A main research field is the development, analysis, improvement and application of numerical methods for equations coming from CFD. These methods are based on finite element or finite volume methods as spatial discretization and implicit temporal discretizations.
Convection-diffusion-reaction equations model, for instance, transport processes which are driven by diffusion or convection. A challenging situation from the point of view of simulations is the c ase of dominant convection, because the solution has scales which cannot be resolved by the used grids (layers). Standard stabilized methods lead to smeared discrete solutions or to solutions possessing spurious oscillations. Spurious oscillations are not acceptable in many applications. On research topic is the development of oscillation-free numerical methods for convection-dominated equations. In addition, numerical methods for nonlinear transport equations with exponential nonlinearities are developed and analyzed.
The Navier--Stokes equations are the fundamental equations of fluid mechanics. A main topic of research are methods for the simulation of turbulent flows, in particular large eddy simulation (LES) and variational multiscale (VMS) methods. The basic difficulty in the simulation of turbulent flows consists also in the impossibility of resolving small flow structures (scales) on the used grids. LES and VMS methods are two approaches for the simulation of large scales, which model the effect of the unresolved small scales onto the large scales via a turbulence model. Another topic of research is the development of discretely mass-conserving discretizations. This property is essential for some of the considered applications.
In these applications, coupled systems of equations have to be solved that contain (among others) the types of equations described above. An example are population balance systems. These are coupled systems consisting of the Navier-Stokes equations (flow), convection-diffusion-reaction equations (chemical reaction, transport of energy) and transport equations for the particle size distribution. These transport equations are defined in a higher dimension than the other equations because the particle size distribution depends also on properties of the particles (internal coordinates). Another example are van Roosbroeck systems which describe the transport of carriers in a semiconductor crystal lattice. The equations of such systems contain exponential nonlinearities. Numerical methods for Stokes--Darcy systems, which model the coupling of free flows and flows in porous media, are investigated, too. In addition, it is studied to which extend reduced order models are able to increase the efficiency of simulations with only an acceptable loss of accuracy.




Flow through an aortic arch






Turbulent flow around a cylinder, velocity (left) and vorticity (right)


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).

  • K. Gärtner, H. Si, A. Rand, N. Walkington, Chapter 11: 3D Delaunay Mesh Generation, in: Combinatorial Scientific Computing, U. Naumann, O. Schenk, eds., Computational Science Series, CRC Computational Science/Chapman & Hall, Boca Raton, 2012, pp. 299--319, (Chapter Published).

  Articles in Refereed Journals

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

  • 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.

  • 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.

  • N. Rotundo, T.-Y. Kim, W. Jiang, L. Heltai, E. Fried, Error estimates of B-spline based finite-element method for the wind-driven ocean circulation, Journal of Scientific Computing, 69 (2016) pp. 430--459.
    Abstract
    We present the error analysis of a B-spline based finite-element approximation of the stream-function formulation of the large scale wind-driven ocean circulation. In particular, we derive optimal error estimates for h-refinement using a Nitsche-type variational formulations of the two simplied linear models of the stationary quasigeostrophic equations, namely the Stommel and Stommel--Munk models. Numerical results on rectangular and embedded geometries confirm the error analysis.

  • 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.

  • 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.

  • 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.

  • 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.

  • G.R. Barrenechea, V. John, P. Knobloch, Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in one dimension, IMA Journal of Numerical Analysis, 34 (2015) pp. 1729--1756.
    Abstract
    Algebraic flux correction schemes are nonlinear discretizations of convection dominated problems. In this work, a scheme from this class is studied for a steady-state convection--diffusion equation in one dimension. It is proved that this scheme satisfies the discrete maximum principle. Also, as it is a nonlinear scheme, the solvability of the linear subproblems arising in a Picard iteration is studied, where positive and negative results are proved. Furthermore, the non-existence of solutions for the nonlinear scheme is proved by means of counterexamples. Therefore, a modification of the method, which ensures the existence of a solution, is proposed. A weak version of the discrete maximum principle is proved for this modified method.

  • CH. Brennecke, A. Linke, Ch. Merdon, J. Schöberl, Optimal and pressure-independent $L^2$ velocity error estimates for a modified Crouzeix--Raviart Stokes element with BDM reconstructions, Journal of Computational Mathematics, 33 (2015) pp. 191--208.
    Abstract
    Nearly all inf-sup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a priori estimates for the velocity error become pressure-dependent, while divergence-free mixed finite elements deliver pressure-independent estimates. A recently introduced new variational crime using lowest-order Raviart-Thomas velocity reconstructions delivers a much more robust modified Crouzeix-Raviart element, obeying an optimal pressure-independent discrete H1 velocity estimate. Refining this approach, a more sophisticated variational crime employing the lowest-order BDM element is proposed, which also allows proving an optimal pressure independent L2 velocity error. Numerical examples confirm the analysis and demonstrate the improved robustness in the Navier-Stokes case.

  • V. John, J. Novo, Analysis of the pressure stabilized Petrov--Galerkin (PSPG) method stabilization for the evolutionary Stokes equations avoiding time-step restrictions, SIAM Journal on Numerical Analysis, 53 (2015) pp. 1005--1031.
    Abstract
    Optimal error estimates for the pressure stabilized Petrov--Galerkin (PSPG) method for the continuous-in-time discretization of the evolutionary Stokes equations are proved in the case of regular solutions. The main result is applicable to higher order finite elements. The error bounds for the pressure depend on the error of the pressure at the initial time. An approach is suggested for choosing the discrete initial velocity in such a way that this error is bounded. The ``instability of the discrete pressure for small time steps'', which is reported in the literature, is discussed on the basis of the analytical results. Numerical studies confirm the theoretical results, showing in particular that this instability does not occur for the proposed initial condition.

  • S. Giere, T. Iliescu, V. John, D. Wells, SUPG reduced order models for convection-dominated convection-diffusion-reaction equations, Computer Methods in Applied Mechanics and Engineering, 289 (2015) pp. 454--474.
    Abstract
    This paper presents a Streamline-Upwind Petrov--Galerkin (SUPG) reduced order model (ROM) based on Proper Orthogonal Decomposition (POD). This ROM is investigated theoretically and numerically for convection-dominated convection-diffusion-reaction equations. The SUPG finite element method was used on realistic meshes for computing the snapshots, leading to some noise in the POD data. Numerical analysis is used to propose the scaling of the stabilization parameter for the SUPG-ROM. Two approaches are used: One based on the underlying finite element discretization and the other one based on the POD truncation. The resulting SUPG-ROMs and the standard Galerkin ROM (G-ROM) are studied numerically. For many settings, the results obtained with the SUPG-ROMs are more accurate. Finally, one of the choices for the stabilization parameter is recommended.

  • A. Caiazzo, R. Guibert, Y. Boudjemline, I.E. Vignon-Clementel, Efficient blood flow simulations for the design of stented valve reducer in enlarged ventricular outflow tracts, Cardiovascular Engineering and Technology, 6 (2015) pp. 485--500.
    Abstract
    Tetralogy of Fallot is a congenital heart disease characterized over time, after the initial repair, by the absence of a functioning pulmonary valve, which causes regurgitation, and by progressive enlargement of the right ventricle and pulmonary arteries. Due to this pathological anatomy, available transcatheter valves are usually too small to be deployed in the enlarged right ventricular outflow tracts (RVOT). To avoid surgical valve replacement, an alternative consists in implanting a reducer prior to or in combination with a transcatheter valve. We describe a computational model to study the effect of a stented valve RVOT reducer on the hemodynamics in enlarged ventricular outflow tracts. To this aim, blood flow in the right ventricular outflow tract is modeled via the incompressible Navier--Stokes equations coupled to a simplified valve model, numerically solved with a standard finite element method and with a reduced order model based on Proper Orthogonal Decomposition (POD). Numerical simulations are based on a patient geometry obtained from medical imaging and boundary conditions tuned according to measurements of inlet flow rates and pressures. Different geometrical models of the reducer are built, varying its length and/or diameter, and compared with the initial device-free state. Simulations thus investigate multiple device configurations and describe the effect of geometry on hemodynamics. Forces exerted on the valve and on the reducer are monitored, varying with geometrical parameters. Results support the thesis that the reducer does not introduce significant pressure gradients, as was found in animal experiments. Finally, we demonstrate how computational complexity can be reduced with POD.

  • A. Caiazzo, G. Montecinos, L.O. Müller, E.M. Haacke, E.F. Toro, Computational haemodynamics in stenotic internal jugular veins, Journal of Mathematical Biology, 70 (2015) pp. 745--772.
    Abstract
    Stenosis in internal jugular veins (IJVs) are frequently associated to pathological venous circulation and insufficient cerebral blood drainage. In this work, we set up a computational framework to assess the relevance of IJV stenoses through numerical simulation, combining medical imaging, patient-specific data and a mathematical model for venous occlusions. Coupling a three-dimensional (3D) description of blood flow in IJVs with a reduced one-dimesional model (1D) for major intracranial veins, we are able to model different anatomical configurations, an aspect of importance to understand the impact of IJV stenosis in intracranial venous haemodynamics. We investigate several stenotic configurations in a physiologic patient-specific regime, quantifying the effect of the stenosis in terms of venous pressure increase and wall shear stress patterns. Simulation results are in qualitative agreement with reported pressure anomalies in pathological cases. Moreover, they demonstrate the potential of the proposed multiscale framework for individual-based studies and computer-aided diagnosis.

  • A. Caiazzo, I. Ramis-Conde, Multiscale modeling of palisade formation in glioblastoma multiforme, Journal of Theoretical Biology, 383 (2015) pp. 145--156.
    Abstract
    Palisades are characteristic tissue aberrations that arise in glioblastomas. Observation of palisades is considered as a clinical indicator of the transition from a noninvasive to an invasive tumour. In this article we propose a computational model to study the influence of genotypic and phenotypic heterogeneity in palisade formation. For this we produced three dimensional realistic simulations, based on a multiscale hybrid model, coupling the evolution of tumour cells and the oxygen diffusion in tissue, that depict the shape of palisades during its formation. Our results can be summarized as the following: (1) we show that cell heterogeneity is a crucial factor in palisade formation and tumour growth; (2) we present results that can explain the observed fact that recursive tumours are more malignant than primary tumours; and (3) the presented simulations can provide to clinicians and biologists for a better understanding of palisades 3D structure as well as glioblastomas growth dynamics

  • A. Linke, Ch. Merdon, Guaranteed energy error estimators for a modified robust Crouzeix--Raviart Stokes element, Journal of Scientific Computing, 64 (2015) pp. 541--558.
    Abstract
    This paper provides guaranteed upper energy error bounds for a modified lowest-order nonconforming Crouzeix--Raviart finite element method for the Stokes equations. The modification from [A. Linke 2014, On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime] is based on the observation that only the divergence-free part of the right-hand side should balance the vector Laplacian. The new method has optimal energy error estimates and can lead to errors that are smaller by several magnitudes, since the estimates are pressure-independent. An efficient a posteriori velocity error estimator for the modified method also should involve only the divergence-free part of the right-hand side. Some designs to approximate the Helmholtz projector are compared and verified by numerical benchmark examples. They show that guaranteed error control for the modified method is possible and almost as sharp as for the unmodified method.

  • A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Computer Methods in Applied Mechanics and Engineering, 268 (2014) pp. 782--800.
    Abstract
    According to the Helmholtz decomposition, the irrotational parts of the momentum balance equations of the incompressible Navier-Stokes equations are balanced by the pressure gradient. Unfortunately, nearly all mixed methods for incompressible flows violate this fundamental property, resulting in the well-known numerical instability of poor mass conservation. The origin of this problem is the lack of L2-orthogonality between discretely divergence-free velocities and irrotational vector fields. In order to cure this, a new variational crime using divergence-free velocity reconstructions is proposed. Applying lowest order Raviart-Thomas velocity reconstructions to the nonconforming Crouzeix-Raviart element allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence-free flow solvers. In the Stokes case, optimal a-priori error estimates for the velocity gradients and the pressure are derived. Moreover, the discrete velocity is independent of the continuous pressure. Several detailed linear and nonlinear numerical examples illustrate the theoretical findings.

  • E. Jenkins, V. John, A. Linke, L.G. Rebholz, On the parameter choice in grad-div stabilization for the Stokes equations, Advances in Computational Mathematics, 40 (2014) pp. 491--516.
    Abstract
    Grad-div stabilization has been proved to be a very useful tool in discretizations of incompressible flow problems. Standard error analysis for inf-sup stable conforming pairs of finite element spaces predicts that the stabilization parameter should be optimally chosen to be O(1). This paper revisits this choice for the Stokes equations on the basis of minimizing the $H^1$ error of the velocity and the $L^2$ error of the pressure. It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle than known so far in the literature. It depends on the used norm, the solution, the family of finite element spaces, and the type of mesh. Depending on the situation, the optimal stabilization parameter might range from being very small to very large. The analytic results are supported by numerical examples.

  • C. Bertoglio, A. Caiazzo, A tangential regularization method for backflow stabilization in hemodynamics, Journal of Computational Physics, 261 (2014) pp. 162--171.
    Abstract
    In computational simulations of fluid flows, instabilities at the Neumann boundaries may appear during backflow regime. It is widely accepted that this is due to the incoming energy at the boundary, coming from the convection term, which cannot be controlled when the velocity field is unknown. We propose a stabilized formulation based on a local regularization of the fluid velocity along the tangential directions on the Neumann boundaries. The stabilization term is proportional to the amount of backflow, and does not require any further assumption on the velocity profile. The perfomance of the method is assessed on a two- and three-dimensional Womersley flows, as well as considering a hemodynamic physiological regime in a patient-specific aortic geometry.

  • R. Eymard, J. Fuhrmann, A. Linke, On MAC schemes on triangular Delaunay meshes, their convergence and application to coupled flow problems, Numerical Methods for Partial Differential Equations. An International Journal, 30 (2014) pp. 1397--1424.
    Abstract
    We study two classical generalized MAC schemes on unstructured triangular Delaunay meshes for the incompressible Stokes and Navier-Stokes equations and prove their convergence for the first time. These generalizations use the duality between Voronoi and triangles of Delaunay meshes, in order to construct two staggered discretization schemes. Both schemes are especially interesting, since compatible finite volume discretizations for coupled convection-diffusion equations can be constructed which preserve discrete maximum principles. In the first scheme, called tangential velocity scheme, the pressures are defined at the vertices of the mesh, and the discrete velocities are tangential to the edges of the triangles. In the second scheme, called normal velocity scheme, the pressures are defined in the triangles, and the discrete velocities are normal to the edges of the triangles. For both schemes, we prove the convergence in $L^2$ for the velocities and the discrete rotations of the velocities for the Stokes and the Navier-Stokes problem. Further, for the normal velocity scheme, we also prove the strong convergence of the pressure in $L^2$. Linear and nonlinear numerical examples illustrate the theoretical predictions.

  • J. DE Frutos, B. Garc'ia-Archilla, V. John, J. Novo, An adaptive SUPG method for evolutionary convection-diffusion equations, Computer Methods in Applied Mechanics and Engineering, 273 (2014) pp. 219--237.
    Abstract
    An adaptive algorithm for the numerical simulation of time-dependent convection-diffusion-reaction equations will be proposed and studied. The algorithm allows the use of the natural extension of any error estimator for the steady-state problem for controlling local refinement and coarsening. The main idea consists in considering the SUPG solution of the evolutionary problem as the SUPG solution of a particular steady-state convection-diffusion problem with data depending on the computed solution. The application of the error estimator is based on a heuristic argument by considering a certain term to be of higher order. This argument is supported in the one-dimensional case by numerical analysis. In the numerical studies, particularly the residual-based error estimator from [18] will be applied, which has proved to be robust in the SUPG norm. The effectivity of this error estimator will be studied and the numerical results (accuracy of the solution, fineness of the meshes) will be compared with results obtained by utilizing the adaptive algorithm proposed in [6].

  • A. Caiazzo, T. Iliescu, V. John, S. Schyschlowa, A numerical investigation of velocity-pressure reduced order models for incompressible flows, Journal of Computational Physics, 259 (2014) pp. 598--616.
    Abstract
    This report has two main goals. First, it numerically investigates three velocity-pressure reduced order models (ROMs) for incompressible flows. The proper orthogonal decomposition (POD) is used to generate the modes. One method computes the ROM pressure solely based on the velocity POD modes, whereas the other two ROMs use pressure modes as well. To the best of the authors' knowledge, one of the latter methods is novel. The second goal is to numerically investigate the impact of the snapshot accuracy on the ROMs accuracy. Numerical studies are performed on a two-dimensional laminar flow past a circular obstacle. It turns out that, both in terms of accuracy and efficiency, the two ROMs that utilize pressure modes are clearly superior to the ROM that uses only velocity modes. The numerical results also show a strong correlation of the accuracy of the snap shots with the accuracy of the ROMs.

  • A. Caiazzo, V. John, U. Wilbrandt, On classical iterative subdomain methods for the Stokes--Darcy problem, Computer & Geosciences, 18 (2014) pp. 711--728.
    Abstract
    Iterative subdomain methods for the Stokes--Darcy problem that use Robin boundary conditions on the interface are reviewed. Their common underlying structure and their main differences are identified. In particular, it is clarified that there are different updating strategies for the interface conditions. For small values of fluid viscosity and hydraulic permeability, which are relevant in applications from geosciences, it is shown in numerical studies that only one of these updating strategies leads to an efficient numerical method, if this strategy is used in combination with appropriate parameters in the Robin boundary conditions. In particular, it is observed that the values of appropriate parameters are larger than those proposed so far. Not only the size but also the ratio of appropriate Robin parameters depends on the coefficients of the problem.

  • V. John, L. Schumacher, A study of isogeometric analysis for scalar convection-diffusion equations, Applied Mathematics Letters, 27 (2014) pp. 43--48.

  • R. Bordás, V. John, E. Schmeyer, D. Thévenin, Numerical methods for the simulation of a coalescence-driven droplet size distribution, Theoretical and Computational Fluid Dynamics. Springer-Verlag, Berlin., 27 (2013) pp. 253--271.
    Abstract
    A droplet size distribution in a turbulent flow field is considered and modeled by means of a population balance system. This paper studies different numerical methods for the 4D population balance equation and their impact on an output of interest, the time-space-averaged droplet size distribution at the outlet which is known from experiments. These methods include different interpolations of the experimental data at the inlet, various discretizations in time and space, and different schemes for computing the aggregation integrals. It will be shown that notable changes in the output of interest might occur. In addition, the efficiency of the studied methods is discussed.

  • G.R. Barrenechea, V. John, P. Knobloch, A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013) pp. 1335--1366.
    Abstract
    An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived which are supported by numerical studies. These studies demonstrate also the reduction of the spurious oscillations.

  • A. Bradji, J. Fuhrmann, Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes, Applications of Mathematics, 58 (2013) pp. 1--38.
    Abstract
    A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems by R. Eymard and coworkers. Thanks to these basic ideas developed for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. Although the numerical scheme stems from the finite volume method, its formulation is based on the discrete version for the weak formulation defined for the heat problem. We derive error estimates for the solution in discrete norm, and an error estimate for an approximation of the gradient, in a general framework in which the discrete bilinear form is satisfying ellipticity. We prove in particular, that, when the discrete flux is calculated using a stabilized discrete gradient, the convergence order is h+k , where h (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumption that the exact solution is twice continuously differentiable in time and space. These error estimates are useful because they allow us to get error estimates for the approximations of the exact solution and its first derivatives.

  • B. Cousins, S. Le Borne, A. Linke, Z. Wang, Efficient linear solvers for incompressible flow simulations using Scott--Vogelius finite elements, Numerical Methods for Partial Differential Equations. An International Journal, 29 (2013) pp. 1217--1237.
    Abstract
    Recent research has shown that in some practically relevant situations like multiphysics flows (Galvin et al., Comput Methods Appl Mech Eng, 2012) divergence-free mixed finite elements may have a significantly smaller discretization error than standard nondivergence-free mixed finite elements. To judge the overall performance of divergence-free mixed finite elements, we investigate linear solvers for the saddle point linear systems arising in Scott-Vogelius finite element implementations of the incompressible Navier-Stokes equations. We investigate both direct and iterative solver methods. Due to discontinuous pressure elements in the case of Scott-Vogelius (SV) elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements such as Taylor-Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes and discuss extensions. For iterative methods, we test augmented Lagrangian and H -LU preconditioners with GMRES, on both full and statically condensed systems. Several numerical experiments are provided that show these classes of solvers are well suited for use with SV elements and could deliver an interesting overall performance in several applications.

  • V. John, J. Novo, A robust SUPG norm a posteriori error estimator for stationary convection-diffusion equations, Computer Methods in Applied Mechanics and Engineering, 255 (2013) pp. 289--305.
    Abstract
    A robust residual-based a posteriori estimator is proposed for the SUPG finite element method applied to stationary convection-diffusion-reaction equations. The error in the natural SUPG norm is estimated. The main concern of this paper is the consideration of the convection-dominated regime. A global upper bound and a local lower bound for the error are derived, where the global upper estimate relies on some hypotheses. Numerical studies demonstrate the robustness of the estimator and the fulfillment of the hypotheses. A comparison to other residual-based estimators with respect to the adaptive grid refinement is also provided.

  • A. Linke, L. Rebholz, On a reduced sparsity stabilization of grad-div type for incompressible flow problems, Computer Methods in Applied Mechanics and Engineering, 261--262 (2013) pp. 142--153.
    Abstract
    We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad-div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad-div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operator's matrices are identical to those of grad-div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad-div stabilization, and in all cases, solutions are found to be very similar.

  • R. Bordás, V. John, E. Schmeyer, D. Thévenin, Measurement and simulation of a droplet population in a turbulent flow field, Computers and Fluids. Pergamon Press, Oxford., 66 (2012) pp. 52--62.
    Abstract
    The interaction of a disperse droplet population (spray) in a turbulent flow field is studied by combining wind tunnel experiments with simulations based on the model of a population balance system. The behavior of the droplets is modeled numerically by a population balance equation. Velocities of the air and of the droplets are determined by non-intrusive measurements. A direct discretization of the 4D equation for the droplet size distribution is used in the simulations. Important components of the numerical algorithm are a variational multiscale method for turbulence modeling, an upwind scheme for the 4D equation and a pre-processing approach to evaluate the aggretation integrals. The simulations of this system accurately predict the modifications of the droplet size distribution from the inlet to the outlet of the measurement section. Since the employed configuration is simple and considering that all measurement data are freely available thanks to an Internet-based repository, the considered experiment is proposed as a benchmark problem for the simulation of disperse two-phase turbulent flows.

  • R. Eymard, Th. Gallouët, R. Herbin, A. Linke, Finite volume schemes for the biharmonic problem on general meshes, Mathematics of Computation, 81 (2012) pp. 2019--2048.
    Abstract
    We propose a finite volume scheme for the approximation of a biharmonic problem with Dirichlet boundary conditions. We prove that the piece-wise constant approximate solution converges to the exact solution, as well as the discrete approximate of the gradient and the discrete approximate of the Laplacian of the exact solution. These results are confirmed by numerical results.

  • K. Galvin, A. Linke, L. Rebholz, N. Wilson, Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection, Computer Methods in Applied Mechanics and Engineering, 237--240 (2012) pp. 166--176.
    Abstract
    We consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation-free forcing in the momentum equation. We provide analysis that suggests for such problems, obtaining accurate solutions necessitates either the use of pointwise divergence-free finite elements (such as Scott-Vogelius), or heavy grad-div stabilization of weakly divergence-free elements. The theory is demonstrated in numerical experiments for a benchmark natural convection problem, where large irrotational forcing occurs with high Rayleigh numbers.

  • W. Hackbusch, V. John, A. Khachatryan, C. Suciu, A numerical method for the simulation of an aggregation-driven population balance system, International Journal for Numerical Methods in Fluids, 69 (2012) pp. 1646--1660.
    Abstract
    A population balance system which models the synthesis of urea is studied in this paper. The equations for the flow field, the mass and the energy balances are given in a three-dimensional domain and the equation for the particle size distribution (PSD) in a four-dimensional domain. This problem is convection-dominated and aggregation-driven. Both features require the application of appropriate numerical methods. This paper presents a numerical approach for simulating the population balance system which is based on finite element schemes, a finite difference method and a modern method to evaluate convolution integrals that appear in the aggregation term. Two experiments are considered and the numerical results are compared with experimental data. Unknown parameters in the aggregation kernel have to be calibrated. For appropriately chosen parameters, good agreements are achieved of the experimental data and the numerical results computed with the proposed method. A detailed study of the computational results reveals the influence of different parts of the aggregation kernel.

  • M. Augustin, A. Caiazzo, A. Fiebach, J. Fuhrmann, V. John, A. Linke, R. Umla, An assessment of discretizations for convection-dominated convection-diffusion equations, Computer Methods in Applied Mechanics and Engineering, 200 (2011) pp. 3395--3409.
    Abstract
    The performance of several numerical schemes for discretizing convection-dominated convection-diffusion equations will be investigated with respect to accuracy and efficiency. Accuracy is considered in measures which are of interest in applications. The study includes an exponentially fitted finite volume scheme, the Streamline-Upwind Petrov--Galerkin (SUPG) finite element method, a spurious oscillations at layers diminishing (SOLD) finite element method, a finite element method with continuous interior penalty (CIP) stabilization, a discontinuous Galerkin (DG) finite element method, and a total variation diminishing finite element method (FEMTVD). A detailed assessment of the schemes based on the Hemker example will be presented.

  • M. Case, V. Ervin, A. Linke, L. Rebholz, N. Wilson, Stable computing with an enhanced physics based scheme for the 3D Navier--Stokes equations, International Journal of Numerical Analysis and Modeling. Wuhan University, Wuhan and Institute for Scientific Computing and Information(ISCI), Edmonton, Alberta. English., 8 (2011) pp. 118--136.
    Abstract
    We study extensions of an earlier developed energy and helicity preserving scheme for the 3D Navier-Stokes equations and apply them to a more general class of problems. The scheme is studied together with stabilizations of grad-div type in order to mitigate the effect of the Bernoulli pressure error on the velocity error. We prove stability, convergence, discuss conservation properties, and present numerical experiments that demonstrate the advantages of the scheme.

  • M. Case, V. Ervin, A. Linke, L. Rebholz, A connection between Scott--Vogelius and grad-div stabilized Taylor--Hood FE approximations of the Navier--Stokes equations, SIAM Journal on Numerical Analysis, 49 (2011) pp. 1461--1481.
    Abstract
    This article studies two methods for obtaining excellent mass conservation in finite element computations of the Navier-Stokes equations using continuous velocity fields. Under mild restrictions, the Scott-Vogelius element pair has recently been shown to be inf-sup stable and have optimal approximation properties, while also providing pointwise mass conservation. We present herein the first numerical tests of this element pair for the time dependent Navier-Stokes equations. We also prove that, again under these mild restrictions, the limit of the grad-div stabilized Taylor-Hood solutions to the Navier-Stokes problem converges to the Scott-Vogelius solution as the stabilization parameter tends to infinity. That is, in this setting, we provide theoretical justification that choosing the parameter large does not destroy the solution. A limiting result is also proven for the general case. Numerical tests are provided which verify the theory, and show how both Scott-Vogelius and grad-div stabilized Taylor-Hood (with large stabilization parameter) elements can provide accurate results with excellent mass conservation for Navier-Stokes approximations.

  • A. Caiazzo, D. Evans, J.-L. Falcone, J. Hegewald, E. Lorenz, ET AL., A complex automata approach for in-stent restenosis: Two-dimensional multiscale modeling and simulations, Journal of Computational Science, 2 (2011) pp. 9--17.

  • J. Fuhrmann, H. Langmach, A. Linke, A numerical method for mass conservative coupling between fluid flow and solute transport, Applied Numerical Mathematics. An IMACS Journal, 61 (2011) pp. 530--553.
    Abstract
    We present a new coupled discretization approach for species transport in an incompressible fluid. The Navier-Stokes equations for the flow are discretized by the divergence-free Scott-Vogelius element on barycentrically refined meshes guaranteeing LBB stability. The convection-diffusion equation for species transport is discretized by the Voronoi finite volume method. In accordance to the continuous setting, due to the exact integration of the normal component of the flow through the Voronoi surfaces, the species concentration fulfills discrete global and local maximum principles. Besides of the the numerical scheme itself, we present important aspects of its implementation. Further, for the case of homogeneous Dirichlet boundary conditions, we give a convergence proof for the coupled scheme. We report results of the application of the scheme to the interpretation of limiting current measurements in an electrochemical flow cell with cylindrical shape.

  • A.L. Bowers, B.R. Cousins, A. Linke, L.G. Rebholz, New connections between finite element formulations of the Navier--Stokes equations, Journal of Computational Physics, 229 (2010) pp. 9020--9025.
    Abstract
    We show the velocity solutions to the convective, skew-symmetric, and rotational Galerkin finite element formulations of the Navier-Stokes equations are identical if Scott-Vogelius elements are used, and thus all three formulations will the same pointwise divergence free solution velocity. A connection is then established between the formulations for grad-div stabilized Taylor-Hood elements: under mild restrictions, the formulations' velocity solutions converge to each other (and to the Scott-Vogelius solution) as the stabilization parameter tends to infinity. Thus the benefits of using Scott-Vogelius elements can be obtained with the less expensive Taylor-Hood elements, and moreover the benefits of all the formulations can be retained if the rotational formulation is used. Numerical examples are provided that confirm the theory.

  • A. Glitzky, J.A. Griepentrog, Discrete Sobolev--Poincaré inequalities for Voronoi finite volume approximations, SIAM Journal on Numerical Analysis, 48 (2010) pp. 372--391.
    Abstract
    We prove a discrete Sobolev-Poincare inequality for functions with arbitrary boundary values on Voronoi finite volume meshes. We use Sobolev's integral representation and estimate weakly singular integrals in the context of finite volumes. We establish the result for star shaped polyhedral domains and generalize it to the finite union of overlapping star shaped domains. In the appendix we prove a discrete Poincare inequality for space dimensions greater or equal to two.

  • V. John, A. Kindl, O.C. Suciu, Finite element LES and VMS methods on tetrahedral meshes, Journal of Computational and Applied Mathematics, (2010) pp. 3095--3102.

  • V. John, A. Kindl, A variational multiscale method for turbulent flow simulation with adaptive large scale space, Journal of Computational Physics, 229 (2010) pp. 301--312.

  • V. John, A. Kindl, Numerical studies of finite element variational multiscale methods for turbulent flow simulations, Computer Methods in Applied Mechanics and Engineering, 199 (2010) pp. 841--852.

  • V. John, J. Rang, Adaptive time step control for the incompressible Navier--Stokes equations, Computer Methods in Applied Mechanics and Engineering, 199 (2010) pp. 514--524.

  • V. John, M. Roland, On the impact of the scheme for solving the higher-dimensional equation in coupled population balance systems, International Journal for Numerical Methods in Engineering, 82 (2010) pp. 1450--1474.

  • S. Bartels, M. Jensen, R. Müller, Discontinuous Galerkin finite element convergence for incompressible miscible displacement problems of low regularity, SIAM Journal on Numerical Analysis, 47 (2009) pp. 3720--3743.

  • P. Rudolph, M. Czupalla, N. Dropka, Ch. Frank-Rotsch, F.-M. Kiessling, O. Klein, B. Lux, W. Miller, U. Rehse, O. Root, Crystal growth from melt in combined heater-magnet modules, Journal of the Korean Crystal Growth and Crystal Technology, 19 (2009) pp. 215-222.

  • A. Glitzky, K. Gärtner, Energy estimates for continuous and discretized electro-reaction-diffusion systems, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 70 (2009) pp. 788--805.
    Abstract
    We consider electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistic relations.
    We investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. Here the essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly.
    The same properties are shown for an implicit time discretized version of the problem. Moreover, we provide a space discretized scheme for the electro-reaction-diffusion system which is dissipative (the free energy decays monotonously). On a fixed grid we use for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species.

  • A. Glitzky, Energy estimates for electro-reaction-diffusion systems with partly fast kinetics, Discrete and Continuous Dynamical Systems, 25 (2009) pp. 159--174.
    Abstract
    We start from a basic model for the transport of charged species in heterostructures containing the mechanisms diffusion, drift and reactions in the domain and at its boundary. Considering limit cases of partly fast kinetics we derive reduced models. This reduction can be interpreted as some kind of projection scheme for the weak formulation of the basic electro--reaction--diffusion system. We verify assertions concerning invariants and steady states and prove the monotone and exponential decay of the free energy along solutions to the reduced problem and to its fully implicit discrete-time version by means of the results of the basic problem. Moreover we make a comparison of prolongated quantities with the solutions to the basic model.

  • V. John, T. Mitkova, M. Roland, K. Sundmacher, L. Tobiska, A. Voigt, Simulations of population balance systems with one internal coordinate using finite element methods, Chemical Engineering Sciences, 64 (2009) pp. 733--741.

  • O. Klein, Ch. Lechner, P.-É. Druet, P. Philip, J. Sprekels, Ch. Frank-Rotsch, F.M. Kiessling, W. Miller, U. Rehse, P. Rudolph, Numerical simulations of the influence of a traveling magnetic field, generated by an internal heater-magnet module, on liquid encapsulated Czochralski crystal growth, Magnetohydrodynamics. Consultants Bureau, New York (US). Consultants Bureau, New York. Translation from: Magnitnaya Gidrodinamika., 45 (2009) pp. 557-567.

  • A. Glitzky, Exponential decay of the free energy for discretized electro-reaction-diffusion systems, Nonlinearity, 21 (2008) pp. 1989--2009.
    Abstract
    Our focus are electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistical relations. We introduce a discretization scheme (in space and fully implicit in time) using a fixed grid but for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species. This scheme has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For the discretized electro-reaction-diffusion system we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly.

  • O. Klein, Ch. Lechner, P.-É. Druet, P. Philip, J. Sprekels, Ch. Frank-Rotsch, F.M. Kiessling, W. Miller, U. Rehse, P. Rudolph, Numerical simulation of Czochralski crystal growth under the influence of a traveling magnetic field generated by an internal heater-magnet module (HMM), Journal of Crystal Growth, 310 (2008) pp. 1523-1532.

  • H. Stephan, A. Khrabustovskyi, Positivity and time behavior of a linear reaction-diffusion system, non-local in space and time, Mathematical Methods in the Applied Sciences, 31 (2008) pp. 1809--1834.
    Abstract
    We consider a general linear reaction-diffusion system in three dimensions and time, containing diffusion (local interaction), jumps (nonlocal interaction) and memory effects. We prove a maximum principle, and positivity of the solution, and investigate its asymptotic behavior. Moreover, we give an explicite expression of the limit of the solution for large times. In order to obtain these results we use the following method: We construct a Riemannian manifold with complicated microstructure depending on a small parameter. We study the asymptotic behavior of the solution of a simple diffusion equation on this manifold as the small parameter tends to zero. It turns out that the homogenized system coincides with the original reaction-diffusion system what allows us to investigate its properties.

  • CH. Lechner, O. Klein, P.-É. Druet, Development of a software for the numerical simulation of VCz growth under the influence of a traveling magnetic field, Journal of Crystal Growth, 303 (2007) pp. 161-164.

  • J. Geiser, O. Klein, P. Philip, Influence of anisotropic thermal conductivity in the apparatus insulation for sublimation growth of SiC: Numerical investigation of heat transfer, Crystal Growth & Design, 6 (2006) pp. 2021--2028.

  • J. Geiser, O. Klein, P. Philip, Transient numerical study of temperature gradients during sublimation growth of SiC: Dependence on apparatus design, Journal of Crystal Growth, 297 (2006) pp. 20-32.

  • O. Klein, F. Luterotti, R. Rossi, Existence and asymptotic analysis of a phase field model for supercooling, Quarterly of Applied Mathematics, 64 (2006) pp. 291-319.

  • P. Philip, O. Klein, Transient conductive-radiative heat transfer: Discrete existence and uniqueness for a finite volume scheme, Mathematical Models & Methods in Applied Sciences, 15 (2005) pp. 227--258.

  • A. Glitzky, R. Hünlich, Global existence result for pair diffusion models, SIAM Journal on Mathematical Analysis, 36 (2005) pp. 1200--1225.

  • A. Glitzky, R. Hünlich, Stationary energy models for semiconductor devices with incompletely ionized impurities, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 85 (2005) pp. 778--792.

  • J. Griepentrog, On the unique solvability of a nonlocal phase separation problem for multicomponent systems, Banach Center Publications, 66 (2004) pp. 153-164.

  • H. Gajewski, I.V. Skrypnik, On unique solvability of nonlocal drift-diffusion-type problems, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 56 (2004) pp. 803--830.

  • H. Gajewski, I.V. Skrypnik, To the uniqueness problem for nonlinear parabolic equations, Discrete and Continuous Dynamical Systems, 10 (2004) pp. 315--336.

  • A. Glitzky, R. Hünlich, Stationary solutions of two-dimensional heterogeneous energy models with multiple species, Banach Center Publications, 66 (2004) pp. 135-151.

  • A. Glitzky, Electro-reaction-diffusion systems with nonlocal constraints, Mathematische Nachrichten, 277 (2004) pp. 14--46.

  • O. Klein, Asymptotic behaviour for a phase-field model with hysteresis in one-dimensional thermo-visco-plasticity, Applications of Mathematics, 49 (2004) pp. 309--341.

  • H. Gajewski, K. Zacharias, On a nonlocal phase separation model, Journal of Mathematical Analysis and Applications, 286 (2003) pp. 11--31.

  • G. Albinus, H. Gajewski, R. Hünlich, Thermodynamic design of energy models of semiconductor devices, Nonlinearity, 15 (2002) pp. 367--383.

  • H. Gajewski, On a nonlocal model of non-isothermal phase separation, Advances in Mathematical Sciences and Applications, 12 (2002) pp. 569--586.

  • A. Glitzky, R. Hünlich, Global properties of pair diffusion models, Advances in Mathematical Sciences and Applications, 11 (2001) pp. 293--321.

  • A. Glitzky, R. Hünlich, Electro-reaction-diffusion systems including cluster reactions of higher order, Mathematische Nachrichten, 216 (2000) pp. 95--118.

  • J.A. Griepentrog, An application of the Implicit Function Theorem to an energy model of the semiconductor theory, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 79 (1999) pp. 43--51.
    Abstract
    In this paper we deal with a mathematical model for the description of heat conduction and carrier transport in semiconductor heterostructures. We solve a coupled system of nonlinear elliptic differential equations consisting of the heat equation with Joule heating as a source, the Poisson equation for the electric field an drift-diffusion equations with temperature dependent coefficients describing the charge and current conservation, subject to general thermal and electrical boundary conditions. We prove the existence and uniqueness of Holder continuous weak solutions near thermodynamic equilibria points using the Implicit Function Theorem. To show the differentiability of maps corresponding to the weak formulation of the problem we use regularity results from the theory of nonsmooth linear elliptic boundary value problems in Sobolev-Campanato spaces.

  Contributions to Collected Editions

  • CH. Brennecke, A. Linke, Ch. Merdon, J. Schöberl, Optimal and pressure-independent $L^2$ velocity error estimates for a modified Crouzeix--Raviart element with BDM reconstructions, in: Finite Volumes for Complex Applications VII -- Methods and Theoretical Aspects -- FVCA 7, Berlin, June 2014, J. Fuhrmann, M. Ohlberger, Ch. Rohde, eds., 77 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2014, pp. 159--167.
    Abstract
    Nearly all inf-sup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a priori estimates for the velocity error become pressure-dependent, while divergence-free mixed finite elements deliver pressure-independent estimates. A recently introduced new variational crime using lowest-order Raviart-Thomas velocity reconstructions delivers a much more robust modified Crouzeix-Raviart element, obeying an optimal pressure-independent discrete H1 velocity estimate. Refining this approach, a more sophisticated variational crime employing the lowest-order BDM element is proposed, which also allows proving an optimal pressureindependent L2 velocity error. Numerical examples confirm the analysis and demonstrate the improved robustness in the Navier-Stokes case.

  • J. Fuhrmann, A. Linke, Ch. Merdon, Coupling of fluid flow and solute transport using a divergence-free reconstruction of the Crouzeix--Raviart element, in: Finite Volumes for Complex Applications VII -- Elliptic, Parabolic and Hyperbolic Problems -- FVCA 7, Berlin, June 2014, J. Fuhrmann, M. Ohlberger, Ch. Rohde, eds., 78 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2014, pp. 587--595.

  • A. Glitzky, J.A. Griepentrog, On discrete Sobolev--Poincaré inequalities for Voronoi finite volume approximations, in: Finite volumes for complex applications VI: Problems and perspectives, J. Fořt, J. Fürst, J. Halama, R. Herbin, F. Hubert, eds., Springer Proceedings in Mathematics 4, Springer, Heidelberg, 2011, pp. 533--541.

  • S. Bartels, R. Müller, Die kalte Zunge, in: Besser als Mathe --- Moderne angewandte Mathematik aus dem MATHEON zum Mitmachen, K. Biermann, M. Grötschel, B. Lutz-Westphal, eds., Reihe: Populär, Vieweg+Teubner, Wiesbaden, 2010, pp. 227--235.

  • M. Jensen, R. Müller, Stable Crank--Nicolson discretisation for incompressible miscible displacement problems of low regularity, in: Numerical Mathematics and Advanced Applications 2009, Part 2, G. Kreiss, P. Lötstedt, A. Målqvist, M. Neytcheva, eds., Springer, Heidelberg et al., pp. 469--477.
    Abstract
    In this article we study the numerical approximation of incompressible miscible displacement problems with a linearised Crank-Nicolson time discretisation, combined with a mixed finite element and discontinuous Galerkin method. At the heart of the analysis is the proof of convergence under low regularity requirements. Numerical experiments demonstrate that the proposed method exhibits second-order convergence for smooth and robustness for rough problems.

  • V. John, E. Schmeyer, On finite element methods for 3D time-dependent convection-diffusion-reaction equations with small diffusion, in: BAIL 2008 --- Boundary and Interior Layers, A. Hegarty, N. Kopteva, E. O'Riordan, M. Stynes, eds., 69 of Lecture Notes in Computational Science and Engineering, Springer, Berlin/Heidelberg, 2009, pp. 173--181.

  • H. Gajewski, H.-Chr. Kaiser, H. Langmach, R. Nürnberg, R.H. Richter, Mathematical modelling and numerical simulation of semiconductor detectors, in: Mathematics --- Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jäger, H.-J. Krebs, eds., Springer, Berlin [u.a.], 2003, pp. 355--364.

  • R. Hünlich, G. Albinus, H. Gajewski, A. Glitzky, W. Röpke, J. Knopke, Modelling and simulation of power devices for high-voltage integrated circuits, in: Mathematics --- Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jäger, H.-J. Krebs, eds., Springer, Berlin [u.a.], 2003, pp. 401--412.

  • U. Bandelow, H. Gajewski, H.-Chr. Kaiser, Modeling combined effects of carrier injection, photon dynamics and heating in Strained Multi-Quantum-Well Laser, in: Physics and Simulation of Optoelectronic Devices VIII, R.H. Binder, P. Blood, M. Osinski, eds., 3944 of Proceedings of SPIE, SPIE, Bellingham, WA, 2000, pp. 301--310.

  • G. Schwarz, E. Schöll, R. Nürnberg, H. Gajewski, Simulation of current filamentation in an extended drift-diffusion model, in: EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, B. Fiedler, K. Gröger, J. Sprekels, eds., 2, World Scientific, Singapore [u. a.], 2000, pp. 1334--1336.

  • H. Gajewski, K. Zacharias, On a reaction-diffusion system modelling chemotaxis, in: EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, B. Fiedler, K. Gröger, J. Sprekels, eds., 2, World Scientific, Singapore [u. a.], 2000, pp. 1098--1103.

  • R. Hünlich, A. Glitzky, On energy estimates for electro-diffusion equations arising in semiconductor technology, in: Partial differential equations. Theory and numerical solution, W. Jäger, J. Nečas, O. John, K. Najzar, eds., 406 of Chapman & Hall Research Notes in Mathematics, Chapman & Hall, Boca Raton, FL, 2000, pp. 158--174.

  • H.-Chr. Kaiser, J. Rehberg, About some mathematical questions concerning the embedding of Schrödinger-Poisson systems into the drift-diffusion model of semiconductor devices, in: EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, B. Fiedler, K. Gröger, J. Sprekels, eds., 2, World Scientific, Singapore [u. a.], 2000, pp. 1328--1333.

  Preprints, Reports, Technical Reports

  • 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.

  • 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, Preprint no. 2316, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2316 .
    PDF (1144 kByte)

  • M. Hintermüller, M. Hinze, Ch. Kahle, T. Keil, A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn--Hilliard--Navier--Stokes system, Preprint no. 2311, WIAS, Berlin, 2016.
    Abstract, PDF (640 kByte)
    This paper is concerned with the development and implementation of an adaptive solution algorithm for the optimal control of a time-discrete Cahn--Hilliard--Navier--Stokes system with variable densities. The free energy density associated to the Cahn--Hilliard system incorporates the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier--Stokes equation. A dual-weighed residual approach for goal-oriented adaptive finite elements is presented which is based on the concept of C-stationarity. The overall error representation depends on primal residual weighted by approximate dual quantities and vice versa as well as various complementary mismatch errors. Details on the numerical realization of the adaptive concept and a report on numerical tests are given.

  • V. John, A. Linke, Ch. Merdon, M. Neilan, L.G. Rebholz, On the divergence constraint in mixed finite element methods for incompressible flows, Preprint no. 2177, WIAS, Berlin, 2015.
    Abstract, PDF (4612 kByte)
    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.

  • 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, Preprint no. 2166, WIAS, Berlin, 2015, DOI 10.20347/WIAS.PREPRINT.2166 .
    Abstract, PDF (649 kByte)
    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.

  Talks, Poster

  • D. Peschka, Modelling and simulation of suspension flow, Graduate Seminar PDE in the Sciences, Universität Bonn, Institut für Angewandte Mathematik, Bonn, January 20, 2017.

  • V. John, Vortragstitel + Datum fehlt, 12th International Workshop on Variational Multiscale and Stabilization Methods (VMS-2017), April 26 - 28, 2017, Edificio Celestino Mutis, Campus Reina Mercedes, Sevilla, Spain.

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

  • D. Peschka, A free boundary problem for the flow of viscous liquid bilayers, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22 - 26, 2016, WIAS Berlin, Berlin, February 26, 2016.

  • D. Peschka, Multi-phase flows with contact lines: Solid vs liquid substrates, Industrial and Applied Mathematics Seminar, University of Oxford, Mathematical Institute, UK, October 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.

  • 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, 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.

  • CH. Merdon, Inverse modeling of thin layer flow cells for detection of solubility transport and reaction coefficients from experimental data, 17th Topical Meeting of the International Society of Electrochemistry Multiscale Analysis of Electrochemical Systems, May 31 - June 3, 2015, Saint Malo Congress Center, France, June 1, 2015.

  • A. Caiazzo, Assessment of Kalman filtering for parameter identification in one-dimensional blood flow model, 4th International Conference on Computational & Mathematical Biomedical Engineering, June 29 - July 1, 2015, Ecole Normale Supérieure de Cachan, Cachan, France, June 29, 2015.

  • V. John, S. Rockel, S. Sobolev, Viscoelastic mantle flow --- Numerical modeling of geological phenomena, 4th Annual GEOSIM Workshop, Potsdam, November 17 - 18, 2015.

  • U. Wilbrandt, Classical iterative subdomain methods for the Stokes--Darcy problem, Rheinisch-Westfälische Technische Hochschule Aachen, Institut für Geometrie und Praktische Mathematik, Aachen, June 11, 2015.

  • A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows, International Conference of Boundary and Interior Layers (BAIL 2014), September 15 - 19, 2014, Carles University Prague, Czech Republic, September 17, 2014.

  • A. Linke, Ch. Merdon, Optimal and pressure-independent $L^2$ velocity error estimates for a modified Crouzeix--Raviart element with BDM reconstructions, The International Symposium of Finite Volumes for Complex Applications VII (FVCA 7), Berlin-Brandenburgische Akademie der Wissenschaften, June 15 - 20, 2014.

  • A. Linke, Ein neues Konstruktionsprinzip zur divergenzfreien Diskretisierung der inkompressiblen Navier-Stokes-Gleichungen, Ruhr-Universität Bochum, Fakultät für Mathematik, July 17, 2014.

  • A. Linke, On the role of the Helmholtz decomposition in incompressible flows and a new variational crime, NonLinear PDE and Applications: Theoretical and Numerical Study, May 5 - 7, 2014, Abdelmalek Essadi University, Tanger, Morocco, May 6, 2014.

  • A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Technische Universität Wien, Institut für Analysis und Scientific Computing, Austria, April 2, 2014.

  • A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Technische Universität Hamburg-Harburg, Institut für Mathematik, January 7, 2014.

  • A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Friedrich-Alexander-Universität Erlangen-Nürnberg, Fachbereich Mathematik, November 20, 2014.

  • A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Georg-August-Universität Göttingen, Institut für Numerische und Angewandte Mathematik, December 9, 2014.

  • V. John, On isogeometric analysis for convection-diffusion and Navier--Stokes equations, International Conference of Boundary and Interior Layers (BAIL 2014), September 15 - 19, 2014, Carles University Prague, Czech Republic, September 15, 2014.

  • J. Fuhrmann, A. Linke, Ch. Merdon, M. Khodayari, H. Baltruschat, Detection of solubility, transport and reaction coefficients from experimental data by inverse modeling of thin layer flow cells, 65th Annual Meeting of the International Society of Electrochemistry, Lausanne, Switzerland, August 31 - September 5, 2014.

  • J. Fuhrmann, A. Linke, Ch. Merdon, Coupling of fluid flow and solute transport using a divergence-free reconstruction of the Crouzeix--Raviart element, The International Symposium of Finite Volumes for Complex Applications VII (FVCA 7), Berlin-Brandenburgische Akademie der Wissenschaften, June 15 - 20, 2014.

  • J. Fuhrmann, Activity based finite volume methods for generalised Nernst--Planck--Poisson systems, The International Symposium of Finite Volumes for Complex Applications VII (FVCA 7), Berlin-Brandenburgische Akademie der Wissenschaften, June 15 - 20, 2014.

  • J. Fuhrmann, Voronoi finite volume methods, XVI. Mathematica-Tag, Berlin, December 9, 2014.

  • V. John, S. Rockel, S. Sobolev, Viscoelastic mantle flow, 3rd Annual GEOSIM Workshop, Potsdam, November 13 - 14, 2014.

  • V. John, Finite element methods for incompressible flow problems, Indian National Program on Differential Equations -- Theory, Computation, and Applications (NPDE-TCA) Advanced Workshop on Finite Element Methods for Navier--Stokes Equations, September 8 - 12, 2014, Indian Institute of Science, Numerical Mathematics & Scientific Computing, Bangalore.

  • V. John, Numerical methods for convection-dominated equations, 2nd Workshop ``Populationsbilanzen'' of the DFG Priority Program SPP 1679 ``Dyn-Sim-FP -- Dynamic Simulation of Interconnected Solids Processes'', May 6 - 7, 2014, Technische Universität Hamburg-Harburg, May 6, 2014.

  • V. John, On the simulation of mantle convection, Symposium on Simulation and Optimization of Extreme Fluids, November 10 - 12, 2014, Internationales Wissenschaftsforum Heidelberg, November 12, 2014.

  • V. John, Turbulent flows and their numerical simulation, Humboldt Kolleg on Interdisciplinary Science: Catalyst for Sustainable Progress, September 4 - 6, 2014, Indian Institute of Science, Numerical Mathematics & Scientific Computing, Bangalore, September 5, 2014.

  • U. Wilbrandt, Iterative subdomain methods for Stokes--Darcy problems, Norddeutsches Kolloquium über Angewandte Analysis und Numerische Mathematik (NoKo), May 9 - 10, 2014, Christian-Albrechts-Universität zu Kiel, May 9, 2014.

  • V. John, On recent analytical and numerical investigations of the SUPG method, Advances in Computational Mechanics (ACM 2013) -- A Conference Celebrating the 70th Birthday of Thomas J.R. Hughes, February 24 - 27, 2013, San Diego, USA, February 27, 2013.

  • A. Linke, Coupled flows and poor mass conservation, Workshop ``Complex grids and fluid flows, conclusion of VFSitCom, National Research Project'', April 2 - 4, 2012, Rhône-Alpes, Lyon, France, April 3, 2012.

  • V. John, On reduced order modeling methods for incompressible flows based proper orthogonal decomposition, 6th Variational Multiscale Methods Workshop (VMS 2012), June 27 - 29, 2012, Christian-Albrechts-Universität zu Kiel, June 28, 2012.

  • V. John, On the analysis and numerical analysis of some turbulence models, Workshop ``Connections Between Regularized and Large-Eddy Simulation Methods for Turbulence'', May 14 - 17, 2012, Banff International Research Station for Mathematical Innovation and Discovery, Canada, May 15, 2012.

  • A. Glitzky, J.A. Griepentrog, Discrete Sobolev--Poincaré inequalities for Voronoi finite volume approximations, Finite Volumes for Complex Applications VI (FVCA 6), Prague, Czech Republic, June 6 - 10, 2011.

  • O.C. Suciu, A numerical method for the simulation of population balance systems with one internal coordinate, 8th International Conference on Large-Scale Scientific Computations, June 6 - 10, 2011, Bulgarian Academy of Sciences, Sozopol, June 7, 2011.

  • A. Caiazzo, J. Fuhrmann, V. John, Finite element-finite volume coupling for simulations of thin porous layer fuel cells, Workshop on Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration, John Radon Institute for Computational and Applied Mathematics, Linz, Austria, October 3 - 7, 2011.

  • A. Caiazzo, Darcy-Stokes coupling for simulation of solute transport in thin-layer channel fuel cells, Workshop on Fluid Dynamics and Porous Media, September 12 - 14, 2011, University of Coimbra, Department of Mathematics, Portugal, September 13, 2011.

  • A. Caiazzo, Model reduction approaches for simulation of cardiovascular stents and pulmonary valve, Laboratory of Modeling and Scientific Computing, Department of Mathematics, Milan, Italy, October 27, 2011.

  • A. Caiazzo, Physical- and mathematical-based reduced order modeling in computational hemodynamics, Wroclaw University of Technology, Institute of Mathematics and Computer Science, Poland, November 16, 2011.

  • E. Schmeyer, V. John, R. Bordás, D. Thévenin, Referenzexperimente im Windkanal, numerische Simulation und Validierung, Annual Meeting 2010 of DFG Priority Program 1276 MetStröm, Zuse Institute Berlin, October 28 - 29, 2010.

  • O.C. Suciu, On numerical methods for simulation of 3D/4D of population balance system, SimParTurS Meeting, Max-Planck-Institut Leipzig, February 17, 2010.

  • V. John, A posteriori optimization of parameters in stabilized methods for convection-diffusion problems, BIRS Workshop ``Nonstandard Discretizations for Fluid Flows'', November 21 - 26, 2010, Banff International Research Station, Canada.

  • V. John, Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations, 6th Variational Multiscale Methods Workshop (VMS 2010), May 27 - 28, 2010, Université de Pau et des Pays de l'Adour, France, May 27, 2010.

  • V. John, Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations, International Conference of Boundary and Interior Layers (BAIL 2010), July 5 - 9, 2010, University of Zaragoza, Spain, July 8, 2010.

  • V. John, On the numerical simulation of population balance systems, Technische Universität Dresden, Fakultät Mathematik und Naturwissenschaften, April 20, 2010.

  • V. John, Variational multiscale methods for the simulation of turbulent flows, Volkswagen Wolfsburg, August 27, 2010.

  • M. Ehrhardt, J. Fuhrmann, A. Linke, Finite volume methods for the simulation of flow cell experiments, Workshop ``New Trends in Model Coupling --- Theory, Numerics & Applications'' (NTMC'09), Paris, France, September 2 - 4, 2009.

  • M. Ehrhardt, A high order finite element method for waves in periodic structures, 9th International Conference on Spectral and High Order Methods (ICOSAHOM09), Minisymposium ``High-order Methods for Linear and Nonlinear Wave Equations'', June 22 - 26, 2009, Norwegian University of Science and Technology, Trondheim, June 24, 2009.

  • V. John, A variational multiscale method for turbulent flow simulation with adaptive large scale space, Workshop on Computational Multiscale Modeling, November 26 - 27, 2009, University of Twente, Enschede, Netherlands, November 27, 2009.

  • V. John, On the numerical simulation of population balance systems, Karlsruher Institut für Technologie, Fakultät für Mathematik, December 9, 2009.

  • H.-Chr. Kaiser, Transient Kohn--Sham theory, Jubiläumssymposium ``Licht -- Materialien -- Modelle'' (100 Jahre Innovation aus Adlershof), Berlin-Adlershof, September 7 - 8, 2009.

  • K. Hoke, Numerical treatment of the Kohn--Sham system for semiconductor devices, Workshop on Mathematical Aspects of Transport in Mesoscopic Systems, Dublin, Ireland, December 4 - 7, 2008.

  • E. Bänsch, H. Berninger, U. Böhm, A. Bronstert, M. Ehrhardt, R. Forster, J. Fuhrmann, R. Klein, R. Kornhuber, A. Linke, A. Owinoh, J. Volkholz, Pakt für Forschung und Innovation: Das Forschungsnetzwerk ``Gekoppelte Strömungsprozesse in Energie- und Umweltforschung'', Show of the Leibniz Association ``Exzellenz durch Vernetzung. Kooperationsprojekte der deutschen Wissenschaftsorganisationen mit Hochschulen im Pakt für Forschung und Innovation'', Berlin, November 12, 2008.

  • M. Ehrhardt, O. Gloger, Th. Dietrich, O. Hellwich, K. Graf, E. Nagel, Level Set Methoden zur Segmentierung von kardiologischen MR-Bildern, 22. Treffpunkt Medizintechnik: Fortschritte in der medizinischen Bildgebung, Charité, Campus Virchow Klinikum Berlin, May 22, 2008.

  • A. Glitzky, Analysis of spin-polarized drift-diffusion models, 79th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2008), session ``Applied Analysis'', March 31 - April 4, 2008, University of Bremen, April 1, 2008.

  • A. Glitzky, Energy estimates for continuous and discretized reaction-diffusion systems in heterostructures, Annual Meeting of the Deutsche Mathematiker-Vereinigung 2008, minisymposium ``Analysis of Reaction-Diffusion Systems with Internal Interfaces'', September 15 - 19, 2008, Friedrich-Alexander-Universität Erlangen-Nürnberg, September 15, 2008.

  • A. Glitzky, Energy estimates for space and time discretized electro-reaction-diffusion systems, Conference on Differential Equations and Applications to Mathematical Biology, June 23 - 27, 2008, Université Le Havre, France, June 26, 2008.

  • H.-Chr. Kaiser, A drift-diffusion model for semiconductors with internal interfaces, Annual Meeting of the Deutsche Mathematiker-Vereinigung 2008, Minisymposium ``Analysis of Reaction-Diffusion Systems with Internal Interfaces'', September 15 - 19, 2008, Friedrich-Alexander-Universität Erlangen-Nürnberg, September 15, 2008.

  • H.-Chr. Kaiser, A thermodynamic approach to transient Kohn--Sham theory, 100th Statistical Mechanics Conference, December 13 - 18, 2008, Rutgers, The State University of New Jersey, New Brunswick, USA, December 16, 2008.

  • H.-Chr. Kaiser, On drift-diffusion Kohn--Sham theory, 79th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2008), session ``Applied Analysis'', March 31 - April 4, 2008, University of Bremen, April 1, 2008.

  • A. Glitzky, Energy estimates for reaction-diffusion processes of charged species, 6th International Congress on Industrial and Applied Mathematics (ICIAM), July 16 - 20, 2007, ETH Zürich, Switzerland, July 16, 2007.

  • H.-Chr. Kaiser, A drift-diffusion model of transient Kohn--Sham theory, First Joint International Meeting between the American Mathematical Society and the Polish Mathematical Society, Special Session ``Mathematics of Large Quantum Systems'', July 31 - August 3, 2007, University of Warsaw, Poland, August 3, 2007.

  • K. Gärtner, A. Glitzky, Th. Koprucki, Analysis and simulation of spin-polarized drift-diffusion models, Evaluation Colloquium of the DFG Priority Program SPP 1285 ``Semiconductor Spintronics'', Bad Honnef, December 14 - 15, 2006.

  • A. Glitzky, Energy estimates for electro-reaction-diffusion systems with partly fast kinetics, 6th AIMS International Conference on Dynamical Systems, Differential Equations & Applications, June 25 - 28, 2006, Université de Poitiers, France, June 27, 2006.

  • A. Glitzky, An application of the Implicit Function Theorem to stationary energy models for semiconductor devices, International Workshop ``Regularity for nonlinear and linear PDEs in nonsmooth domains'', September 4 - 7, 2005, Universität Stuttgart, Hirschegg, Austria, September 5, 2005.

  • A. Glitzky, Stationary energy models for semicoductor devices with incompletely ionized impurities, 2nd Joint Meeting of AMS, DMV, ÖMG, June 16 - 19, 2005, Johannes Gutenberg Universität, Mainz, June 19, 2005.

  • H. Gajewski, R. Hünlich, H.-Chr. Kaiser, M. Baro, Quantum mechanical and macroscopic models for optoelectronic devices, DFG Research Center sc Matheon, Technische Universität Berlin, July 19, 2004.

  • H. Gajewski, Zur Numerik des Ladungsträgertransports in Halbleiterbauelementen, Technische Universität München, Institut fär Technische Elektrophysik, February 5, 2004.

  • M. Baro, H. Gajewski, R. Hünlich, H.-Chr. Kaiser, Optoelektronische Bauelemente: mikroskopische & makroskopische Modelle, MathInside --- Überall ist Mathematik, event of the DFG Research Center ``Mathematics for Key Technologies'' on the occasion of the Open Day of Urania, Berlin, September 13, 2003 - December 3, 2004.

  • A. Glitzky, R. Hünlich, Stationary solutions of two-dimensional heterogeneous energy models with multiple species, Nonlocal Elliptic and Parabolic Problems, September 9 - 11, 2003, Bedlewo, Poland, September 10, 2003.

  • H.-Chr. Kaiser, Classical solutions of van Roosbroeck's equations with discontinuous coefficients and mixed boundary conditions on two-dimensional space domains, 19th GAMM Seminar Leipzig on High-dimensional problems --- Numerical treatment and applications, January 23 - 25, 2003, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, January 25, 2003.