## Doktorandenseminar des WIAS

# Numerische Mathematik und Wissenschaftliches Rechnen

Forschungsgruppe

2016
(2015,
2014,
2013,
2012,
2011,
2010,
2009,
2008,
2007,
2006,
2005,
2004,
2003,
2002)
## Seminar Numerische Mathematik / Numerical mathematics seminars

aktuelles Programm / current program

**
Donnerstag, 06. 04. 2017, 14:00 Uhr (ESH)
**

Prof. D. Silvester(University of Manchester, GB)

Accurate time-integration strategies for modelling incompressible flow bifurcationsEigenvalue analysis is a well-established tool for stability analysis of dynamical systems. However, there are situations where eigenvalues miss some important features of physical models. For example, in models of incompressible fluid dynamics, there are examples where linear stability analysis predicts stability but transient simulations exhibit significant growth of infinitesimal perturbations. In this study, we show that an approach similar to pseudo-spectral analysis can be performed inexpensively using stochastic collocation methods and the results can be used to provide quantitive information about the nature and probability of instability.

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Donnerstag, 30. 03. 2017, 14:00 Uhr (ESH)
**

Prof. J.H.M. ten Thije Boonkkamp(Eindhoven University of Technology, Netherlands)

Complete flux schemes fOR conservation laws of advection-diffusion-reaction typComplete flux schemes are recently developed numerical flux approximation schemes for conservation laws of advection-diffusion-reaction type; see e.g. [1, 2]. The basic complete flux scheme is derived from a local one-dimensional boundary value problem for the entire equation, including the source term. Consequently, the integral representation of the flux contains a homogeneous and an inhomogeneous part, corresponding to the advection-diffusion operator and the source term, respectively. Suitable quadrature rules give the numerical flux. For time-dependent problems, the time derivative is considered a source term and is included in the inhomogeneous flux, resulting in an implicit semi-discretisation. The implicit system proves to have much smaller dissipation and dispersion errors than the standard semidiscrete system, especially for dominant advection. Just as for scalar equations, for coupled systems of conservation laws, the complete flux approximation is derived from a local system boundary value problem, this way incorporating the coupling between the constituent equations in the discretization. Also in the system case, the numerical flux (vector) is the superpostion of a homogeneous and an inhomogeneous component, corresponding to the advection-diffusion operator and the source term vector, respectively. The scheme is applied to multispecies diffusion and satisfies the mass constraint exactly.

References:

[1] J.H.M. ten Thije Boonkkamp and M.J.H. Anthonissen, ``The finite volume-complete flux scheme for advection-diffusion-reaction equations'', J. Sci. Comput., 46, 47--70, (2011).

[2] J.H.M. ten Thije Boonkkamp, J. van Dijk, L. Liu and K.S.C. Peerenboom, ``Extension of the complete flux scheme to systems of comservation laws'', J. Sci. Comput., 53, 552?568, (2012).

**
Dienstag, 28. 03. 2017, 13:30 Uhr (ESH)
**

G. Pitton(SISSA, Italien)

Accelerating augmented and deflated Krylov space methods for convection-diffusion problemsIn this talk I will recall some basic notions of augmented and deflated Krylov space methods for the iterative solution of linear systems. Then I will discuss a few strategies to apply these techniques to the solution of linear systems coming from nonlinear convection-diffusion equations . In particular, I will argue that in some cases it may be convenient to exploit some alternative recycling strategies based on the SVD selection of previous solutions. Some numerical tests in scalar nonlinear convection-diffusion problems discretized with Finite Elements and Spectral Elements will be discussed.

und 14:30 Uhr

Prof. L. Heltai(SISSA, Italien)

Immersed Finite Element Methods for interface and fluid structure interaction problems: An overview and some recent resultsImmersed Finite Element Methods (IFEM) are an evolution of the original Immersed Boundary Element Method (IBM) developed by Peskin in the early seventies for the simulation of complex Fluid Structure Interaction (FSI) problems. In the IBM, the coupled FSI problem is discretised using a single (uniformly discretised) background fluid solver, where the presence of the solid is taken into account by adding appropriate forcing terms in the fluid equation. Approximated Dirac delta distributions are used to interpolate between the Lagrangian and the Eulerian framework in the original formulation by Peskin, while a variational formulation was introduced by Boffi and Gastaldi (2003), and later generalised by Heltai and Costanzo (2012). By carefully exploiting the variational definition of the Dirac distribution, it is possible to reformulate the discrete Finite Element problem using non-matching discretisations without recurring to Dirac delta approximation.

One of the key issues that kept people from adopting IBM or IFEM techniques is related to the loss in accuracy attributed to the non-matching nature of the discretisation between the fluid and the solid domains, leading to solvers that converge only sub-optimally.

In this talk I will present a brief overview of Immersed Finite Element Methods, and will present some recent results that exploit techniques introduced by D?Angelo and Quarteroni (2012), to show that, for the variational finite element formulation, the loss in accuracy is only restricted to a thin layer of elements around the solid-fluid interface, and that optimal error estimates in all norms are recovered if one uses appropriate weighted norms when measuring the error.

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Donnerstag, 16. 03. 2017, 14:00 Uhr (ESH)
**

M. Cicuttin(CERMICS - ENPC, Frankreich)

Implementation of Discontinuous Skeletal methods on arbitrary-dimensional, polytopal meshes using generic programmingDiscontinuous Skeletal methods are devised at the mathematical level in a dimension-independent and cell-shape-independent fashion. Their implementation, at least in principle, should conserve this feature: a single piece of code should be able to work in any space dimension and to deal with any cell shape. It is not common, however, to see software packages taking this approach. In the vast majority of the cases, the codes are capable to run only on few very specific kinds of mesh, or only in 1D or 2D or 3D. On the one hand, this can happen simply because a fully general tool is not always needed. On the other hand, the programming languages commonly used by the scientific computing community (in particular Fortran and Matlab) are not easily amenable to an implementation which is generic and efficient at the same time. The usual (and natural) approach, in conventional languages, is to have different versions of the code, for example one specialized for 1D, one for 2D and one for 3D applications, making the overall maintenance of the codes rather cumbersome. The same considerations generally apply to the handling of mesh cells with various shapes, i.e., codes written in conventional languages generally support only a limited (and set in advance) number of cell shapes.

Generic programming offers a valuable tool to address the above issues: by writing the code generically, it is possible to avoid making any assumption neither on the dimension (1D, 2D, 3D) of the problem, nor on the kind of mesh. In some sense, writing generic code resembles writing pseudocode: the compiler will take care of giving the correct meaning to each basic operation. As a result, with generic programming there will be still differents versions of the code, but they will be generated by the compiler, and not by the programmer. As these considerations suggest, generic programming is a static technique: if correctly realized, the abstractions do not penalize the performance at runtime, because they will leave no trace in the generated code.

In this talk we will discuss how the Hybrid High Order method is implemented atop of DiSk++, a newly developed library for the generic implementation of Discontinuous Skeletal methods.

**
Donnerstag, 02. 03. 2017, 14:00 Uhr (ESH)
**

Dr. L. O. Müller(Norwegian University of Science and Technology)

A local time stepping solver for one-dimensional blood flowWe present a finite volume solver for one-dimensional blood flow simulations in networks of elastic and viscoelastic vessels, featuring high-order space-time accuracy and local time stepping (LTS). The solver is built on:

(i) a high-order finite-volume type numerical scheme,

(ii) a high-order treatment of the numerical solution at internal vertexes of the network (junctions);

(iii) an accurate LTS strategy.

Several applications of the method will be presented. First, we apply the LTS scheme to the Anatomically Detailed Arterial Network model (ADAN), comprising 2142 arterial vessels. Second, we show results of a computational study where the ADAN model is coupled to automatically generated microvascular networks in order to elucidate aspects on the pathopysiology of small vessel disease for cerebral arteries.

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Donnerstag, 09. 02. 2017, 14:00 Uhr (ESH)
**

Prof. M. Eikerling(Simon Fraser University, Kanada)

Theory and modeling of materials for electrochemical energy systemsThe ever-escalating need for highly efficient and environmentally benign energy technology drives research on materials for fuel cells, supercapacitors, batteries, electrolysers, and other electrochemical systems. In this realm, physical-mathematical theory, modeling, and simulation provide increasingly powerful tools to unravel how multifunctional electrochemical materials come to life during self-organization, how they live and operate, e.g., by breathing in oxygen and breathing out water vapor or by shuttling ions across electrolytes, and how they age and fail because of normal wear-and-tear or improper use. The introductory part of the presentation will give a sweeping perspective of research forays in theory and modeling. Thereafter, two topics will be presented and discussed in detail: modeling approaches to study the interplay of interfacial charging phenomena, fluid flow, ion transport, and electrochemical reaction in nanoporous electrodes; and statistical physics-based modeling of degradation, aging, and failure in particle-based electrodes and fibrillar membranes.

und 15:15 Uhr

Prof. G. Lube und Ph. Schröder(Georg-August-Universität Göttingen)

Pressure-robust error estimates of exactly divergence-free FEM for time-dependent incompressible flowsThe talk focusses on the analysis of a conforming finite element method for the time-dependent incompressible Navier--Stokes model. For divergence-free approximations, in a time-continuous formulation, we prove error estimates for the velocity that hold independently of both pressure and Reynolds number. A key aspect is the use of the discrete Stokes projection for the error splitting. Optionally, edge stabilization can be included in the case of dominant convection. Emphasizing the importance of conservation properties, the theoretical results are complemented with numerical simulations of vortex dynamics and laminar boundary layer flow.

**
Donnerstag, 02. 02. 2017, 14:00 Uhr (ESH) **
[Dr. P. A. Zegeling
(Utrecht University, Niederlande) entfällt]

Dr. J. Mura(Pontificia Universidad Catolica de Chile, Chile)

An automatic method to estimate 3D Pulse Wave Velocity from 4D-flow MRI dataIn this talk will be presented a novel method to automatically construct a continuous Pulse Wave Velocity map using 4D-Flow MRI data, based on the observation that in curved vessels, the propagation of velocity wavefronts do not necessarily follow perpendicular planes to some symmetry axis, but intricate shapes that strongly depends on the arterial morphology. This observation is considered to estimate continuous 1D PWV from velocities acquired with 4D-flow MRI data and projected back to 3D for better visualizations. This technique was assessed with in-silico and in-vitro phantoms, volunteers, and Fontan patients, showing a good agreement with expected values.

**
Donnerstag, 26. 01. 2017, 14:00 Uhr (ESH)
**

Dr. F. Dassi(Politecnico di Milano, Italien)

The Virtual Element Method in three dimensionsThe Virtual Element Method (VEM) is sharing a good degree of success in the recent years and its robustness and flexibility are already numerically provided for the two dimensional case. In three dimensions the results are currently restricted to the lowest order although the theory of this higher dimensional case is already developed in literature, see for instance [1, 2]. This talk is the first step towards a deeper numerical analysis of VEM in 3D [3]. After a first review of the scheme, we show a series of numerical results that validate this new method in three dimensions. To achieve this goal, we consider standard reaction-diffusion equations solved on several polyhedral meshes with different VEM order.

References: [1] L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo. Basic principles of virtual element methods. Mathematical Models and Methods in Applied Sciences, 23(01):199--214, 2013.

[2] L. Beirao da Veiga, F. Brezzi, L. D. Marini, and A. Russo. The hitchhiker's guide to the virtual element method. Mathematical Models and Methods in Applied Sciences, 24(08):1541--1573, 2014.

[3] L. Beirao da Veiga, F. Dassi, and A. Russo. Numerical investigations for three dimensional virtual elements of arbitrary order. to appear.

**
Donnerstag, 12. 01. 2017, 14:00 Uhr (ESH)
**

Dr. A. Gassmann(Leibniz-Instituts für Atmosphärenphysik, Kühlungsborn)

Fluid dynamics on icosahedral staggered gridsIn the last decade, some weather and climate modeling centers started to develop atmospheric models that reside on tessellations of the icosahedron. The resulting hexagonal or triangular meshes are essential for two reasons: first, the numerical difficulties arising from CFL restrictions near the poles are avoided, and second, the lower boundary (land structure and ocean) is represented by approximately equally sized areas.

C-staggered meshes are common in atmospheric modeling because they allow for good wave propagation properties. However, this staggering, which positions the mass points at the grid box centers and the velocity points at the grid box edges, generates other problems for tessellations of the icosahedron, which will be discussed in the talk.

The first problem is the overspecification of velocity components in comparison to the mass components. This problem can only be solved for hexagonal C-grid meshes by defining discretization procedures which guarantee the linear dependency of velocity components during time stepping. This applies to the Coriolis term and the momentum diffusion term. Such methods are understood by the community since the work of Thuburn (2008).

The second problem is the grid deformation in the vicinity of the 12 pentagon grid boxes. It will be discussed that this deformation is responsible for non-convergence of some measures which are needed for the evaluation of the friction tensor. The formulation of the friction in dependency on vorticity and divergence instead on strain and shear deformations avoids this problem. However, a direct numerical integration by parts which delivers the frictional heating is then no longer possible. Results with different approaches for the friction term will be presented.