# Numerische Mathematik und Wissenschaftliches RechnenForschungsgruppe

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## Seminar Numerische Mathematik / Numerical mathematics seminars aktuelles Programm / current program

Donnerstag, 9. 10. 2014, 14:00 Uhr (ESH)

Ph. Lederer   (TU Wien)
Hybrid Discontinuous Galerkin (HDG) methods for the incompressible Navier--Stokes Equations

n this talk we discuss a high order Hybrid Discontiuous Galerkin (HDG) method for the incompressible Navier--Stokes equations. The HDG method approximates the solution by independent polynomials on elements and faces. First, we derive the HDG method for the Poisson equation. The same techniques are then applied to the Stokes equation using normal-continuous Raviart--Thomas Finite Elements. We discuss two tricks to reduce the costs of the inter-element trace space and will give some 2D and 3D numerical examples using the convection and rotational formulation of the Navier--Stokes equations.

Donnerstag, 2. 10. 2014, 16:00 Uhr (ESH)

Adaptive moving meshes in large eddy simulation for turbulent flows

In the last years considerable progress has been made in the development of Large Eddy Simulation (LES) for turbulent flows. The characteristic length scale of the turbulent fluctuation varies substantially over the computational domain and has to be resolved by an appropriate numerical grid. We propose to adjust the grid size in an LES by adaptive moving meshes. The monitor function, which is the main ingredient of a moving mesh method, is determined with respect to a quantity of interest (QoI). These QoIs can be physically motivated, like vorticity, turbulent kinetic energy or enstrophy, as well as mathematically motivated, like solution gradient or some adjoint-based error estimator. The main advantage of mesh moving methods is that during the integration process the mesh topology is preserved and no new degrees of freedom are added and therefore the data structures are preserved as well. I will present results for real-life engineering and meteorological applications.

Dienstag, 30. 9. 2014, 10:00 Uhr (ESH)

Prof. Weizhang Huang   (University of Kansas, USA)
Computation of eigenvalue problems with anisotropic diffusion operators

Anisotropic eigenvalue problems can come from application of the Laplace-Beltrami operator to geometric shape analysis and imaging segmentation. They also arise from stability and sensitivity analysis and construction of special solutions for anisotropic diffusion partial differential equations. The latter appears in many areas of science and engineering including plasma physics, petroleum engineering, and image processing. This talk is concerned with finite element computation of those eigenvalue problems, with emphasis on anisotropic mesh adaptation and preservation of basic structures. Both analytical analysis and numerical results are presented.

Donnerstag, 25. 9. 2014, 14:00 Uhr (ESH)

Dr. J. Pellerin   (WIAS Berlin)
Accounting for the geometrical complexity of geological structural models in Voronoi-based meshing methods

Depending on the specific method used to build a 3D structural model, and on the exact purpose of this model, its mesh must be adapted so that it enforces criteria on element types, maximum number of elements, and mesh quality. Meshing methods developed for applications others than geomodeling forbid any modification of the input model, that may be desirable in geomodeling to better control the number of elements in the final mesh and their quality.
The objective of this thesis is to develop meshing methods that fulfill this requirement to better manage the geometrical complexity of B-Rep geological structural models. An analysis of the sources of geometrical complexity in those models is first proposed. The introduced measures are a first step toward the definition of tools allowing objective comparisons of structural models and permit to characterize the model zones that are more complicated to mesh. We then introduce two original meshing methods based on Voronoi diagrams: the first for surface remeshing, the second for hybrid gridding. The key ideas of these methods are identical: (1) the use of a centroidal Voronoi optimization to have a globally controlled number of elements of good quality, and (2) combinatorial considerations to locally build the final mesh while sometimes modifying the initial model. The surface remeshing method is automatic and permits to simplify a model at a given resolution. The gridding method generates a hybrid volumetric mesh. Prisms and pyramids fill the very thin layers of the model while the remaining regions are filled with tetrahedra.

Donnerstag, 10. 7. 2014, 14:00 Uhr (ESH)

S. K. Ganesan   (Indian Institute of Science, Bangalore)
SUPG finite element method for PDEs in time-dependent domains

In this talk, a stabilized finite element scheme for a convection dominated transient scalar equation in a time-dependent domain will be presented. The deformation of the domain is handled by the arbitrary Lagrangian--Eulerian (ALE) approach, whereas the Streamline Upwind Petrov--Galerkin (SUPG) finite element method is used for the spatial discretization. Further, the backward Euler method is used for the temporal discretization. The derived stability estimates of the proposed numerical scheme will be discussed. Moreover, numerical results obtained for boundary/interior layer problems in a time-dependent domain using the proposed numerical scheme will also be presented.

Mittwoch, 25. 6. 2014, 14:00 Uhr (Raum: 405/406)

Prof. L. Rebholz   (Clemson University, USA)
A connection between coupled and penalty projection timestepping schemes with FE spatial discretization

We prove that in finite element settings where the divergence-free subspace of the velocity space has optimal approximation properties, the solution of Chorin/Temam projection methods for Navier-Stokes equations equipped with grad-div stabilization with parameter $\gamma$, converge to the associated coupled method solution with rate $\gamma^{-1}$ as $\gamma\rightarrow \infty$. We prove this first for backward Euler schemes, and then extend the results to BDF2 schemes, and finally to schemes with outflow boundary conditions. Several numerical experiments are given which verify the convergence rate, and show how using projection methods in this setting with large grad-div stabilization parameters can dramatically improve accuracy.
Joint work with A. Linke (WIAS), M. Neilan (Pitt), and N. Wilson (Michigan Tech)

Donnerstag, 12. 6. 2014, 10:00 Uhr (ESH)

A. Fiebach   (Physikalisch-Technische Bundesanstalt Berlin)
An introduction to semiconductor device simulation

The simulation of fabrication processes for microelectronic devices is one possible application of the methods studied in the thesis. The simulation of the electrical op- eration is crucial in the development of semiconductor devices. Drift terms have to be added and the resulting drift-diffusion model, formulated by van Roosbroeck, is a well accepted mathematical description of charged carrier transport in semiconduc- tor devices. In the first part of the talk the model is introduced and the qualitative properties of the equations will be summarized. Next, a discretization in space and time of the system is derived which preserves the physical properties of the continuous problem. The application of the numerical method to a CCD sensor illustrates the working principle of many electronic cameras.
Results of the thesis
The talk gives an short overview of the results of my thesis. First a model for reaction- diffusion systems with reversible reactions is introduced which is suitable for hetero- geneous materials. Then a Voronoi finite volume method in space and implicit Euler in time discretization is introduced. Next, the new analytical results concerning the discretized problem obtained in the thesis will be summarized. A numerical example will demonstrate the stability of the method.

und 12. 6. 2014, 14:00 Uhr (ESH)

Elena Queirolo   (EPFL, Switzerland)
Isogeometrical Analysis for Navier-Stokes equations

Isogeometrical analysis is a new numerical method to discretize PDEs, based on NURBS. Such setting is more flexible that the usual FEM and has not yet been studied in depts. In this project, the construction of coupled spaces satisfying the Stokes and Navier-Stokes inf-sup condition is the main focus.

Dienstag, 27. 5. 2014, 13:30 Uhr (ESH)

Prof. G. Barrenechea   (University of Strathclude, UK)
Curing inf-sup deficiencies: Two quick examples

One rather standard way of justifying stabilised finite element methods is to prove that they cure the inf-sup deficiency associated to a given pair of finite element spaces. In this talk, I will describe how this study can be made in order to derive a stabilised finite element method (in the spirit of theminimal stabilisation approach'' by F. Brezzi and M. Fortin). Then, I will present the application of this idea to two different problems. More precisely, in the first part of the talk I will present a stabilised finite element method for the Reissner-Mindlin plate problem, and in the second part I will briefly present a stabilised finite element method for a fictitious domain formulation. The work presented in this talk has been carried out in collaboration with T. Barrios (UCSC, Concepcion, Chile), F. Chouly (Besancon, France), and A. Wachtel (Strathclyde).
'A paso lento, como bostezando, como quien besa el barrio al irlo pisando'

Donnerstag, 22. 5. 2014, 14:00 Uhr (Raum:405/406)

Dr. H. Stephan   (WIAS Berlin)
Deterministic and stochastic evolution equations and their connections

The aim of the talk is to show the connections of some analytic and stochastic concepts. All is presented from an analytical point of view. We give an overview of some evolution equations describing deterministic and stochastic problems in classical physics. We discuss the connections of the following equations and systems (given by its authors): Newton, Lagrange, Hamilton, Liouville (deterministic problems) and Langevin, Fokker-Planck, Kolmogorov-Chapman (stochastic problems).

Donnerstag, 10. 4. 2014, 14:00 Uhr (Raum: 406)

Stabilization of finite element approximations to the Stokes and Oseen equations

When considering the numerical approximation of the Navier-Stokes equations by means of mixed finite elements one can found two types on instabilities. On the one hand, it is well known that the standard Galerkin finite element method suffers from instabilities caused by the dominance of convection. On the other hand, stable mixed finite element approximations to the Stokes and Navier-Stokes equations are required to satisfy a discrete inf-sup condition. In this talk we study both kinds of instabilities. In the first part of the talk, we consider the stabilization in the convection dominated regime by means of SUPG/grad-div stabilized methods using LBB stable elements. We revise the existing literature pointing out some open questions. In the second part of the talk, we consider non LBB stable elements and analyze the so called pressure stabilized Petrov-Galerkin method for the continuous in time discretization of the evolutionary Stokes equations. We show some recent advances that avoid the so called instability of the discrete pressure for small time steps that has been reported in the literature.

Donnerstag, 27. 03. 2014, 11:00 Uhr (Raum 405/406)

S. Molnos   (Technische Universität Berlin)
Controlling transversal instabilities of two-dimensional travelling waves in reaction-diffusion systems

und 27. 3. 2014, 14:00 Uhr (ESH)

Dr. A. Caiazzo   (WIAS Berlin)
Multiscale modeling of weakly compressible elastic materials in harmonic regime

This talk focuses on the modeling of elastic materials composed by an incompressible elastic matrix and small compressible gaseous inclusions, under a time harmonic excitation. In a biomedical context, this model describes the dynamics of a biological tissue (e.g. liver) when wave analysis methods (such as Magnetic Resonance Elastography) are used to estimate tissue properties. Due to the multiscale nature of the problem, direct numerical simulations are prohibitive. After extending a recently proposed homogenized model [Baffico et al. SIAM MMS, 2008] which describes the solid-gas mixture as a compressible material in terms of an effective elasticity tensor, we derive and validate numerically analytical approximations for the effective elastic coefficients in terms of macroscopic parameters only. This simplified description is used to to set up an inverse problem for the estimation of the tissue porosity, using the mechanical response to external harmonic excitations.

Mittwoch, 26. 3. 2014, 13:00 Uhr (Raum 405/406)

Dr. J.-F. Mennemann   (Universität Wien)
Simulation of nanoscale electronic devices using the Schrödinger equation with open boundary conditions

Dienstag, 25. 3. 2014, 13:30 Uhr (ESH)

J. Neumann   (WIAS Berlin)
Stochastic bounds for quantities of interest in groundwater flow with uncertain data

Unreliable or unknown data is a hard problem for real world applications of partial differential equations. In recent years, uncertainty quantification with Monte Carlo and multilevel Monte Carlo methods became feasible as affordable computational power increased. This talk aims to present a method to tackle the approximation error for quantities of interest in the spatial as well as the stochastic dimensions for the Darcy model problem used to describe flow in porous media. In order to do so, bounds are derived for the spatial FEM error in the deterministic case and then carried over to the two sampling methods mentioned above. Here, the arising stochastic error is taken care off.

Donnerstag, 20. 3. 2014, 14:00 Uhr (ESH)

I. Ramis-Conde   (IMACI, Spain)
Multiscale modeling of palisade formation in glioblastoma

The talk present a hybrid model for the formation of palisades, multi-cellular circular structures that can be localized around occluded vascular vessels in glioblastomas (a malignant primary brain tumor). Observation suggests the following formation process. Initially the tumour grows around the vasculature in a cylindrical shape. This structure facilitates the tumour supply of oxygen and nutrients. The cells that are closest positioned to the vessel obtain have a better access to nutrients and oxygen than those positioned at the outer parts. When the pressure exerted by the tumor cells on the vessel cause vessel collapse, the cells closest to the vessel loose oxygen and nutrient supply, becoming hypoxic and, if the tumour geometry or the irrigating vasculature do not change, they die. On the other end, the cells which are not immersed in the tumour mass can activate migration mechanisms that allow them to invade the nearby tissue, in the search for available oxygen and nutrients. The collective migration results in the formation of a palisade. Observations of palisades and pseudopalisades under microscopy are constrained to two dimensional layers as a consequence of the experimental setups, due to the difficulties represented by observing glioblastomas in-vivo. We present a mathematical model and three dimensional numerical simulations coupling the evolution of tumor cells and the oxygen diffusion in the surrounding tissue, that depict the shape of palisades during their formation.

Donnerstag, 13. 3. 2014, 14:00 Uhr (ESH)

L. O. Müller   (Universita degli Studi di Trento, Italy)
Well-balanced finite volume schemes for one-dimensional blood flow models: Application to a closed-loop model for the cardiovascular system

The presence of so-called geometrical source terms in governing partial differential equations, which in the case of one-dimensional blood flow arise for varying mechanical and geometrical properties of vessels, causes a relevant problem if explicit numerical schemes are used. In fact, a naive discretization of these source terms will result in a numerical scheme which is unable to reproduce steady solutions, i. e. the scheme will not be well-balanced. A well-known way to deal with this problem consists in reformulating the original equations and solving the resulting non-conservative PDE system. We will first present a reformulation of the classic one-dimensional blood flow model, proposed [Toro & Siviglia, 2013]. After introducing the concept of path-conservative to solve this problem correctly [Müller & Toro, 2013]. The presented numerical schemes are applied to a closed-loop model for the cardiovascular system with emphasis in the venous system [Müller & Toro, 2014], constructed having the Chronic Cerebro-Spinal Venous Insuciency as main application.

Donnerstag, 20. 2. 2014, 14:00 Uhr (ESH)

PHD A. Fabri   (GeometryFactory)
CGAL - The Gomputational Geometry Algorithms Library

The CGAL C++ library, developed by the CGAL Open Source Project, offers geometric data structures and algorithms that are reliable, efficient, easy to use, and easy to integrate in existing software.
In this talk I will give an overview on what is currently available in CGAL, as well as what is under development. We will see algorithms from the areas 2D vector graphics (e.g., Boolean operations on B{ezier curves, off sets, polyline simplification, and geometry on the sphere), point set processing (e.g., normal estimation, denoising, shape detection, and surface reconstruction) surface mesh processing (e.g., Boolean operations, simplification, deformation, segmen tation, and skeletonization), and mesh generation (e.g., surface and volume mesh generation from 3D images, implicit functions, or polyhedral surfaces, anisotropic mesh generation, and mesh generation in periodic spaces).
In the second half of the talk I will cover non-geometric topics: First, the exact geometric computing paradigm that makes CGAL reliable without sac rificing efficiency. Then, the generic programming paradigm that facilitates integration into existing software. Finally, organizational issues, such as how the CGAL project works internally, how students can get involved through our participation in the Google Summer of Code, and how research groups can become project partners.

Dienstag, 18. 2. 2014, 13:30 Uhr (ESH)

Dr. C. Bertoglio   (TU München)
Forward and inverse modeling of the cardiovascular system: How PDE's can help to improve clinical outcomes

Biophysical computational models have the potential of extracting from clinical data additional metrics not directly visible, which could be used for more accurate diagnosis and better understanding of disease progress. This data-model coupling relies on a good balance among the type of data, model complexity and the data assimilation techniques. Motivated by the application of mathematical and computational modeling in medicine, we will present some results relevant to the cardiovascular system, where different PDE's arise:
(i) Solid Mechanics (heart, blood vessels);
(ii) Fluid Mechanics (blood flows);
(iii) Reaction-Diffusion Systems (cardiac electrophysiology);
(iv) Harmonic interpolations (geometrical models for fiber orientations in heart tissue)
For these problems, we will discuss the mathematical models involved, the type of clinical data available, and the solution of inverse problems in order to personalize these models and to obtain precise clinical indicators of the patient status. Concrete example with real clinical data will be shown.

Donnerstag, 23. 1. 2014, 14:00 Uhr (ESH)

J. Pellerin   (ENSG-Universite de Lorraine, France)
Accounting for the geometrical complexity of geological models in meshing methods based on Voronoi diagrams

Depending on the specific method used to build a 3D structural model and on the exact purpose of this model, its mesh must be adapted so that it enforces criteria on element type, maximum number of elements, and mesh quality. Meshing methods developed for applications different from geomodeling forbid any modification of the input model. This may be desirable in geomodeling to better control the number of elements in the final mesh and their quality. The objective of my thesis is to develop meshing methods that fulfill this requirement to better manage the geometrical complexity of geological structural models. An analysis of the sources of geometrical complexity sources in those models is first proposed. The introduced measures are a first step toward the definition of tools allowing objective comparisons of structural models and permit to characterize the model zones that are more complicated to mesh. We then introduce two original meshing methods based on Voronoi diagrams: one for surface remeshing and the second one for hybrid gridding. The key ideas of these methods are identical: (1) the use of a centroidal Voronoi optimization to have a globally controlled number of elements of good quality, and (2) combinatorial considerations to locally build the final mesh while sometimes modifying the initial model. The surface remeshing method is automatic and permit to simplify a model at a given resolution. The gridding method generates a hybrid volumetric mesh. Prisms and pyramids fill the very thin layers of the model while the remaining regions are filled with tetrahedra.