaptation strategy, we first provide a series of experimental tests for piecewise linear interpolation of known functions. We then applied this approach in an adaptive finite element solution of partial di erential equations. Both tests are performed on two-dimensional domains in which adaptive triangular meshes are generated. We compared these results with the ones obtained by the software BAMG - a metric-based adaptive mesh generator. The errors measured in the L_2 norm are comparable. Moreover, our meshes captured the anisotropy more accurately than the meshes of BAMG.

  • W. Dreyer, C. Guhlke, Sharp limit of the viscous Cahn--Hilliard equation and thermodynamic consistency, Continuum Mechanics and Thermodynamics, 29 (2017) pp. 913--934.
    Abstract
    Diffuse and sharp interface models represent two alternatives to describe phase transitions with an interface between two coexisting phases. The two model classes can be independently formulated. Thus there arises the problem whether the sharp limit of the diffuse model fits into the setting of a corresponding sharp interface model. We call a diffuse model admissible if its sharp limit produces interfacial jump conditions that are consistent with the balance equations and the 2nd law of thermodynamics for sharp interfaces. We use special cases of the viscous Cahn-Hilliard equation to show that there are admissible as well as non-admissible diffuse interface models.

  • P. Farrell, A. Linke, Uniform second order convergence of a complete flux scheme on unstructured 1D grids for a singularly perturbed advection-diffusion equation and some multidimensional extensions, Journal of Scientific Computing, 72 (2017) pp. 373--395, DOI 10.1007/s10915-017-0361-7 .
    Abstract
    The accurate and efficient discretization of singularly perturbed advection-diffusion equations on arbitrary 2D and 3D domains remains an open problem. An interesting approach to tackle this problem is the complete flux scheme (CFS) proposed by G. D. Thiart and further investigated by J. ten Thije Boonkkamp. For the CFS, uniform second order convergence has been proven on structured grids. We extend a version of the CFS to unstructured grids for a steady singularly perturbed advection-diffusion equation. By construction, the novel finite volume scheme is nodally exact in 1D for piecewise constant source terms. This property allows to use elegant continuous arguments in order to prove uniform second order convergence on unstructured one-dimensional grids. Numerical results verify the predicted bounds and suggest that by aligning the finite volume grid along the velocity field uniform second order convergence can be obtained in higher space dimensions as well.

  • U. Wilbrandt, C. Bartsch, N. Ahmed, N. Alia, F. Anker, L. Blank, A. Caiazzo, S. Ganesan, S. Giere, G. Matthies, R. Meesala, A. Shamim, J. Venkatensan, V. John, ParMooN -- A modernized program package based on mapped finite elements, Computers & Mathematics with Applications. An International Journal, 74 (2017) pp. 74--88, DOI 10.1016/j.camwa.2016.12.020 .

  • S. Giere, V. John, Towards physically admissible reduced-order solutions for convection-diffusion problems, Applied Mathematics Letters, 73 (2017) pp. 78--83, DOI 10.1016/j.aml.2017.03.022 .

  • V. Wiedmeyer, F. Anker, C. Bartsch, A. Voigt, V. John, K. Sundmacher, Continuous crystallization in a helically-coiled flow tube: Analysis of flow field, residence time behavior and crystal growth, Industrial and Engineering Chemistry Research, 56 (2017) pp. 3699--3712, DOI 10.1021/acs.iecr.6b04279 .

  • G.R. Barrenechea, V. John, P. Knobloch, An algebraic flux correction scheme satisfying the discrete maximum principle and linearity preservation on general meshes, Mathematical Models & Methods in Applied Sciences, 27 (2017) pp. 525--548, DOI 10.1142/S0218202517500087 .

  • J. Bulling, V. John, P. Knobloch, Isogeometric analysis for flows around a cylinder, Applied Mathematics Letters, 63 (2017) pp. 65--70.

  • P.L. Lederer, A. Linke, Ch. Merdon, J. Schöberl, Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements, SIAM Journal on Numerical Analysis, 55 (2017) pp. 1291--1314.
    Abstract
    Classical inf-sup stable mixed finite elements for the incompressible (Navier--)Stokes equations are not pressure-robust, i.e., their velocity errors depend on the continuous pressure. How-ever, a modification only in the right hand side of a Stokes discretization is able to reestablish pressure-robustness, as shown recently for several inf-sup stable Stokes elements with discontinuous discrete pressures. In this contribution, this idea is extended to low and high order Taylor--Hood and mini elements, which have continuous discrete pressures. For the modification of the right hand side a velocity reconstruction operator is constructed that maps discretely divergence-free test functions to exactly divergence-free ones. The reconstruction is based on local H(div)-conforming flux equilibration on vertex patches, and fulfills certain orthogonality properties to provide consistency and optimal a-priori error estimates. Numerical examples for the incompressible Stokes and Navier--Stokes equations confirm that the new pressure-robust Taylor--Hood and mini elements converge with optimal order and outperform signi--cantly the classical versions of those elements when the continuous pressure is comparably large.

  • N. Ahmed, S. Becher, G. Matthies, Higher-order discontinuous Galerkin time stepping and local projection stabilization techniques for the transient Stokes problem, Computer Methods in Applied Mechanics and Engineering, 313 (2017) pp. 28--52.
    Abstract
    We introduce and analyze discontinuous Galerkin time discretizations coupled with continuous finite element methods based on equal-order interpolation in space for velocity and pressure in transient Stokes problems. Spatial stability of the pressure is ensured by adding a stabilization term based on local projection. We present error estimates for the semi-discrete problem after discretization in space only and for the fully discrete problem. The fully discrete pressure shows an instability in the limit of small time step length. Numerical tests are presented which confirm our theoretical results including the pressure instability.

  • N. Ahmed, T.Ch. Rebollo, V. John, S. Rubino, A review of variational multiscale methods for the simulation of turbulent incompressible flows, Archives of Computational Methods in Engineering. State of the Art Reviews, 24 (2017) pp. 115--164.
    Abstract
    Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: They are based on the variational formulation of the incompressible Navier--Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used mathematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.

  • N. Ahmed, T.Ch. Rebollo, V. John, S. Rubino, Analysis of a full space-time discretization of the Navier--Stokes equations by a local projection stabilization method, IMA Journal of Numerical Analysis, 37 (2017) pp. 1437--1467, DOI https://doi.org/10.1093/imanum/drw048 .
    Abstract
    A finite element error analysis of a local projection stabilization (LPS) method for the time-dependent Navier--Stokes equations is presented. The focus is on the high-order term-by-term stabilization method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard LPS methods is replaced by an interpolation-stabilized structure. The main contribution is on proving, theoretically and numerically, the optimal convergence order of the arising fully discrete scheme. In addition, the asymptotic energy balance is obtained for slightly smooth flows. Numerical studies support the analytical results and illustrate the potential of the method for the simulation of turbulent flows. Smooth unsteady flows are simulated with optimal order of accuracy.

  • N. Ahmed, On the grad-div stabilization for the steady Oseen and Navier--Stokes equations, Calcolo. A Quarterly on Numerical Analysis and Theory of Computation, 54 (2017) pp. 471--501, DOI 10.1007/s10092-016-0194-z .
    Abstract
    This paper studies the parameter choice in the grad-div stabilization applied to the generalized problems of Oseen type. Stabilization parameters based on minimizing the H1(Ω) error of the velocity are derived which do not depend on the viscosity parameter. For the proposed parameter choices, the H1(Ω) error of the velocity is derived that shows a direct dependence on the viscosity parameter. Differences and common features to the situation for the Stokes equations are discussed. Numerical studies are presented which confirm the theoretical results. Moreover, for the Navier- Stokes equations, numerical simulations were performed on a two-dimensional ow past a circular cylinder. It turns out, for the MINI element, that the best results can be obtained without grad-div stabilization.

  • F. Dassi, P. Farrell, H. Si, A novel surface remeshing scheme via higher dimensional embedding and radial basis functions, SIAM Journal on Scientific Computing, 39 (2017) pp. B522--B547, DOI 10.1137/16M1077015 .
    Abstract
    Many applications heavily rely on piecewise triangular meshes to describe complex surface geometries. High-quality meshes significantly improve numerical simulations. In practice, however, one often has to deal with several challenges. Some regions in the initial mesh may be overrefined, others too coarse. Additionally, the triangles may be too thin or not properly oriented. We present a novel mesh adaptation procedure which greatly improves the problematic input mesh and overcomes all of these drawbacks. By coupling surface reconstruction via radial basis functions with the higher dimensional embedding surface remeshing technique, we can automatically generate anisotropic meshes. Moreover, we are not only able to fill or coarsen certain mesh regions but also align the triangles according to the curvature of the reconstructed surface. This yields an acceptable trade-off between computational complexity and accuracy.

  • P. Farrell, K. Gillow, H. Wendland, Multilevel interpolation of divergence-free vector fields, IMA Journal of Numerical Analysis, 37 (2017) pp. 332--353, DOI 10.1093/imanum/drw006 .
    Abstract
    We introduce a multilevel technique for interpolating scattered data of divergence-free vector fields with the help of matrix-valued compactly supported kernels. The support radius at a given level is linked to the mesh norm of the data set at that level. There are at least three advantages of this method: no grid structure is necessary for the implementation, the multilevel approach is computationally cheaper than solving a large one-shot system and the interpolant is guaranteed to be analytically divergence-free. Furthermore, though we will not pursue this here, our multiscale approach is able to represent multiple scales in the data if present. We will prove convergence of the scheme, stability estimates and give a numerical example.

  • P. Farrell, Th. Koprucki, J. Fuhrmann, Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond Boltzmann statistics, Journal of Computational Physics, 346 (2017) pp. 497--513, DOI 10.1016/j.jcp.2017.06.023 .
    Abstract
    For a Voronoï finite volume discretization of the van Roosbroeck system with general charge carrier statistics we compare three thermodynamically consistent numerical fluxes known in the literature. We discuss an extension of the Scharfetter--Gummel scheme to non-Boltzmann (e.g. Fermi--Dirac) statistics. It is based on the analytical solution of a two-point boundary value problem obtained by projecting the continuous differential equation onto the interval between neighboring collocation points. Hence, it serves as a reference flux. The exact solution of the boundary value problem can be approximated by computationally cheaper fluxes which modify certain physical quantities. One alternative scheme averages the nonlinear diffusion (caused by the non-Boltzmann nature of the problem), another one modifies the effective density of states. To study the differences between these three schemes, we analyze the Taylor expansions, derive an error estimate, visualize the flux error and show how the schemes perform for a carefully designed p-i-n benchmark simulation. We present strong evidence that the flux discretization based on averaging the nonlinear diffusion has an edge over the scheme based on modifying the effective density of states.

  • V. John, A. Linke, Ch. Merdon, M. Neilan, L.G. Rebholz, On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Review, 59 (2017) pp. 492--544, DOI 10.1137/15M1047696 .
    Abstract
    The divergence constraint of the incompressible Navier--Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This paper reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, $bH(mathrmdiv)$-conforming finite elements, and mixed methods with an appropriate reconstruction of the test functions. Numerical examples illustrate both the potential effects of using non-robust discretizations and the improvements obtained by utilizing pressure-robust discretizations.

  • A. Linke, Ch. Merdon, W. Wollner, Optimal L2 velocity error estimate for a modified pressure-robust Crouzeix--Raviart Stokes element, IMA Journal of Numerical Analysis, 37 (2017) pp. 354--374.
    Abstract
    Recently, a novel approach for the robust discretization of the incompressible Stokes equations was proposed that slightly modifies the nonconforming Crouzeix--Raviart element such that its velocity error becomes pressure-independent. The modification results in an O(h) consistency error that allows straightforward proofs for the optimal convergence of the discrete energy norm of the velocity and of the L2 norm of the pressure. However, though the optimal convergence of the velocity in the L2 norm was observed numerically, it appeared to be nontrivial to prove. In this contribution, this gap is closed. Moreover, the dependence of the error estimates on the discrete inf-sup constant is traced in detail, which shows that classical error estimates are extremely pessimistic on domains with large aspect ratios. Numerical experiments in 2D and 3D illustrate the theoretical findings.

  • M. Eigel, Ch. Merdon, J. Neumann, An adaptive multilevel Monte--Carlo method with stochastic bounds for quantities of interest in groundwater flow with uncertain data, SIAM/ASA Journal on Uncertainty Quantification, 4 (2016) pp. 1219--1245.
    Abstract
    The focus of this work is the introduction of some computable a posteriori error control to the popular multilevel Monte Carlo sampling for PDE with stochastic data. We are especially interested in applications in the geosciences such as groundwater flow with rather rough stochastic fields for the conductive permeability. With a spatial discretisation based on finite elements, a goal functional is defined which encodes the quantity of interest. The devised goal-oriented error estimator enables to determine guaranteed a posteriori error bounds for this quantity. In particular, it allows for the adaptive refinement of the mesh hierarchy used in the multilevel Monte Carlo simulation. In addition to controlling the deterministic error, we also suggest how to treat the stochastic error in probability. Numerical experiments illustrate the performance of the presented adaptive algorithm for a posteriori error control in multilevel Monte Carlo methods. These include a localised goal with problem-adapted meshes and a slit domain example. The latter demonstrates the refinement of regions with low solution regularity based on an inexpensive explicit error estimator in the multilevel algorithm.

  • M. Eigel, Ch. Merdon, Equilibration a posteriori error estimation for convection-diffusion-reaction problems, Journal of Scientific Computing, 67 (2016) pp. 747--768.
    Abstract
    We study a posteriori error estimates for convection-diffusion-reaction problems with possibly dominating convection or reaction and inhomogeneous boundary conditions. For the conforming FEM discretisation with streamline diffusion stabilisation (SDM), we derive robust and efficient error estimators based on the reconstruction of equilibrated fluxes in an admissible discrete subspace of H (div, Ω). Error estimators of this type have become popular recently since they provide guaranteed error bounds without further unknown constants. The estimators can be improved significantly by some postprocessing and divergence correction technique. For an extension of the energy norm by a dual norm of some part of the differential operator, complete independence from the coefficients of the problem is achieved.

    Numerical benchmarks illustrate the very good performance of the error estimators in the convection dominated and the singularly perturbed cases.

  • M. Eigel, Ch. Merdon, Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order Galerkin finite element methods, SIAM/ASA Journal on Uncertainty Quantification, 4 (2016) pp. 1372--1397.
    Abstract
    Equilibration error estimators have been shown to commonly lead to very accurate guaranteed error bounds in the a posteriori error control of finite element methods for second order elliptic equations. Here, we extend previous results by the design of equilibrated fluxes for higher-order finite element methods with nonconstant coefficients and illustrate the favourable performance of different variants of the error estimator within two deterministic benchmark settings. After the introduction of the respective parametric problem with stochastic coefficients and the stochastic Galerkin FEM discretisation, a novel a posteriori error estimator for the stochastic error in the energy norm is devised. The error estimation is based on the stochastic residual and its decomposition into approximation residuals and a truncation error of the stochastic discretisation. Importantly, by using the derived deterministic equilibration techniques for the approximation residuals, the computable error bound is guaranteed for the considered class of problems. An adaptive algorithm allows the simultaneous refinement of the deterministic mesh and the stochastic discretisation in anisotropic Legendre polynomial chaos. Several stochastic benchmark problems illustrate the efficiency of the adaptive process.

  • A. Linke, G. Matthies, L. Tobiska, Robust arbitrary order mixed finite element methods for the incompressible Stokes equations with pressure independent velocity errors, ESAIM: Mathematical Modelling and Numerical Analysis, 50 (2016) pp. 289--309.
    Abstract
    Standard mixed finite element methods for the incompressible Navier-Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H1-conforming, divergence-free mixed finite element methods is rather difficult. Instead, we present a novel approach for the construction of arbitrary order mixed finite element methods which deliver pressure-independent velocity errors. The approach does not change the trial functions but replaces discretely divergence-free test functions in some operators of the weak formulation by divergence-free ones. This modification is applied to inf-sup stable conforming and nonconforming mixed finite element methods of arbitrary order in two and three dimensions. Optimal estimates for the incompressible Stokes equations are proved for the H1 and L2 errors of the velocity and the L2 error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results.

  • M. Shi, G. Printsypar, P.H.H. Duong, V.M. Calo, O. Iliev, S.P. Nunes, 3D morphology design for forward osmosis, Journal of Membrane Science, 516 (2016) pp. 172--184.

  • G.R. Barrenechea, V. John, P. Knobloch, Analysis of algebraic flux correction schemes, SIAM Journal on Numerical Analysis, 54 (2016) pp. 2427--2451.
    Abstract
    A family of algebraic flux correction schemes for linear boundary value problems in any space dimension is studied. These methods' main feature is that they limit the fluxes along each one of the edges of the triangulation, and we suppose that the limiters used are symmetric. For an abstract problem, the existence of a solution, existence and uniqueness of the solution of a linearized problem, and an a priori error estimate, are proved under rather general assumptions on the limiters. For a particular (but standard in practice) choice of the limiters, it is shown that a local discrete maximum principle holds. The theory developed for the abstract problem is applied to convection-diffusion-reaction equations, where in particular an error estimate is derived. Numerical studies show its sharpness.

  • C. Bertoglio, A. Caiazzo, A Stokes-residual backflow stabilization method applied to physiological flows, Journal of Computational Physics, 313 (2016) pp. 260--278.
    Abstract
    In computational fluid dynamics incoming flow at open boundaries, or emphbackflow, often yields to unphysical instabilities for high Reynolds numbers. It is widely accepted that this is due to the incoming energy arising from the convection term, which cannot be empha priori controlled when the velocity field is unknown at the boundary. In order to improve the robustness of the numerical simulations, we propose a stabilized formulation based on a penalization of the residual of a weak Stokes problem on the open boundary, whose viscous part controls the incoming convective energy, while the inertial term contributes to the kinetic energy. We also present different strategies for the approximation of the boundary pressure gradient, which is needed for defining the stabilization term. The method has the advantage that it does not require neither artificial modifications or extensions of the computational domain. Moreover, it is consistent with the Womersley solution. We illustrate our approach on numerical examples  - both academic and real-life -  relevant to blood and respiratory flows. The results also show that the stabilization parameter can be reduced with the mesh size.

  • P. Bringmann, C. Carstensen, Ch. Merdon, Guaranteed error control for the pseudostress approximation of the Stokes equations, Numerical Methods for Partial Differential Equations. An International Journal, 32 (2016) pp. 1411--1432.
    Abstract
    The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in $L^2$. Any standard mixed finite element function space can be utilized for this mixed formulation, e.g. the Raviart-Thomas discretization which is related to the Crouzeix-Raviart nonconforming finite element scheme in the lowest-order case. The effective and guaranteed a posteriori error control for this nonconforming velocity-oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf-sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy.

  • A. Ern, D. Di Pietro, A. Linke, F. Schieweck, A discontinuous skeletal method for the viscosity-dependent Stokes problem, Computer Methods in Applied Mechanics and Engineering, 306 (2016) pp. 175--195.
    Abstract
    We devise and analyze arbitrary-order nonconforming methods for the discretization of the viscosity-dependent Stokes equations on simplicial meshes. We keep track explicitly of the viscosity and aim at pressure-robust schemes that can deal with the practically relevant case of body forces with large curl-free part in a way that the discrete velocity error is not spoiled by large pressures. The method is inspired from the recent Hybrid High-Order (HHO) methods for linear elasticity. After elimination of the auxiliary variables by static condensation, the linear system to be solved involves only discrete face-based velocities, which are polynomials of degree k >=0, and cell-wise constant pressures. Our main result is a pressure-independent energy-error estimate on the velocity of order (k+1). The main ingredient to achieve pressure-independence is the use of a divergence-preserving velocity reconstruction operator in the discretization of the body forces. We also prove an L2-pressure estimate of order (k+1) and an L2-velocity estimate of order (k+2), the latter under elliptic regularity. The local mass and momentum conservation properties of the discretization are also established. Finally, two- and three-dimensional numerical results are presented to support the analysis.

  • A. Fiebach, A. Glitzky, A. Linke, Convergence of an implicit Voronoi finite volume method for reaction-diffusion problems, Numerical Methods for Partial Differential Equations. An International Journal, 32 (2016) pp. 141--174.
    Abstract
    We investigate the convergence of an implicit Voronoi finite volume method for reaction- diffusion problems including nonlinear diffusion in two space dimensions. The model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. The numerical scheme uses boundary conforming Delaunay meshes and preserves positivity and the dissipative property of the continuous system. Starting from a result on the global stability of the scheme (uniform, mesh-independent global upper and lower bounds), we prove strong convergence of the chemical activities and their gradients to a weak solution of the continuous problem. In order to illustrate the preservation of qualitative properties by the numerical scheme, we present a long-term simulation of the Michaelis-Menten-Henri system. Especially, we investigate the decay properties of the relative free energy and the evolution of the dissipation rate over several magnitudes of time, and obtain experimental orders of convergence for these quantities.

  • W. Huang, L. Kamenski, J. Lang, Stability of explicit one-step methods for P1-finite element approximation of linear diffusion equations on anisotropic meshes, SIAM Journal on Numerical Analysis, 54 (2016) pp. 1612--1634.
    Abstract
    We study the stability of explicit Runge-Kutta integration schemes for the linear finite element approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and any diffusion matrix within a factor of 2 (d + 1), where d is the spatial dimension. Both full mass matrix and mass lumping are considered. The bound reveals that the stability condition is affected by two factors. The first one depends on the number of mesh elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The other factor reflects the effects of the interplay of the mesh geometry and the diffusion matrix. It is shown that it is not the mesh geometry itself but the mesh geometry in relation to the diffusion matrix that is crucial to the stability of explicit methods. When the mesh is uniform in the metric specified by the inverse of the diffusion matrix, the stability condition is comparable to the situation with the Laplace operator on a uniform mesh. Numerical results are presented to verify the theoretical findings.

  • M. Khodayari, P. Reinsberg, A.A. Abd-El-Latif, Ch. Merdon, J. Fuhrmann, H. Baltruschat, Determining solubility and diffusivity by using a flow cell coupled to a mass spectrometer, ChemPhysChem, 17 (2016) pp. 1647--1655.

  • J. DE Frutos, B. Garc'ia-Archilla, V. John, J. Novo, Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements, Journal of Scientific Computing, 66 (2016) pp. 991--1024.
    Abstract
    The approximation of the time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin method with grad-div stabilization is studied. The main goal is to prove that adding a grad-div stabilization term to the Galerkin approximation has a stabilizing effect for small viscosity. Both the continuous-in-time and the fully discrete case (backward Euler method, the two-step BDF, and Crank--Nicolson schemes) are analyzed. In fact, error bounds are obtained that do not depend on the inverse of the viscosity in the case where the solution is sufficiently smooth. The bounds for the divergence of the velocity as well as for the pressure are optimal. The analysis is based on the use of a specific Stokes projection. Numerical studies support the analytical results.

  • J. DE Frutos, V. John, J. Novo, Projection methods for incompressible flow problems with WENO finite difference schemes, Journal of Computational Physics, 309 (2016) pp. 368--386.
    Abstract
    Weighted essentially non-oscillatory (WENO) finite difference schemes have been recommended in a competitive study of discretizations for scalar evolutionary convection-diffusion equations [20]. This paper explores the applicability of these schemes for the simulation of incompressible flows. To this end, WENO schemes are used in several non-incremental and incremental projection methods for the incompressible Navier-Stokes equations. Velocity and pressure are discretized on the same grid. A pressure stabilization Petrov-Galerkin (PSPG) type of stabilization is introduced in the incremental schemes to account for the violation of the discrete inf-sup condition. Algorithmic aspects of the proposed schemes are discussed. The schemes are studied on several examples with different features. It is shown that the WENO finite difference idea can be transferred to the simulation of incompressible flows. Some shortcomings of the methods, which are due to the splitting in projection schemes, become also obvious.

  • N. Ahmed, G. Matthies, Numerical study of SUPG and LPS methods combined with higher order variational time discretization schemes applied to time-dependent convection-diffusion-reaction equations, Journal of Scientific Computing, 67 (2016) pp. 988--1018.
    Abstract
    This paper considers the numerical solution of time-dependent convection-diffusion-reaction equations. We shall employ combinations of streamline-upwind Petrov-Galerkin (SUPG) and local projection stabilization (LPS) methods in space with the higher order variational time discretization schemes. In particular, we consider time discretizations by discontinuous Galerkin (dG) methods and continuous Galerkin-Petrov (cGP) methods. Several numerical tests have been performed to assess the accuracy of combinations of spatial and temporal discretization schemes. Furthermore, the dependence of the results on the stabilization parameters of the spatial discretizations are discussed. Finally the long-time behavior of overshoots and undershoots is investigated.

  • A. Caiazzo, R. Guibert, I.E. Vignon-Clementel, A reduced-order modeling for efficient design study of artificial valve in enlarged ventricular outflow tracts, Computer Methods in Biomechanics and Biomedical Engineering, 19 (2016) pp. 1314--1318.
    Abstract
    A computational approach is proposed for efficient design study of a reducer stent to be percutaneously implanted in enlarged right ventricular outflow tracts (RVOT). The need for such a device is driven by the absence of bovine or artificial valves which could be implanted in these RVOT to replace the absent or incompetent native valve, as is often the case over time after Tetralogy of Fallot repair. Hemodynamics are simulated in the stented RVOT via a reduce order model based on proper orthogonal decomposition (POD), while the artificial valve is modeled as a thin resistive surface. The reduced order model is obtained from the numerical solution on a reference device configuration, then varying the geometrical parameters (diameter) for design purposes. To validate the approach, forces exerted on the valve and on the reducer are monitored, varying with geometrical parameters, and compared with the results of full CFD simulations. Such an approach could also be useful for uncertainty quantification.

  • F. Dassi, L. Kamenski, H. Si, Tetrahedral mesh improvement using moving mesh smoothing and lazy searching flips, Procedia Engineering, 163 (2016) pp. 302--314.
    Abstract
    In this paper we combine two new smoothing and flipping techniques. The moving mesh smoothing is based on the integration of an ordinary differential coming from a given functional. The lazy flip technique is a reversible edge removal algorithm to automatically search flips for local quality improvement. On itself, these strategies already provide good mesh improvement, but their combination achieves astonishing results which have not been reported so far. Provided numerical examples show that we can obtain final tetrahedral meshes with dihedral angles between 40 and 123 degrees. We compare the new method with other publicly available mesh improving codes.

  • J. Fuhrmann, A numerical strategy for Nernst--Planck systems with solvation effect, Fuel Cells, 16 (2016) pp. 704--714.

  • J. Fuhrmann, A. Linke, Ch. Merdon, F. Neumann, T. Streckenbach, H. Baltruschat, M. Khodayari, Inverse modeling of thin layer flow cells for detection of solubility, transport and reaction coefficients from experimental data, Electrochimica Acta, 211 (2016) pp. 1--10.
    Abstract
    Thin layer flow cells are used in electrochemical research as experimental devices which allow to perform investigations of electrocatalytic surface reactions under controlled conditions using reasonably small electrolyte volumes. The paper introduces a general approach to simulate the complete cell using accurate numerical simulation of the coupled flow, transport and reaction processes in a flow cell. The approach is based on a mass conservative coupling of a divergence-free finite element method for fluid flow and a stable finite volume method for mass transport. It allows to perform stable and efficient forward simulations that comply with the physical bounds namely mass conservation and maximum principles for the involved species. In this context, several recent approaches to obtain divergence-free velocities from finite element simulations are discussed. In order to perform parameter identification, the forward simulation method is coupled to standard optimization tools. After an assessment of the inverse modeling approach using known realistic data, first results of the identification of solubility and transport data for O2 dissolved in organic electrolytes are presented. A plausibility study for a more complex situation with surface reactions concludes the paper and shows possible extensions of the scope of the presented numerical tools.

  • V. John, K. Kaiser, J. Novo, Finite element methods for the incompressible Stokes equations with variable viscosity, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 96 (2016) pp. 205--216.
    Abstract
    Finite element error estimates are derived for the incompressible Stokes equations with variable viscosity. The ratio of the supremum and the infimum of the viscosity appears in the error bounds. Numerical studies show that this ratio can be observed sometimes. However, often the numerical results show a weaker dependency on the viscosity.

  • A. Linke, Ch. Merdon, On velocity errors due to irrotational forces in the Navier--Stokes momentum balance, Journal of Computational Physics, 313 (2016) pp. 654--661.
    Abstract
    This contribution studies the influence of the pressure on the velocity error in finite element discretisations of the Navier--Stokes equations. Three simple benchmark problems that are all close to real-world applications convey that the pressure can be comparably large and is not to be underestimated. For widely used finite element methods like the Taylor--Hood finite element method, such relatively large pressures can lead to spurious oscillations and arbitrarily large errors in the velocity, even if the exact velocity is in the ansatz space. Only mixed finite element methods, whose velocity error is pressure-independent, like the Scott--Vogelius finite element method can avoid this influence.

  • A. Linke, Ch. Merdon, Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier--Stokes equations, Computer Methods in Applied Mechanics and Engineering, 311 (2016) pp. 304--326.
    Abstract
    Recently, it was understood how to repair a certain L2-orthogonality of discretely-divergence-free vector fields and gradient fields such that the velocity error of inf-sup stable discretizations for the incompressible Stokes equations becomes pressure-independent. These new 'pressure-robust' Stokes discretizations deliver a small velocity error, whenever the continuous velocity field can be well approximated on a given grid. On the contrary, classical inf-sup stable Stokes discretizations can guarantee a small velocity error only, when both the velocity and the pressure field can be approximated well, simultaneously.
    In this contribution, 'pressure-robustness' is extended to the time-dependent Navier--Stokes equations. In particular, steady and time-dependent potential flows are shown to build an entire class of benchmarks, where pressure-robust discretizations can outperform classical approaches significantly. Speedups will be explained by a new theoretical concept, the 'discrete Helmholtz projector' of an inf-sup stable discretization. Moreover, different discrete nonlinear convection terms are discussed, and skew-symmetric pressure-robust discretizations are proposed.

Contributions to Collected Editions

  • M. Liero, J. Fuhrmann, A. Glitzky, Th. Koprucki, A. Fischer, S. Reineke, Modeling and simulation of electrothermal feedback in large-area organic leds, in: Proceedings of ``the 17th International Conference on Numerical Simulation of Optoelectronic Devices'', J. Piprek, M. Willatzen, eds., IEEE Conference Publications Management Group, Piscataway, 2017, pp. 105--106, DOI 10.1109/NUSOD.2017.8010013 .

  • N. Kumar, J.H.M. Ten Thije Boonkkamp, B. Koren, A. Linke, A nonlinear flux approximation scheme for the viscous Burgers equation, in: Finite Volumes for Complex Applications VIII -- Hyperbolic, Elliptic and Parabolic Problems -- FVCA 8, Lille, France, June 2017, C. Cances, P. Omnes, eds., 200 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2017, pp. 457--465.

  • N. Ahmed, A. Linke, Ch. Merdon, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, in: Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects FVCA 8, Lille, France, June 2017, C. Cances, P. Omnes, eds., 199 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2017, pp. 351--359.

  • P. Farrell, A. Linke, Uniform second order convergence of a complete flux scheme on nonuniform 1D grids, in: Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects FVCA 8, Lille, France, June 2017, C. Cances, P. Omnes, eds., 199 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2017, pp. 303--310.

  • P. Farrell, Th. Koprucki, J. Fuhrmann, Comparison of consistent flux discretizations for drift diffusion beyond boltzmann statistics, in: Proceedings of ``the 17th International Conference on Numerical Simulation of Optoelectronic Devices'', J. Piprek, M. Willatzen, eds., IEEE Conference Publications Management Group, Piscataway, 2017, pp. 219--220, DOI 10.1109/NUSOD.2017.8010070 .

  • J. Fuhrmann, A. Glitzky, M. Liero, Hybrid finite-volume/finite-element schemes for p(x)-Laplace thermistor models, in: Finite Volumes for Complex Applications VIII -- Hyperbolic, Elliptic and Parabolic Problems -- FVCA 8, Lille, France, June 2017, C. Cancès, P. Omnes, eds., 200 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2017, pp. 397--405, DOI 10.1007/978-3-319-57394-6_42 .

  • J. Fuhrmann, A. Glitzky, M. Liero, Hybrid finite-volume/finite-element schemes for $p(x)$-Laplace thermistor models, in: Finite Volumes for Complex Applications VIII -- Hyperbolic, Elliptic and Parabolic Problems -- FVCA 8, Lille, France, June 2017, C. Cancès, P. Omnes, eds., 200 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2017, pp. 397--405.
    Abstract
    We introduce an empirical PDE model for the electrothermal description of organic semiconductor devices by means of current and heat flow. The current flow equation is of p(x)-Laplace type, where the piecewise constant exponent p(x) takes the non-Ohmic behavior of the organic layers into account. Moreover, the electrical conductivity contains an Arrhenius-type temperature law. We present a hybrid finite-volume/finite-element discretization scheme for the coupled system, discuss a favorite discretization of the p(x)-Laplacian at hetero interfaces, and explain how path following methods are applied to simulate S-shaped current-voltage relations resulting from the interplay of self-heating and heat flow.

  • J. Fuhrmann, C. Guhlke, A finite volume scheme for Nernst--Planck--Poisson systems with Ion size and solvation effects, in: Finite Volumes for Complex Applications VIII -- Hyperbolic, Elliptic and Parabolic Problems -- FVCA 8, Lille, France, June 2017, C. Cancès, P. Omnes, eds., 200 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2017, pp. 497--505, DOI 10.1007/978-3-319-57394-6_52 .

  • M. Radziunas, A. Zeghuzi, J. Fuhrmann, Th. Koprucki, H.-J. Wünsche, H. Wenzel, U. Bandelow, Efficient coupling of inhomogeneous current spreading and electro-optical models for simulation of dynamics in broad-area semiconductor lasers, in: Proceedings of ``the 17th International Conference on Numerical Simulation of Optoelectronic Devices'', J. Piprek, M. Willatzen, eds., IEEE Conference Publications Management Group, Piscataway, 2017, pp. 231--232.

  • S. Ganesan, V. John, G. Matthies, R. Meesala, A. Shamim, U. Wilbrandt, An object oriented parallel finite element scheme for computations of PDEs: Design and implementation, in: 2016 IEEE 23rd International Conference on High Performance Computing Workshops (PDF only), pp. 106--115, DOI 10.1109/HiPCW.2016.19 .

  • A. Caiazzo, J. Mura, A two-scale homogenization approach for the estimation of porosity in elastic media subject area, in: Trends in Differential Equations and Applications, F.O. Gallego, M.V. Redondo Neble, J.R.R. Galván, eds., 8 of SEMA SIMAI Springer Series, Springer International Publishing Switzerland, Cham, 2016, pp. 89--105.

  • H. Si, N. Goerigk, On tetrahedralisations of reduced Chazelle polyhedra with interior Steiner points, in: 25th International Meshing Roundtable, S. Canann, S. Owen, H. Si, eds., 163 of Procedia Engineering, Elsevier, Amsterdam, 2016, pp. 33--45.
    Abstract
    The polyhedron constructed by Chazelle, known as Chazelle polyhedron [4], is an important example in many partitioning problems. In this paper, we study the problem of tetrahedralising a Chazelle polyhedron without modifying its exterior boundary. It is motivated by a crucial step in 3d finite element mesh generation in which a set of arbitrary boundary constraints (edges or faces) need to be entirely preserved. We first reduce the volume of a Chazelle polyhedron by removing the regions that are tetrahedralisable. This leads to a 3d polyhedron which may not be tetrahedralisable unless extra points, so-called Steiner points, are added. We call it a reduced Chazelle polyhedron. We define a set of interior Steiner points that ensures the existence of a tetrahedralisation of the reduced Chazelle polyhedron. Our proof uses a natural correspondence that any sequence of edge flips converting one triangulation of a convex polygon into another gives a tetrahedralization of a 3d polyhedron which have the two triangulations as its boundary. Finally, we exhibit a larger family of reduced Chazelle polyhedra which includes the same combinatorial structure of the Schönhardt polyhedron. Our placement of interior Steiner points also applies to tetrahedralise polyhedra in this family.

Preprints, Reports, Technical Reports

  • P.W. Schroeder, Ch. Lehrenfeld, A. Linke, G. Lube, Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier--Stokes equations, Preprint no. 2436, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2436 .
    Abstract, PDF (798 kByte)
    Inf-sup stable FEM applied to time-dependent incompressible Navier--Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, Re-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on an essential regularity assumption for the gradient of the velocity, which is discussed in detail. In the sense of best practice, we review and establish pressure- and Re-semi-robust estimates for pointwise divergence-free H1-conforming FEM (like Scott--Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free H(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.

  • S. Mohammadi, Ch. D'alonzo, L. Ruthotto, J. Polzehl, I. Ellerbrock, M.F. Callaghan, N. Weiskopf, K. Tabelow, Simultaneous adaptive smoothing of relaxometry and quantitative magnetization transfer mapping, Preprint no. 2432, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2432 .
    Abstract, PDF (3888 kByte)
    Attempts for in-vivo histology require a high spatial resolution that comes with the price of a decreased signal-to-noise ratio. We present a novel iterative and multi-scale smoothing method for quantitative Magnetic Resonance Imaging (MRI) data that yield proton density, apparent transverse and longitudinal relaxation, and magnetization transfer maps. The method is based on the propagation-separation approach. The adaptivity of the procedure avoids the inherent bias from blurring subtle features in the calculated maps that is common for non-adaptive smoothing approaches. The characteristics of the methods were evaluated on a high-resolution data set (500 μ isotropic) from a single subject and quantified on data from a multi-subject study. The results show that the adaptive method is able to increase the signal-to-noise ratio in the calculated quantitative maps while largely avoiding the bias that is otherwise introduced by spatially blurring values across tissue borders. As a consequence, it preserves the intensity contrast between white and gray matter and the thin cortical ribbon.

  • P. Farrell, M. Patriarca, J. Fuhrmann, Th. Koprucki, Comparison of thermodynamically consistent charge carrier flux discretizations for Fermi--Dirac and Gauss--Fermi statistics, Preprint no. 2424, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2424 .
    Abstract, PDF (1572 kByte)
    We compare three thermodynamically consistent Scharfetter--Gummel schemes for different distribution functions for the carrier densities, including the Fermi--Dirac integral of order 1/2 and the Gauss--Fermi integral. The most accurate (but unfortunately also most costly) generalized Scharfetter--Gummel scheme requires the solution of an integral equation. We propose a new method to solve this integral equation numerically based on Gauss quadrature and Newton's method. We discuss the quality of this approximation and plot the resulting currents for Fermi--Dirac and Gauss--Fermi statistics. Finally, by comparing two modified (diffusion-enhanced and inverse activity based) Scharfetter--Gummel schemes with the more accurate generalized scheme, we show that the diffusion-enhanced ansatz leads to considerably lower flux errors, confirming previous results (J. Comp. Phys. 346:497-513, 2017).

  • M. Radziunas, A. Zeghuzi, J. Fuhrmann, Th. Koprucki, H.-J. Wünsche, H. Wenzel, U. Bandelow, Efficient coupling of inhomogeneous current spreading and dynamic electro-optical models for broad-area edge-emitting semiconductor devices, Preprint no. 2421, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2421 .
    Abstract, PDF (246 kByte)
    We extend a 2 (space) + 1 (time)-dimensional traveling wave model for broad-area edge-emitting semiconductor lasers by a model for inhomogeneous current spreading from the contact to the active zone of the laser. To speedup the performance of the device simulations, we suggest and discuss several approximations of the inhomogeneous current density in the active zone.

  • M. Liero, J. Fuhrmann, A. Glitzky, Th. Koprucki, A. Fischer, S. Reineke, 3D electrothermal simulations of organic LEDs showing negative differential resistance, Preprint no. 2420, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2420 .
    Abstract, PDF (2587 kByte)
    Organic semiconductor devices show a pronounced interplay between temperature-activated conductivity and self-heating which in particular causes inhomogeneities in the brightness of large-area OLEDs at high power. We consider a 3D thermistor model based on partial differential equations for the electrothermal behavior of organic devices and introduce an extension to multiple layers with nonlinear conductivity laws, which also take the diode-like behavior in recombination zones into account. We present a numerical simulation study for a red OLED using a finite-volume approximation of this model. The appearance of S-shaped current-voltage characteristics with regions of negative differential resistance in a measured device can be quantitatively reproduced. Furthermore, this simulation study reveals a propagation of spatial zones of negative differential resistance in the electron and hole transport layers toward the contact.

  • V. John, P. Knobloch, J. Novo, Finite elements for scalar convection-dominated equations and incompressible flow problems -- A never ending story?, Preprint no. 2410, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2410 .
    Abstract, PDF (283 kByte)
    The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important problems in these fields are discussed.

  • N. Ahmed, C. Bartsch, V. John, U. Wilbrandt, An assessment of solvers for saddle point problems emerging from the incompressible Navier--Stokes equations, Preprint no. 2408, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2408 .
    Abstract, PDF (1086 kByte)
    Efficient incompressible flow simulations, using inf-sup stable pairs of finite element spaces, require the application of efficient solvers for the arising linear saddle point problems. This paper presents an assessment of different solvers: the sparse direct solver UMFPACK, the flexible GMRES (FGMRES) method with different coupled multigrid preconditioners, and FGMRES with Least Squares Commutator (LSC) preconditioners. The assessment is performed for steady-state and time-dependent flows around cylinders in 2d and 3d. Several pairs of inf-sup stable finite element spaces with second order velocity and first order pressure are used. It turns out that for the steady-state problems often FGMRES with an appropriate multigrid preconditioner was the most efficient method on finer grids. For the time-dependent problems, FGMRES with LSC preconditioners that use an inexact iterative solution of the velocity subproblem worked best for smaller time steps.

  • V. John, S. Kaya, J. Novo, Finite element error analysis of a mantle convection model, Preprint no. 2403, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2403 .
    Abstract, PDF (539 kByte)
    A mantle convection model consisting of the stationary Stokes equations and a time-dependent convection-diffusion equation for the temperature is studied. The Stokes problem is discretized with a conforming inf-sup stable pair of finite element spaces and the temperature equation is stabilized with the SUPG method. Finite element error estimates are derived which show the dependency of the error of the solution of one problem on the error of the solution of the other equation. The dependency of the error bounds on the coefficients of the problem is monitored.

  • N. Ahmed, A. Linke, Ch. Merdon, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, Preprint no. 2402, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2402 .
    Abstract, PDF (479 kByte)
    In this contribution, classical mixed methods for the incompressible Navier-Stokes equations that relax the divergence constraint and are discretely inf-sup stable, are reviewed. Though the relaxation of the divergence constraint was claimed to be harmless since the beginning of the 1970ies, Poisson locking is just replaced by another more subtle kind of locking phenomenon, which is sometimes called poor mass conservation. Indeed, divergence-free mixed methods and classical mixed methods behave qualitatively in a different way: divergence-free mixed methods are pressure-robust, which means that, e.g., their velocity error is independent of the continuous pressure. The lack of pressure-robustness in classical mixed methods can be traced back to a consistency error of an appropriately defined discrete Helmholtz projector. Numerical analysis and numerical examples reveal that really locking-free mixed methods must be discretely inf-sup stable and pressure-robust, simultaneously. Further, a recent discovery shows that locking-free, pressure-robust mixed methods do not have to be divergence-free. Indeed, relaxing the divergence constraint in the velocity trial functions is harmless, if the relaxation of the divergence constraint in some velocity test functions is repaired, accordingly.

  • W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part III: Compactness and convergence, Preprint no. 2397, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2397 .
    Abstract, PDF (327 kByte)
    We consider an improved Nernst--Planck--Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross--diffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper, which continues the investigations of [DDGG17a, DDGG17b], we prove the compactness of the solution vector, and existence and convergence for the approximation schemes. We point at simple structural PDE arguments as an adequate substitute to the Aubin--Lions compactness Lemma and its generalisations: These familiar techniques attain their limit in the context of our model in which the relationship between time derivatives (transport) and diffusion gradients is highly non linear.

  • W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part II: Approximation and a priori estimates, Preprint no. 2396, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2396 .
    Abstract, PDF (355 kByte)
    We consider an improved Nernst--Planck--Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross--diffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper, which continues the investigation of [DDGG17a], we derive for thermodynamically consistent approximation schemes the natural uniform estimates associated with the dissipations. Our results essentially improve our former study [DDGG16], in particular the a priori estimates concerning the relative chemical potentials.

  • W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part I: Derivation of the model and survey of the results, Preprint no. 2395, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2395 .
    Abstract, PDF (343 kByte)
    We consider an improved Nernst--Planck--Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross--diffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper we establish the existence of a global--in--time weak solution for the full model, allowing for a general structure of the mobility tensor and for chemical reactions with highly non linear rates in the bulk and on the active boundary. We characterise the singular states of the system, showing that the chemical species can vanish only globally in space, and that this phenomenon must be concentrated in a compact set of measure zero in time. With respect to our former study [DDGG16], we also essentially improve the a priori estimates, in particular concerning the relative chemical potentials.

  • A. Linke, Ch. Merdon, M. Neilan, F. Neumann, Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes problem, Preprint no. 2374, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2374 .
    Abstract, PDF (334 kByte)
    Nearly all classical inf-sup stable mixed finite element methods for the incompressible Stokes equations are not pressure-robust, i.e., the velocity error is dependent on the pressure. However, recent results show that pressure-robustness can be recovered by a non-standard discretization of the right hand side alone. This variational crime introduces a consistency error in the method which can be estimated in a straightforward manner provided that the exact velocity solution is sufficiently smooth. The purpose of this paper is to analyze the pressure-robust scheme with low regularity. The numerical analysis applies divergence-free H¹-conforming Stokes finite element methods as a theoretical tool. As an example, pressure-robust velocity and pressure a-priori error estimates will be presented for the (first order) nonconforming Crouzeix--Raviart element. A key feature in the analysis is the dependence of the errors on the Helmholtz projector of the right hand side data, and not on the entire data term. Numerical examples illustrate the theoretical results.

  • F. Dassi, L. Kamenski, P. Farrell, H. Si, Tetrahedral mesh improvement using moving mesh smoothing, lazy searching flips, and RBF surface reconstruction, Preprint no. 2373, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2373 .
    Abstract, PDF (3335 kByte)
    Given a tetrahedral mesh and objective functionals measuring the mesh quality which take into account the shape, size, and orientation of the mesh elements, our aim is to improve the mesh quality as much as possible. In this paper, we combine the moving mesh smoothing, based on the integration of an ordinary differential equation coming from a given functional, with the lazy flip technique, a reversible edge removal algorithm to modify the mesh connectivity. Moreover, we utilize radial basis function (RBF) surface reconstruction to improve tetrahedral meshes with curved boundary surfaces. Numerical tests show that the combination of these techniques into a mesh improvement framework achieves results which are comparable and even better than the previously reported ones.

  • C. Bertoglio, A. Caiazzo, Y. Bazilevs, M. Braack, M. Esmaily-Moghadam, V. Gravemeier, A.L. Marsden, O. Pironneau, I.E. Vignon-Clementel, W.A. Wall, Benchmark problems for numerical treatment of backflow at open boundaries, Preprint no. 2372, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2372 .
    Abstract, PDF (3076 kByte)
    In computational fluid dynamics, incoming velocity at open boundaries, or backflow, often yields to unphysical instabilities already for moderate Reynolds numbers. Several treatments to overcome these backflow instabilities have been proposed in the literature. However, these approaches have not yet been compared in detail in terms of accuracy in different physiological regimes, in particular due to the difficulty to generate stable reference solutions apart from analytical forms. In this work, we present a set of benchmark problems in order to compare different methods in different backflow regimes (with a full reversal flow and with propagating vortices after a stenosis). The examples are implemented in FreeFem++ and the source code is openly available, making them a solid basis for future method developments.

  • N. Ahmed, A. Linke, Ch. Merdon, On really locking-free mixed finite element methods for the transient incompressible Stokes equations, Preprint no. 2368, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2368 .
    Abstract, PDF (388 kByte)
    Inf-sup stable mixed methods for the steady incompressible Stokes equations that relax the divergence constraint are often claimed to deliver locking-free discretizations. However, this relaxation leads to a pressure-dependent contribution in the velocity error, which is proportional to the inverse of the viscosity, thus giving rise to a (different) locking phenomenon. However, a recently proposed modification of the right hand side alone leads to a discretization that is really locking-free, i.e., its velocity error converges with optimal order and is independent of the pressure and the smallness of the viscosity. In this contribution, we extend this approach to the transient incompressible Stokes equations, where besides the right hand side also the velocity time derivative requires an improved space discretization. Semi-discrete and fully-discrete a-priori velocity and pressure error estimates are derived, which show beautiful robustness properties. Two numerical examples illustrate the superior accuracy of pressure-robust space discretizations in the case of small viscosities.

  • N. Ahmed, V. John, G. Matthies, J. Novo, A local projection stabilization/continuous Galerkin--Petrov method for incompressible flow problems, Preprint no. 2347, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2347 .
    Abstract, PDF (601 kByte)
    The local projection stabilization (LPS) method in space is consid-ered to approximate the evolutionary Oseen equations. Optimal error bounds independent of the viscosity parameter are obtained in the continuous-in-time case for the approximations of both velocity and pressure. In addition, the fully discrete case in combination with higher order continuous Galerkin--Petrov (cGP) methods is studied. Error estimates of order k + 1 are proved, where k denotes the polynomial degree in time, assuming that the convective term is time-independent. Numerical results show that the predicted order is also achieved in the general case of time-dependent convective terms.

  • W. Dreyer, P. Friz, P. Gajewski, C. Guhlke, M. Maurelli, Stochastic model for LFP-electrodes, Preprint no. 2329, WIAS, Berlin, 2016.
    Abstract, PDF (1531 kByte)
    In the framework of non-equilibrium thermodynamics we derive a new model for porous electrodes. The model is applied to LiFePO4 (LFP) electrodes consisting of many LFP particles of nanometer size. The phase transition from a lithium-poor to a lithium-rich phase within LFP electrodes is controlled by surface fluctuations leading to a system of stochastic differential equations. The model is capable to derive an explicit relation between battery voltage and current that is controlled by thermodynamic state variables. This voltage-current relation reveals that in thin LFP electrodes lithium intercalation from the particle surfaces into the LFP particles is the principal rate limiting process. There are only two constant kinetic parameters in the model describing the intercalation rate and the fluctuation strength, respectively. The model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate limiting capacity. Moreover we study the impact of both the particle size distribution and the active surface area on the voltagecharge characteristics of the electrode. Finally we carefully discuss the phase transition for varying charging/discharging rates.

  • N. Ahmed, G. Matthies, Numerical studies of higher order variational time stepping schemes for evolutionary Navier--Stokes equations, Preprint no. 2322, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2322 .
    Abstract, PDF (1552 kByte)
    We present in this paper numerical studies of higher order variational time stepping schemes com-bined with finite element methods for simulations of the evolutionary Navier--Stokes equations. In particular, conforming inf-sup stable pairs of finite element spaces for approximating velocity and pressure are used as spatial discretization while continuous Galerkin--Petrov methods (cGP) and discontinuous Galerkin (dG) methods are applied as higher order variational time discretizations. Numerical results for the well-known problem of incompressible flows around a circle will be presented.

  • W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Existence of weak solutions for improved Nernst--Planck--Poisson models of compressible reacting electrolytes, Preprint no. 2291, WIAS, Berlin, 2016.
    Abstract, PDF (638 kByte)

    We consider an improved Nernst-Planck-Poisson model for compressible electrolytes first proposed by Dreyer et al. in 2013. The model takes into account the elastic deformation of the medium. In particular, large pressure contributions near electrochemical interfaces induce an inherent coupling of mass and momentum transport. The model consists of convection-diffusion-reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity and the Poisson equation for the electrical potential. Cross-diffusion phenomena occur due to the principle of mass conservation. Moreover, the diffusion matrix (mobility matrix) has a zero eigenvalue, meaning that the system is degenerate parabolic. In this paper we establish the existence of a global-in- time weak solution for the full model, allowing for cross-diffusion and an arbitrary number of chemical reactions in the bulk and on the active boundary.
  • A. Wahab, T. Abbas, N. Ahmed, Q.M. Zaigham Zia, Detection of electromagnetic inclusions using topological sensitivity, Preprint no. 2285, WIAS, Berlin, 2016.
    Abstract, PDF (2179 kByte)
    In this article a topological sensitivity framework for far field detection of a diametrically small electromagnetic inclusion is established. The cases of single and multiple measurements of the electric far field scattering amplitude at a fixed frequency are taken into account. The performance of the algorithm is analyzed theoretically in terms of its resolution and sensitivity for locating an inclusion. The stability of the framework with respect to measurement and medium noises is discussed. Moreover, the quantitative results for signal-to-noise ratio are presented. A few numerical results are presented to illustrate the detection capabilities of the proposed framework with single and multiple measurements.

  • P. Farrell, N. Rotundo, D.H. Doan, M. Kantner, J. Fuhrmann, Th. Koprucki, Numerical methods for drift-diffusion models, Preprint no. 2263, WIAS, Berlin, 2016.
    Abstract, PDF (4768 kByte)
    The van Roosbroeck system describes the semi-classical transport of free electrons and holes in a self-consistent electric field using a drift-diffusion approximation. It became the standard model to describe the current flow in semiconductor devices at macroscopic scale. Typical devices modeled by these equations range from diodes, transistors, LEDs, solar cells and lasers to quantum nanostructures and organic semiconductors. The report provides an introduction into numerical methods for the van Roosbroeck system. The main focus lies on the Scharfetter-Gummel finite volume discretization scheme and recent efforts to generalize this approach to general statistical distribution functions.

  • A. Caiazzo, F. Caforio, G. Montecinos, L.O. Müller, P.J. Blanco, E.F. Toro, Assessment of reduced order Kalman filter for parameter identification in one-dimensional blood flow models using experimental data, Preprint no. 2248, WIAS, Berlin, 2016.
    Abstract, PDF (8646 kByte)
    This work presents a detailed investigation of a parameter estimation approach based on the reduced order unscented Kalman filter (ROUKF) in the context of one-dimensional blood flow models. In particular, the main aims of this study are (i) to investigate the effect of using real measurements vs. synthetic data (i.e., numerical results of the same in silico model, perturbed with white noise) for the estimation and (ii) to identify potential difficulties and limitations of the approach in clinically realistic applications in order to assess the applicability of the filter to such setups. For these purposes, our numerical study is based on the in vitro model of the arterial network described by [Alastruey et al. 2011, J. Biomech. bf 44], for which experimental flow and pressure measurements are available at few selected locations. In order to mimic clinically relevant situations, we focus on the estimation of terminal resistances and arterial wall parameters related to vessel mechanics (Young's modulus and thickness) using few experimental observations (at most a single pressure or flow measurement per vessel). In all cases, we first perform a theoretical identifiability analysis based on the generalized sensitivity function, comparing then the results obtained with the ROUKF, using either synthetic or experimental data, to results obtained using reference parameters and to available measurements.

  • W. Huang, L. Kamenski, On the mesh nonsingularity of the moving mesh PDE method, Preprint no. 2218, WIAS, Berlin, 2016.
    Abstract, PDF (594 kByte)
    The moving mesh PDE (MMPDE) method for variational mesh generation and adaptation is studied theoretically at the discrete level, in particular the nonsingularity of the obtained meshes. Meshing functionals are discretized geometrically and the MMPDE is formulated as a modified gradient system of the corresponding discrete functionals for the location of mesh vertices. It is shown that if the meshing functional satisfies a coercivity condition, then the mesh of the semi-discrete MMPDE is nonsingular for all time if it is nonsingular initially. Moreover, the altitudes and volumes of its elements are bounded below by positive numbers depending only on the number of elements, the metric tensor, and the initial mesh. Furthermore, the value of the discrete meshing functional is convergent as time increases, which can be used as a stopping criterion in computation. Finally, the mesh trajectory has limiting meshes which are critical points of the discrete functional. The convergence of the mesh trajectory can be guaranteed when a stronger condition is placed on the meshing functional. Two meshing functionals based on alignment and equidistribution are known to satisfy the coercivity condition. The results also hold for fully discrete systems of the MMPDE provided that the time step is sufficiently small and a numerical scheme preserving the property of monotonically decreasing energy is used for the temporal discretization of the semi-discrete MMPDE. Numerical examples are presented

  • J. Borchardt, P. Mathé, G. Printsypar, Calibration methods for gas turbine performance models, Technical Report no. 16, WIAS, Berlin, 2016, DOI 10.20347/WIAS.TECHREPORT.16 .
    Abstract
    The WIAS software package BOP is used to simulate gas turbine models. In order to make accurate predictions the underlying models need to be calibrated. This study compares different strategies of model calibration. These are the deterministic optimization tools as non-linear least squares (MSO) and the sparsity promoting variant LASSO, but also the probabilistic (Bayesian) calibration. The latter allows for the quantification of the inherent uncertainty, and it gives rise to a surrogate uncertainty measure in the MSO tool. The implementation details are accompanied with a numerical case study, which highlights the advantages and drawbacks of each of the proposed calibration methods.

  • H. Gajewski, M. Liero, R. Nürnberg, H. Stephan, WIAS-TeSCA -- Two-dimensional semi-conductor analysis package, Technical Report no. 14, WIAS, Berlin, 2016.
    Abstract
    WIAS-TeSCA (Two-dimensional semiconductor analysis package) is a simulation tool for the numerical simulation of charge transfer processes in semiconductor structures, especially in semiconductor lasers. It is based on the drift-diffusion model and considers a multitude of additional physical effects, like optical radiation, temperature influences and the kinetics of deep impurities. Its efficiency is based on the analytic study of the strongly nonlinear system of partial differential equations -- the van Roosbroeck system -- which describes the electron and hole currents. Very efficient numerical procedures for both the stationary and transient simulation have been implemented.
    WIAS-TeSCA has been successfully used in the research and industrial development of new electronic and optoelectronic semiconductor devices such as transistors, diodes, sensors, detectors and lasers and has already proved its worth many times in the planning and optimization of these devices. It covers a broad spectrum of applications, from hetero-bipolar transistor (mobile telephone systems, computer networks) through high-voltage transistors (power electronics) and semiconductor laser diodes (fiber optic communication systems, medical technology) to radiation detectors (space research, high energy physics).
    WIAS-TeSCA is an efficient simulation tool for analyzing and designing modern semiconductor devices with a broad range of performance that has proved successful in solving many practical problems. Particularly, it offers the possibility to calculate self-consistently the interplay of electronic, optical and thermic effects.

Talks, Poster

  • W. Dreyer, J. Fuhrmann, P. Gajewski, C. Guhlke, M. Landstorfer, M. Maurelli, R. Müller, Stochastic model for LiFePO4-electrodes, ModVal14 - 14th Symposium on Fuel Cell and Battery Modeling and Experimental Validation, Karlsruhe, March 2 - 3, 2017.

  • P. Farrell, Numerical Solution of PDEs via RBFs and FVM with focus on semiconductor problems, Technische Universität Hamburg, Institut für Mathematik, Harburg, January 6, 2017.

  • CH. Merdon, A novel concept for the discretisation of the coupled Nernst--Planck--Poisson--Navier--Stokes system, 14th Symposium on Fuel Cell Modelling and Experimental Validation (MODVAL 14), March 2 - 3, 2017, Karlsruher Institut für Technologie, Institut für Angewandte Materialien, Karlsruhe, Germany, March 3, 2017.

  • CH. Merdon, Druckrobuste Finite-Elemente-Methoden für die Navier-Stokes-Gleichungen, Universität Paderborn, Institut für Mathematik, April 25, 2017.

  • CH. Merdon, Pressure-robustness in mixed finite element discretisations for the Navier--Stokes equations, Universität des Saarlandes, Fakultät für Mathematik und Informatik, July 12, 2017.

  • N. Kumar, J.H.M. Ten Thije Boonkkamp, B. Koren, A. Linke, A nonlinear flux approximation scheme for the viscous Burgers equation., 8th International Symposium on Finite Volumes for Complex Applications (FVCA 8), Villeneuve d'Ascq, France, June 12 - 16, 2017.

  • N. Ahmed, Higher-order discontinuous Galerkin time discretizations for the evolutionary Navier--Stokes equations, Technische Universität Dresden, Institut für Numerische Mathematik, March 9, 2017.

  • N. Ahmed, On really locking-free mixed finite element methods for the transient incompressible Stokes equations, CASM International Conference on Applied Mathematics, May 22 - 24, 2017, Lahore University of Management Sciences, Centre for Advanced Studies in Mathematics, Pakistan, May 22, 2017.

  • C. Bartsch, A mixed stochastic -- Numeric algorithm for transported interacting particles, 38th Northern German Colloquium on Applied Analysis and Numerical Mathematics (NoKo 2017), May 4 - 5, 2017, Technische Universität Hamburg, Institut für Mathematik, May 5, 2017.

  • C. Bartsch, A mixed stochastic-deterministic approach to particles interacting in a flow, SIAM Conference on Mathematical and Computational Issues in the Geosciences, September 11 - 14, 2017, Friedrich-Alexander-Universität Erlangen-Nürnberg, September 14, 2017.

  • C. Bartsch, ParMooN -- A parallel finite element solver, Part I, Indian Institute of Science, Supercomputer Education and Research Centre, Bangalore, India, March 16, 2017.

  • A. Caiazzo, Estimation of cardiovascular system parameters from real data, 2nd Leibniz MMS Days 2017, February 22 - 23, 2017, Technische Informationsbibliothek, Hannover, February 22, 2017.

  • A. Caiazzo, Homogenization methods for weakly compressible elastic materials forward and inverse problem, Workshop on Numerical Inverse and Stochastic Homogenization, February 13 - 17, 2017, Universität Bonn, Hausdorff Research Institute for Mathematics, February 17, 2017.

  • W. Dreyer, Space-Time transformations and the principle of material objectivity, May 3 - 5, 2017, Technische Universität Darmstadt, Fachbereich Mathematik.

  • W. Dreyer, Thermodynamically consistent modeling of fluids, 2nd Leibniz MMS Days 2017, February 22 - 24, 2017, Technische Informationsbibliothek, Hannover, February 23, 2017.

  • P. Farrell, How do electrons move in space? Flux discretizations for non-Boltzmann statistics, SIAM Conference on Computational Science and Engineering (CSE17), February 27 - March 3, 2017, Hilton Atlanta, Georgia, USA, March 1, 2017.

  • J. Fuhrmann, A. Glitzky, M. Liero, Hybrid finite-volume/finite-element schemes for p(x)-laplace thermistor models, 8th International Symposium on Finite Volumes for Complex Applications (FVCA 8), Villeneuve d'Ascq, France, June 15, 2017.

  • J. Fuhrmann, A finite volume scheme for Nernst-Planck-Poisson systems with ion size and solvation effects, 8th International Symposium on Finite Volumes for Complex Applications (FVCA 8), June 12 - 16, 2017, Université Lille 1, Villeneuve d'Ascq, France, June 14, 2017.

  • J. Fuhrmann, Ionic mixtures with volume constraints: Models and numerical approaches, International workshop on liquid metal battery fluid dynamics, May 15 - 17, 2017, Dresden, May 16, 2017.

  • J. Fuhrmann, Strategien für die Verwertung wissenschaftlicher Software: praktische Erfahrungen und aktuelle Entwicklungen, Frühjahrstreffen des Arbeitskreises Wissenstransfer der Leibniz-Gemeinschaft, Leibniz-Zentrum für Marine Tropenforschung, Bremen, May 10, 2017.

  • J. Fuhrmann, tba, 30th Chemnitz FEM Symposium, September 25 - 27, 2017, Bundesinstitut für Erwachsenenbildung, St. Wolfgang / Strobl, Austria.

  • V. John, Analytical and numerical results for algebraic flux correction schemes, 12th International Workshop on Variational Multiscale and Stabilization Methods (VMS-2017), April 26 - 28, 2017, Edificio Celestino Mutis, Campus Reina Mercedes, Sevilla, Spain, April 26, 2017.

  • V. John, Finite element methods for incompressible flow problems, May 14 - 18, 2017, Beijing Computational Science Research Center, Applied and Computational Mathematics, Beijing, China.

  • V. John, Variational Multiscale (VMS) Methods for the simulation of turbulent incompressible flows, Chinese Academy of Science, Beijing, China, May 10, 2017.

  • V. John, Variational Multiscale (VMS) Methods for the simulation of turbulent incompressible flows, Peking University, Beijing, China, May 11, 2017.

  • V. John, Variational multiscale (VMS) methods for the Simulation of turbulent incompressible flows, Mahindra-Ecole-Centrale Hyderabad, Bangalore, India, March 9, 2017.

  • V. John, Variational multiscale (VMS) methods for the Simulation of turbulent incompressible flows, Indian Institute of Science, Supercomputer Education and Research Centre, Bangalore, India, March 16, 2017.

  • V. John, tba, 30th Chemnitz FEM Symposium, September 25 - 27, 2017, Bundesinstitut für Erwachsenenbildung, St. Wolfgang / Strobl, Austria.

  • M. Liero, A. Glitzky, Th. Koprucki, J. Fuhrmann, 3D electrothermal simulations of organic LEDs showing negative differential resistance, Multiscale Modelling of Organic Semiconductors: From Elementary Processes to Devices, Grenoble, France, September 12 - 15, 2017.

  • A. Linke, On new developments in the discretization theory for PDEs, possibly relevant for CFD & GFD, 2nd Leibniz MMS Days 2017, February 22 - 24, 2017, Technische Informationsbibliothek, Hannover, February 23, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, 12th International Workshop on Variational Multiscale and Stabilization Methods (VMS-2017), April 26 - 28, 2017, Edificio Celestino Mutis, Campus Reina Mercedes, Sevilla, Spain, April 27, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, 8th International Symposium on Finite Volumes for Complex Applications (FVCA 8), June 12 - 16, 2017, Université Lille 1, Villeneuve d'Ascq, France, June 13, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, CASM International Conference on Applied Mathematics, May 22 - 24, 2017, Lahore University of Management Sciences, Centre for Advanced Studies in Mathematics, Pakistan, May 23, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, 30th Chemnitz FEM Symposium, September 25 - 27, 2017, Bundesinstitut für Erwachsenenbildung, St. Wolfgang / Strobl, Austria.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, Universität der Bundeswehr München, Institut für Mathematik und Bauinformatik, Neubiberg, January 18, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, Technische Universität Dortmund, Institut für Angewandte Mathematik, March 23, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, Freie Universität Berlin, Institut für Mathematik, May 3, 2017.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, Technische Universität Darmstadt, Fachereich Mathematik, July 20, 2017.

  • CH. Merdon, Pressure-robust finite element methods for the Navier--Stokes equations, GAMM Workshop on Numerical Analysis, November 1 - 2, 2017, Rheinisch-Westfälische Technische Hochschule Aachen, November 2, 2017.

  • CH. Merdon, Pressure-robust mixed finite element methods for the Navier--Stokes equations, scMatheon Workshop RMMM 8 - Berlin 2017, Reliable Methods of Mathematical Modeling, July 31 - August 3, 2017, Humboldt-Universität zu Berlin, August 2, 2017.

  • H. Si, An introduction to delaunay-based mesh generation and adaptation, 10th National Symposium on Geometric Design and Computing (GDC 2017), August 12 - 14, 2017, Shandong Business School, Yantai, China, August 12, 2017.

  • H. Si, An introduction to delaunay-based mesh generation and adaptation, Beijing Computational Science Reserach Center, Beijing, China, August 3, 2017.

  • H. Si, Challenges in Tetrahedral Mesh Generation, Parallel Mesh Partitioning and Adaptation, October 17 - 19, 2017, Inria Bordeaux - Sud-Oues, Research Centre, France, October 18, 2017.

  • H. Si, Mathematical problems in tetrahedral mesh generation, Dalian University, School of Software and Technology, Dalian, China, August 10, 2017.

  • H. Si, n.n., 26th International Meshing Roundtable and User Forum, September 18 - 22, 2017, Crowne Plaza Fira Center, Barcelona, Spain.

  • K. Tabelow, Ch. D'alonzo, L. Ruthotto, M.F. Callaghan, N. Weiskopf, J. Polzehl, S. Mohammadi, Removing the estimation bias due to the noise floor in multi-parameter maps, The International Society for Magnetic Resonance in Medicine (ISMRM) 25th Annual Meeting /& Exhibition, Honolulu, USA, April 22 - 27, 2017.

  • K. Tabelow, Ch. D'alonzo, J. Polzehl, Toward in-vivo histology of the brain, 2nd Leibniz MMs Days2017, Hannover, February 22 - 24, 2017.

  • U. Wilbrandt, ParMooN -- A parallel finite element solver, Part II, Indian Institute of Science, Supercomputer Education and Research Centre, Bangalore, India, March 16, 2017.

  • J. Pellerin, Simultaneous meshing and simplification of complex 3D geometrical models using Voronoi diagrams, Sorbonne Universités, Institut du calcul et de la simulation, Paris, France, January 28, 2016.

  • G. Printsypar, Modeling of membranes for forward osmosis at different scales, Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM, Strömungs- und Materialsimulation, Kaiserslautern, May 19, 2016.

  • N. Alia, V. John, Optimal control of ladle stirring, 1st Leibniz MMS Mini Workshop on CFD & GFD, WIAS Berlin, September 8 - 9, 2016.

  • F. Dassi, A novel anisotropic mesh adaption strategy, Politecnico di Milano, Dipartimento di Matematica ``F. Brioschi'', Milano, Italy, February 23, 2016.

  • CH. Merdon, J. Fuhrmann, A. Linke, A.A. Abd-El-Latif, M. Khodayari, P. Reinsberg, H. Baltruschat, Inverse modelling of thin layer flow cells and RRDEs, The 67th Annual Meeting of the International Society of Electrochemistry, Den Haag, Netherlands, August 21 - 26, 2016.

  • CH. Merdon, Inverse modelling of transport and reaction processes in thin-layer flow cells, 13th Symposium for Fuel Cell and Battery Modeling and Experimental Validation (MODVAL 13), March 22 - 23, 2016, Lausanne, Switzerland, March 23, 2016.

  • CH. Merdon, Pressure-robust 1st and 2nd order finite element methods for Navier--Stokes discretisations, ``Variational Multiscale and Stabilization Methods'' (VMS2016), March 16 - 18, 2016, Otto-von-Guericke Universität Magdeburg, Magdeburg, March 17, 2016.

  • CH. Merdon, Pressure-robust 1st and 2nd-order finite element methods for Navier--Stokes discretisations, 37. Norddeutsches Kolloquium über Angewandte Analysis und Numerische Mathematik (NoKo 2016), April 22 - 23, 2016, Universität zu Lübeck, Lübeck, April 22, 2016.

  • CH. Merdon, Pressure-robust finite element methods for the Navier--Stokes equations and mass conservative coupling to transport processes, Friedrich-Alexander Universität Erlangen-Nürnberg, Fachbereich Mathematik, May 11, 2016.

  • CH. Merdon, Pressure-robust mixed finite element methods for transient Navier--Stokes discretisations, Technische Universität Wien, Institut für Analysis und Scientific Computing, Austria, December 14, 2016.

  • R. Müller, W. Dreyer, J. Fuhrmann, C. Guhlke, New insights into Butler--Volmer kinetics from thermodynamic modeling, The 67th Annual Meeting of the International Society of Electrochemistry, Den Haag, Netherlands, August 21 - 26, 2016.

  • S. Giere, A walk to a random forest, Seminar Numerische Mathematik, WIAS, Berlin, October 13, 2016.

  • J. Pellerin, Simultaneous meshing and simplification of complex 3D geometrical models using Voronoi diagrams, Université Catholique de Louvain, Institute of Mechanics, Materials and Civil Engineering, Louvain-la-Neuve, Belgium, June 30, 2016.

  • N. Ahmed, A review of VMS methods for the simulation of turbulent incompressible flows, International Conference on Differential Equations and Applications, May 26 - 28, 2016, Lahore University of Management Sciences, Pakistan, May 27, 2016.

  • N. Ahmed, On the grad-div stabilization for the steady Oseen and Navier--Stokes evaluations, International Conference of Boundary and Interior Layers (BAIL 2016), August 15 - 19, 2016, Beijing Computational Science Research Center, Beijing, China, August 15, 2016.

  • C. Bartsch, An assessment of solvers for saddle point problems emerging from the incompressible Navier--Stokes equations, GAMM workshop on Computational Science and Engineering, September 8 - 9, 2016, Universität Kassel, Kassel, September 9, 2016.

  • A. Caiazzo, A comparative study of backflow stabilization methods, 7th European Congress of Mathematics (7ECM), July 18 - 22, 2016, Technische Universität Berlin, Berlin, July 19, 2016.

  • A. Caiazzo, Backflow stabilization methods for open boundaries, Christian-Albrechts-Universität zu Kiel, Angewandte Mathematik, Kiel, May 19, 2016.

  • W. Dreyer, Subtle properties of the barycentric velocity in mixture models, 1st Leibniz MMS Mini Workshop on CFD & GFD, September 8, 2016, WIAS, Berlin, September 8, 2016.

  • P. Farrell, Finite volume schemes for non-Boltzmann statistics, Friedrich-Alexander Universität Erlangen-Nürnberg, Fachbereich Mathematik, May 12, 2016.

  • P. Farrell, Multilevel collocation with radial basis function, Friedrich-Alexander Universität Erlangen-Nürnberg, Fachbereich Mathematik, May 12, 2016.

  • P. Farrell, Scharfetter--Gummel schemes for Non-Boltzmann statistics, Conference on Scientific Computing (ALGORITMY 2016), March 14 - 18, 2016, Slovak University of Technology, Department of Mathematics and Descriptive Geometry, Podbanské, Slovakia, March 17, 2016.

  • P. Farrell, Scharfetter--Gummel schemes for non-Boltzmann statistics, The 19th European Conference on Mathematics for Industry (ECMI2016), Minisymposium 23 ``Charge Transport in Semiconductor Materials: Emerging and Established Mathematical Topics'', June 13 - 17, 2016, Universidade de Santiago de Compostela, Spain, June 14, 2016.

  • P. Farrell, Scharfetter--Gummel schemes for non-Boltzmann statistics, Seminar Numerische Mathematik, WIAS Berlin, September 22, 2016.

  • J. Fuhrmann, Ch. Merdon, A thermodynamically consistent numerical approach to Nernst--Planck--Poisson systems with volume constraints, The 67th Annual Meeting of the International Society of Electrochemistry, Den Haag, Netherlands, August 21 - 26, 2016.

  • J. Fuhrmann, W. Dreyer, C. Guhlke, M. Landstorfer, R. Müller, A. Linke, Ch. Merdon, Modeling and numerics for electrochemical systems, Micro Battery and Capacitive Energy Harvesting Materials -- Results of the MatFlexEnd Project, Universität Wien, Austria, September 19, 2016.

  • J. Fuhrmann, A. Linke, Ch. Merdon, M. Khodayari , H. Baltruschat, Detection of solubility, transport and reaction coefficients from experimental data by inverse modelling of thin layer flow cells, 1st Leibniz MMS Mini Workshop on CFD & GFD, WIAS Berlin, September 8 - 9, 2016.

  • J. Fuhrmann, A. Linke, Ch. Merdon, W. Dreyer, C. Guhle, M. Landstorfer, R. Müller, Numerical methods for electrochemical systems, 2nd Graz Battery Days, Graz, Austria, September 27 - 28, 2016.

  • J. Fuhrmann, Computational assessment of the derivation of the Butler--Volmer kinetics as a limit case of the Nernst--Planck equations with surface reactions, 13th Symposium for Fuel Cell and Battery Modeling and Experimental Validation (MODVAL 13), March 22 - 23, 2016, Lausanne, Switzerland, March 23, 2016.

  • J. Fuhrmann, Models and numerics for Nernst--Planck--Poisson systems with volume constraints, Helmholtz Institut, Fakultät für Naturwissenschaften, Ulm, September 1, 2016.

  • J. Fuhrmann, Nernst-Planck-Poisson-Systems with volume constraints: modeling and numerics, Conference on Scientific Computing (ALGORITMY 2016), March 14 - 18, 2016, Slovak University of Technology, Department of Mathematics and Descriptive Geometry, Podbanské, Slovakia, March 17, 2016.

  • J. Fuhrmann, Numerical methods for generalized Drift-Diffusion models in electrochemical devices and semiconductors, 7th European Congress of Mathematics (7ECM), July 18 - 22, 2016, Technische Universität Berlin, Berlin, July 22, 2016.

  • C. Guhlke, W. Dreyer, R. Müller, M. Landstorfer, J. Fuhrmann, Beyond Newman's battery model, 2nd Graz Battery Days, Graz, Austria, September 27 - 28, 2016.

  • C. Guhlke, J. Fuhrmann, W. Dreyer, R. Müller, M. Landstorfer, Modeling of batteries, Batterieforum Deutschland 2016, Berlin, April 6 - 8, 2016.

  • V. John, A review of variational multiscale methods for the simulation of turbulent incompressible flows, 15th Mathematics of Finite Elements and Applications, June 14 - 17, 2016, Brunel University London, London, UK, June 17, 2016.

  • V. John, A review of variational multiscale methods for the simulation of turbulent incompressible flows, ``Variational Multiscale and Stabilization Methods'' (VMS2016), March 16 - 18, 2016, Otto-von-Guericke Universität Magdeburg, Magdeburg, March 17, 2016.

  • V. John, A review of variational multiscale methods for the simulation of turbulent incompressible flows, International Conference of Boundary and Interior Layers (BAIL 2016), August 15 - 19, 2016, Beijing Computational Science Research Center, Beijing, China, August 15, 2016.

  • V. John, A review of variational multiscale methods for the simulation of turbulent incompressible flows, 7th American Mathematical Society Meeting, Special Session on Above and Beyond Fluid Flow Studies, October 8 - 9, 2016, Denver, Colorado, USA, October 9, 2016.

  • V. John, A survey on the analysis and numerical analysis of some turbulence models, Technische Universität Darmstadt, Fachbereich Mathematik, January 20, 2016.

  • V. John, Analytical and numerical results for algebraic flux correction schemes, Conference on Recent Advances in Analysis and Numerics of Hyperbolic Conservation Laws, September 8 - 10, 2016, Otto-von-Guericke Universität Magdeburg, September 9, 2016.

  • V. John, Ein weites Feld -- Wissenschaftliche Beiträge von Prof. Dr. Lutz Tobiska, Festkolloquium aus Anlass des 65. Geburtstags von Prof. Dr. Lutz Tobiska, Universität Magdeburg, Institut für Analysis und Numerik, March 31, 2016.

  • V. John, On the divergence constraint in mixed finite element methods for incompressible flows, 5th European Seminar on Computing (ESCO 2016), June 5 - 10, 2016, Pilsen, Czech Republic, June 7, 2016.

  • V. John, On the divergence constraint in mixed finite element methods for incompressible flows, Beijing Computational Science Research Center, China, August 23, 2016.

  • V. John, The role of the pressure in finite element methods for incompressible flow problems, Summer School 2016 ``Fluids under Pressure'' and Workshop, August 29 - September 2, 2016, Nečas Center for Mathematical Modeling, Prague, Czech Republic.

  • L. Kamenski, Tetrahedral mesh improvement using moving mesh smoothing and lazy searching flips, 25th International Meshing Roundtable, September 27 - 30, 2016, DoubleTree by Hilton Washington, DC, USA, September 28, 2016.

  • A. Linke, Ch. Merdon, Pressure-robustness and acceleration of Navier--Stokes solvers, 1st Leibniz MMS Days, WIAS Berlin, January 27 - 29, 2016.

  • A. Linke, Ch. Merdon, Pressure-robustness and acceleration of Navier--Stokes solvers, 37th Northern German Colloquium on Applied Analysis and Numerical Mathematics (NoKo 2016), Universität zu Lübeck, April 22 - 23, 2016.

  • A. Linke, Robust discretization of advection-diffusion-reaction equations and the incompressible Navier--Stokes equations, Technische Universiteit Eindhoven, Department of Mathematics and Computer Science, Netherlands, November 24, 2016.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stockes equations, Universität Rostock, Leibniz-Institut für Atmosphärenphysik, Kühlungsborn, July 14, 2016.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, Numerical Analysis and Predictability of Fluid Motion, May 3 - 4, 2016, Institute for Mathematics and its Applications, Pittsburgh, USA, May 4, 2016.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, 15th Mathematics of Finite Elements and Applications, June 14 - 17, 2016, Brunel University London, London, UK, June 17, 2016.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, ``Variational Multiscale and Stabilization Methods'' (VMS2016), March 16 - 18, 2016, Otto-von-Guericke Universität Magdeburg, Magdeburg, March 17, 2016.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, Workshop ``Modelling, Model Reduction, and Optimization of Flows'', September 26 - 30, 2016, Shanghai University, China, September 27, 2016.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, Advanced Numerical Methods: Recent Developments, Analysis, and Applications, October 3 - 7, 2016, Institut Henri Poincaré, Paris, France, October 6, 2016.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, 1st Leibniz MMS Mini Workshop on CFD & GFD, September 8 - 9, 2016, WIAS, Berlin, September 8, 2016.

  • A. Linke, Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations, Technische Universität Kaiserslautern, Scientific Computing, December 15, 2016.

  • H. Si, An introduction to Delaunay-based mesh generation and adaptation, University of Kansas, Department of Mathematics, Lawrence, USA, October 5, 2016.

  • H. Si, An introduction to Delaunay-based mesh generation and adaptation, State University of New York, Department of Computer Science, Stony Brook, USA, October 11, 2016.

  • H. Si, On 3D irreducible and indecomposable polyhedra and the number of interior Steiner points, International Conference ``Numerical Geometry, Grid Generation and Scientific Computing'' (NUMGRID 2016), October 31 - November 2, 2016, Russian Academy of Sciences, Federal Research Center of Information and Control, Moscow, October 31, 2016.

  • H. Si, Some geometric problems in tetrahedral mesh generation, Fifth Workshop on Grid Generation for Numerical Computations (Tetrahedron V), July 4 - 5, 2016, University of Liège, Montefiore Institute, Department of Electrical Engineering and Computer Science, Belgium, July 5, 2016.

  • H. Si, TetGen, a Delaunay-based quality tetrahedral mesh generator, Old Dominion University, Department of Computer Science, Norfolk, USA, October 7, 2016.

  • K. Tabelow, Ch. D'alonzo, J. Polzehl, M.F. Callaghan, L. Ruthotto, N. Weiskopf, S. Mohammadi, How to achieve very high resolution quantitative MRI at 3T?, 22th Annual Meeting of the Organization of Human Brain Mapping (OHBM 2016), Geneva, Switzerland, June 26 - 30, 2016.

External Preprints

  • J. Fuhrmann, Zugang zu und Nachnutzung von wissenschaftlicher Software, Report, Deutsches GeoForschungsZentrum GFZ, 2017, DOI 10.2312/lis.17.01 .

  • G.R. Barrenechea, V. John, P. Knobloch, An algebraic flux correction scheme satisfying the discrete maximum principle and linearity preservation on general meshes, Preprint no. 2016-06, Nečas Center for Mathematical Modeling, 2016.