## Doktorandenseminar des WIAS

# Numerische Mathematik und Wissenschaftliches Rechnen

Forschungsgruppe

zurück zum aktuellen Programm

## Seminar Numerische Mathematik / Numerical mathematics seminars

aktuelles Programm / current program

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Donnerstag, 30. 11. 2017, 14:00 Uhr (ESH)
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M. Pavelka(Charles University, Tschechische Republik)

42=GENERIC --- Unified Hamiltonian description of solids and fluidsGENERIC (General equation for non-equilibrium reversible-irreversible coupling) provides evolution equations for many physical theories: fluid mechanics, kinetic theory, mechanics of solids, plasticity, non-Newtonian fluid mechanics, electromagnetic field coupled with matter, dynamics of mixtures, etc. The equations consist of a reversible Hamiltonian part, generated by a Poisson bracket and energy, and by an irreversible gradient part, generated by a dissipation potential and entropy. The purpose of this talk is to provide introduction into GENERIC and discuss some specific applications.

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Dienstag, 07. 11. 2017, 13:30 Uhr (ESH)
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J. Novo(Universidad Autonoma de Madrid, Spanien)

Quasi-optimal methods to approximate the incompressible Navier--Stokes squationsIn this talk we will consider different methods to approximate the incompressible Navier--Stokes squations. Our main concern is to get error bounds with constants independent on inverse powers of the viscosity parameter. We will consider inf-sup stable methods with grad-div stabilization, non inf-sup stable methods with local projection stabilization and fully discrete schemes based on projection methods in time plus grad-div stabilization in space. Some numerical experiments will be shown to check the optimality of the theoretical rates of convergence of the methods.

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Donnerstag, 19. 10. 2017, 14:00 Uhr (ESH)
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L. Rebholz(Clemson University, USA)

On conservation laws of Navier--Stokes Galerkin discretizationsWe study conservation properties of Galerkin methods for the incompressible Navier--Stokes equations, without the divergence constraint strongly enforced. In typical discretizations such as the mixed finite element method, the conservation of mass is enforced only weakly, and this leads to discrete solutions which may not conserve energy, momentum, angular momentum, helicity, or vorticity, even though the physics of the Navier-Stokes equations dictate that they should. We aim in this work to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint, and doing so leads us to a new formulation that conserves each of energy, momentum, angular momentum, enstrophy in 2D, helicity and vorticity (for reference, the usual convective formulation does not conserve most of these quantities). Several numerical experiments are performed, which verify the theory and test the new formulation.

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Dienstag, 26. 09. 2017, 13:30 Uhr (WIAS-ESH)
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Y. Ren(Dalian University of Technology, China)

On tetrahedralisations containing knotted and linked line segmentsThis talk considers a set of twisted line segments in 3d such that they form a knot (a closed curve) or a link of two closed curves. Such line segments appear on the boundary of a family of 3d

indecomposablepolyhedra (like the Schönhardt polyhedron) whose interior cannot be tetrahedralised without additional vertices added. On the other hand, a 3d (non-convex) polyhedron whose boundary contains such line segments may still be decomposable as long as the twist is not too large. It is therefore interesting to consider the question: when there exists a tetrahedralisation contains a given set of knotted or linked line segments?

In this talk, we studied a simplified question with the assumption that all vertices of the line segments are in convex position. It is straightforward to show that no tetrahedralisation of6vertices (the three-line-segments case) can contain a trefoil knot. When the number of twisted line segments is larger than3, it is necessary to create new interior edges to form a tetrahedralisation. We provided a detailed analysis for the case of a set of4twisted line segments. We show that the addition of a pair of new interior edges decomposes the original knot (or link) into two links (or knots) with less crossing numbers. This leads to a crucial condition on the orientation of pairs of new interior edges which determines whether this set is decomposable or not. We then prove a new theorem about the decomposability for a set ofn (n ≥ 3)twisted line segments. This theorem implies that the family of polyhedra generalised from the Schönhardt polyhedron by Rambau are all indecomposable.

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Dienstag, 19. 09. 2017, 15:00 Uhr (ESH)
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M. Patriarca(University of Rome ``Tor Vergata'', Italy)

Discontinuous finite element discretization for semiclassical particle transportIn order to understand how semiconductor devices operate, it is very common to solve a Poisson equation coupled with drift-diffusion equations for electrons and holes. We present several discontinuous approaches to solving the Poisson equation: non-symmetric interior penalty, lifting operator and a mixed approach among finite volume method and discontinuous Gal{\"e}rkin finite element method. We discuss the advantages and drawbacks of these approaches in order to extend the best discretization method to the drift-diffusion model. The stationary case is implemented in our software simulator TiberCAD. Simulation results are shown to highlight the problems encountered.

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Donnerstag, 14. 09. 2017, 14:00 Uhr (Raum 405-406)
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Prof. V. Garanzha(Russian Academy of Sciences, Moskau)

Construction of quasi-isometric elastic deformations in mesh generation problemsTheory of quasi-isometric (QI) mappings allows to reformulate in unified terms the intuitive idea of quasi-uniform meshes and presents natural multidimensional generalization of equidistribution principle. Suprisingly theory of QI mapping is not very rich. Rigorous results about existence of QI mappings are available only in 2d, see works by Reshetnyak, Bakelman, Bonk and Lang, Burago on parameterization of 2d manifolds of bounded curvature. In 2000 Garanzha suggested stored energy with guaranteed QI minimizer. Multidimensional "existence theorems" (Garanzha, 2005, 2010) essentially follow formulations of existence theorems of finite hyperelasticity due to John Ball: minimum of stored energy is attained on admssible mapping provided that at least one admissible mapping exists. It is not known how to find this initial admissible mapping. Another unsolved problem is lack of discrete convergence: one cannot prove that solution of discretized variational problem converges to exact minimizer. In order to construct QI mappings one has to solve continuation problem which minimizes certain evaluation of quasi-isometry constant. While providing rigorous foundations and best results in terms of distortion, this aproach is too complicated for solving engineering problems. Simple solution to this problem is based on introducing spatial weight distributions which premultiply density of stored energy (distortion measure) and sharply increase in the zones where small distortion is required. Strategies for devising these weight functions are discussed as applied to construction of smooth and orthogonal boundary layer meshes with precise stretching control, to construction of prismatic mesh layer using elastic springback technique, to surface flattening and parameterization and to construction of adaptive meshes.

und 15:00 Uhr

Prof. Na Lei(Dalian University of Technology, China)

Quadrilateral and hexahedral mesh generation based on surface foliation theoryFor the purpose of isogeometric analysis, one of the most common ways is to construct structured hexahedral meshes, which have regular tensor product structure, and fit them by volumetric T-Splines. This theoretic work proposes a novel surface quadrilateral meshing method, colorable quad-mesh, which leads to the structured hexahedral mesh of the enclosed volume for high genus surfaces. The work proves the equivalence relations among colorable quad-meshes, finite measured foliations and Strebel differentials on surfaces. This trinity theorem lays down the theoretic foundation for quadrilateral/hexahedral mesh generation, and leads to practical, automatic algorithms. The work proposes the following algorithm: the user inputs a set of disjoint, simple loops on a high genus surface, and specifies a height parameter for each loop; a unique Strebel differential is computed with the combinatorial type and the heights prescribed by the user?s input; the Strebel differential assigns a flat metric on the surface and decomposes the surface into cylinders; a colorable quad-mesh is generated by splitting each cylinder into two quadrilaterals, followed by subdivision; the surface cylindrical decomposition is extended inward to produce a solid cylindrical decomposition of the volume; the hexahedral meshing is generated for each volumetric cylinder and then glued together to form a globally consistent hex-mesh.

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Donnerstag, 31. 08. 2017, 14:00 Uhr (ESH)
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C. Bartsch(WIAS Berlin)

A mixed stochastic-deterministic approach to particles interacting in a flowWe present a new approach to the solution of coupled particle population balance systems. Such population balance systems consist of particles transported by a moving fluid. The particles interact with each other and with the surrounding fluid. Population balance systems aim at keeping track of both particle- and fluid quantities. In our approach, particle-particle interactions and particle-fluid interactions are solved with a stochastic simulation algorithm, while a finite element flow solver is used for fluid-fluid interactions. This best-of-two-worlds approach enables the efficient treatment even of such particle-fluid systems, where the particles are described by multiple internal coordinates. A simulation of a flow crystallizer will be presented as a proof of concept.

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Dienstag, 18. 07. 2017, 13:30 Uhr (ESH)
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Dr. D. Peschka(WIAS Berlin)

Variational structure of viscous flows with contact line motionIn this talk I will present variational approach to the free boundary problem of fluid motion over solid and liquid substrates, focussing on algorithmic approaches for the treatment of contact line motion.

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Donnerstag, 01. 06. 2017, 14:00 Uhr (Raum 405/406)
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Dr. S. Ganesan(Indian Institute of Science, Bangalore)

Stabilized three-field formulation of viscoelastic fluid flowsViscoelastic flows can be found in a wide range of industrial and commercial applications such as enhanced oil recovery, pesticide deposition, medicinal/pharmaceutical sprays, drug delivery, injection molding, polymer melts, inkjet printing, additive manufacturing, cosmetics industry and food processing. Computations of viscoelastic fluid flows involve simultaneous solutions of the conformation stress tensor equation together with the momentum and mass balance equations. The constitutive equation is advection dominant at high Weissenberg numbers, which necessitates the use of stabilized numerical schemes to avoid local and global oscillations in the numerical solution. The common stabilization schemes used in the context of finite element computations of viscoelastic fluid flows in the literature are Streamline-upwind/Petrov--Galerkin~(SUPG), discontinuous Galerkin~(DG), discrete Elastic Viscous Stress Splitting~(DEVSS)~[1], Galerkin Least Square~(GLS)~[2] and Variational Multiscale method~(VMS)~[3].

In this talk, a three-field formulation based on the one-level Local Projection Stabilization~(LPS)~[4, 5] will be presented for computations of Oldroyd-B viscoelastic fluid flows with high Weissenberg numbers. One-level LPS is based on the enriched approximation space and the discontinuous projection space, where both spaces are defined on the same mesh. It allows us to use equal order interpolation spaces for the velocity and viscoelastic stress, whereas inf-sup stable finite elements are used for the velocity and pressure. Further, the coupled system of equations are solved in a monolithic approach. Since LPS stabilization terms are assembled only once, the proposed numerical scheme is computationally efficient in comparison with residual based methods. Numerical studies that show optimal order of convergence in the respective norms will be presented. Further, the numerical results of two benchmark problems: flow past a cylinder in a wide channel and lid-driven cavity flow will be given. Moreover, the effects of elasticity and inertia on the flow dynamics will be discussed.

References:

[1] R. Guenette, A. Fortin, A new mixed finite element method for computing viscoelastic flows, J. Non-Newtonian Fluid Mech., 60 (1995), 27--52.

[2] O. M. Coronado, D. Arora, M. Behr, M. Pasquali, Four-field Galerkin/least-squares formulation for viscoelastic fluids, J. Non-Newtonian Fluid Mech., 140 (2006), 132--144.

[3] E. Castillo, R. Codina, Variational multi-scale stabilized formulation for the stationary three-field incompressible viscoelastic flow problem, Comput. Methods Appl. Mech. Engrg., 279 (2014), 579--605.

[4] G. Matthies, P. Skrzypacz, L. Tobiska, A unified convergence analysis for local projection stabilisations applied to the oseen problem, M2AN Math. Model. Numer. Anal., 41 (2007), 713--742.

[5] S. Ganesan, G. Matthies, L. Tobiska, Local projection stabilization of equal order interpolation applied to the Stokes problem, Math. Comp., 77 (2008), 2039--2060.

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Dienstag, 30. 05. 2017, 13:30 Uhr (ESH)
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Prof. C. Gräser(Freie Universität Berlin)

Solving nonsmooth PDEs in DuneThe design of efficient numerical software for the solution of PDEs is a challenging task because different applications may have very different requirements. As a consequence a large set of libraries has been developed, each with its own focus but in general not compatible with each other.

The central idea of the Dune (distributed unified numerics environment) framework is to change the perspective and to design interfaces that allow to represent a large range of different implementations. This idea is applied on many scales leading to a broad spectrum of interfaces ranging from low level grid operations to high level simulation and code generation frameworks. Based on these principle Dune has evolved to a modular and flexible framework with constantly growing user base and a developer group spread over many institutions.

The talk consists of two parts: The first part will present the Dune framework while the second part briefly outlines how this is used to solve nonsmooth PDEs originating e.g. from applications in material science and biomechanics with state of the art multigrid methods.

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Dienstag, 11. 04. 2017, 13:30 Uhr (ESH)
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Prof. J. Chen(Zheijian University, China)

Automatic and parallel mesh generation: Recent advancesMesh generation is recognized as the main performance bottleneck in simulations involving complicated geometric definitions and/or complex physical behaviors. Automatic generation of high-quality meshes remains a challenging issue. Besides, quite a few simulations are now demanding meshes consisting in hundreds of millions of elements or more. It could be very difficult and inefficient to create such big meshes by sequential algorithms due to the bottlenecks of memory usage and computing time.

In this talk, Dr. Chen will present three technologies developed by his research group recently, as part of his efforts to develop a completely automatic and parallel meshing system for real-world simulations. These technologies are: (1) Automatic creation of a feature-aware and gradient-limited sizing function; (2) Automatic generation of high-quality boundary layer elements using vector fields computed by the boundary element method; (3) Distributed and/or multi-threaded parallel mesh generation and mesh quality improvement. Mathematic models behind these technologies will be investigated in this talk, plus important algorithmic details. Meshing examples are selected to show that the developed technologies are applicable to configurations of a complication level experienced in industry.

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

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

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

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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.

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

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