Forschungsgruppe ''Numerische Mathematik und Wissenschaftliches Rechnen''

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

Donnerstag, 29. 11. 2012, 14:00 Uhr (ESH)

Mag. rer. nat. A. Kucher   (University of Graz, Austria)
Directive-based hardware accelerator frameworks for hybrid CPU/many-Core platforms

GPUs as general purpose many-core hardware accelerators already are well adopted in scientific and high performance computing. With CUDA and OpenCL there are two mature language-based frameworks available for GPU programming. They give excellent control over the hardware, but apart from potential compatibility problems with upcoming many-core hardware (e.g. Xeon Phi), both frameworks share common disadvantages. It is hard and time consuming to program them efficiently and it can be even harder to successfully integrate and maintain GPU specific code in large (existing) applications. This is particularly true for hybrid CPU/GPU applications, as CPU- and GPU-specific implementations are required. OpenACC and HMPP are two compiler--based hardware accelerator frameworks with an abstract programming model, similar to OpenMP, that allow many-core programming and the porting of existing sequential codes for many-core hardware accelerators by means of meta--information in form of preprocessor directives. Depending on the target architecture, a code generator uses these directives to generate hardware accelerator code automatically. In this talk we will discuss these frameworks and their programming model. We will evaluate them in terms of performance, usability and applicability, where we particularly set a focus on their limitations.

Donnerstag, 15. 11. 2012, 14:00 Uhr (ESH)

Dr. P. Berg   (Norwegain University of Science and Technology)
Dynamics and formation of nanopores in polymer electrolyte membranes

Polymer electrolyte membranes (PEM) play a critical role as the electrolyte of choice in PEM fuels cells. The latter are electro-chemical energy devices that convert hydrogen and oxygen into water and heat while useful electrical energy is generated. In this application, the PEM serves as an ion exchange membrane that facilitates proton and water flow through nano-scale pores whose shape changes dynamically, determined by the operating conditions. Understanding the dynamics of this electro-kinetic flow is key to improving the performance of the PEM and, hence, the fuel cell. In this talk, we will discuss the flow of protons and water in PEM nano-pores through the use of mathematical modeling and simulation. In particular, we employ Poisson-Nernst-Planck equations coupled to Stokes flow, meaning a continuum approach. In addition, water uptake isotherms of PEM will be analyzed with the help of a simplified pressure equilibrium model for pore formation. This research is motivated by experimental data in the literature, the interpretation of which is still an unresolved problem. It also touches upon other technology areas where membranes play a pivotal role such as reverse osmosis or osmotic power generation.

[1] Berg and Findlay, Proc. Roy. Soc. A, 467, pp. 3157-3169 (2011)
[2] Eikerling and Berg, Soft Matter, 7, pp. 5976-5990 (2011)
[3] Mauritz and Moore, Chem. Rev., 104, pp. 4535-4585 (2004)

Donnerstag, 08. 11. 2012, 14:00 Uhr (ESH)

Prof. J. Novo   (Universidad Autonoma de Madrid, Spain)
Postprocessing the Galerkin method for the Navier-Stokes equations: A review

The so called postprocessed Galerkin method was introduced in 1999 for Fourier spectral methods in connection with approximate inertial manifolds. The method was later extended to mixed finite element based methods. In this talk we will make a survey about the results of postprocessing concerning mixed finite element methods including different aspects such that the required regularity of solutions, the treatment of the fully discrete case and the connections with two grid mixed finite element methods.

Donnerstag, 18. 10. 2012, 14:00 Uhr (ESH)

Prof. L. Rebholz   (Clemson University, USA)
Efficient, unconditionally stable, and optimally accurate FE algorithms for approximate deconvolution models of fluid flow

This paper addresses an open question of how to devise numerical schemes for approximate deconvolution fluid flow models that are efficient, unconditionally stable, and optimally accurate. We propose, analyze and test a scheme for these models that has each of these properties for the case of homogeneous Dirichlet velocity boundary conditions. There are several important components to the derivation, both at the continuous and discrete levels, which allow for these properties to hold. The proofs of unconditional stability and optimal convergence are carried out through the use of a special choice of test function and some technical estimates. Numerical tests are provided that confirm the effectiveness of the scheme.

Dienstag, 09. 10. 2012, 13:30 Uhr (ESH)

Prof. S. Le Borne   (Technische Universität Hamburg-Harburg)
Hierarchical matrices: Some theory and some applications in preconditioning discretized fluid flow problems

Hierarchical ($\mathcal{H}$-) matrices, first introduced in 1999 by W. Hackbusch, provide a data-sparse technique for the storage and arithmetic of dense matrices. In this talk, we begin with an introduction of the basic underlying idea of the technique of $\mathcal{H}$-matrices. We will continue with a brief history of the development of $\mathcal{H}$-matrices, highlighting the major advancements since their initial introduction. By now, the $\mathcal{H}$-matrix arithmetic has reached a relatively mature state, and $\mathcal{H}$-matrix techniques are entering into an increasing number of application fields. As representatives of application fields for $\mathcal{H}$-matrices, we will show two examples where $\mathcal{H}$-matrix techniques are exploited in the numerical solution of partial differential equations. Whereas $\mathcal{H}$-matrix preconditioners could be computed in an entirely blackbox manner, it is usually advantageous to include information that is specific to the application or even discretization at hand. We will illustrate this distinction between blackbox and problem-specific $\mathcal{H}$-matrix preconditioners for our two examples.

Donnerstag, 20. 09. 2012, 14:00 Uhr (ESH)

H. Vu Nguyen   (King Abdullah University of Science and Technology (KAUST))
The physics of wind erosion: a fundamental topic in climate research

Wind erosion is a fundamental process in the Earth system. Its physics are complex, involving several other processes. It also has sophisticated interactions with other Earth system components.The research on wind erosion has made considerable progress in recent years but much remains to be done. This talk will provide an overview of the main topics in this field, including saltation, dust (emission, transport and deposition) and sand dunes from the climate science point of view. Current theory as well as numerical modeling approaches are presented together with open problems such as the electrification of sand particles. Extra-terrestrial wind erosion on Mars and other celestial bodies will also be briefly covered.

Montag, 27. 08. 2012, 14:00 Uhr (ESH)

D. Kourounis   (Universita della Svizzera italiana)
The Constrained Pressure Residual (CPR) preconditioning strategy in reservoir simulation

The Constrained Pressure Residual (CPR) preconditioning is one of the most appreciated techniques used in reservoir simulation. Its robustness is due to its particular designed tailored to capture fundamental mechanisms governing the flow in porous media. The design and implementation of CPR are presented for preconditioning the Jacobians that naturally arise from the applications of Newton's method in the forward simulation as well as the transposed Jacobians that need to be inverted during the adjoint problem for the calculation of sensitivities.

Mittwoch, 01. 08. 2012, 15:15 Uhr (ESH)

Remi Joubaud   (CERMICS - Ecole des Ponts ParisTech, France)
Continuous model for equilibrium electrolytes: Non ideality and phase separation

We are interested in equilibrium properties of confined electrolytes surrounded by charged solid walls. The problem is formulated in terms of the electrostatic potential and the ionic concentrations of the constituants which have prescribed spatial mean values. In a first part, we will present our main result which asserts the existence and uniqueness of the saddle point of the free energy functional and its characterization as a solution of a system of conservation equations. Numerical illustrations of this setting based on finite elements discretization and a Newton-Raphson algorithm are presented. In a second part, we will discuss several mathematical and numerical aspects of the case where the free-energy functional is no longer a convex functional of the concentrations. This case is particularly relevant for divalent and trivalent ions. For this physical setting, phase separation between diluted and condensed phase can occur for high surface charge density.

Donnerstag, 26. 07. 2012, 14:00 Uhr (ESH)

Prof. J. Shewchuk   (University of California at Berkeley, USA)
Weighted delaunay triangulations and restricted Delaunay triangulations in guaranteed-quality mesh generation

Many tasks in scientific computing and computer graphics require surface meshes or volume meshes composed of high-quality triangles or tetrahedra. Some mesh generation algorithms use the mathematical properties of Delaunay triangulations to offer guarantees on the quality of the meshes they produce. I discuss mesh generation algorithms that take advantage of two variants of the Delaunay triangulation to resolve specific problems. Three-dimensional domains whose polygonal boundaries meet at small angles are particularly difficult to mesh with high-quality tetrahedra; weighted Delaunay triangulations provide a way to ensure that a mesh will conform to the shape of the domain. Curved surfaces embedded in three dimensions are difficult to mesh because of the difficulty of ensuring that a piecewise linear mesh will be a topologically and geometrically accurate representation of a surface; restricted Delaunay triangulations, coupled with a theory of surface sampling, provide a way to guarantee this accuracy along with a guarantee of high-quality triangles.
The theory and algorithms in this talk will appear in a forthcoming book, ``Delaunay Mesh Generation'', by Siu-Wing Cheng, Tamal Dey, and myself.

Donnerstag, 05. 07. 2012, 14:00 Uhr (ESH)

A. Fiebach   (WIAS Berlin)
Uniform global bounds for solutions of an implicit Voronoi finite volume method for reaction-diffusion problems

We consider discretizations for reaction-diffusion systems with nonlinear diffusion in two space dimensions. The applied model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. We propose an implicit Voronoi finite volume discretization on regular Delaunay meshes that allows to prove uniform, mesh-independent global upper and lower $L^\infty$ bounds for the chemical potentials. These bounds provide the main step for a convergence analysis for the full discretized nonlinear evolution problem. The fundamental ideas are energy estimates, a discrete Moser iteration and the use of discrete Gagliardo-Nirenberg inequalities. For the proof of the Gagliardo-Nirenberg inequalities we exploit that the discrete Voronoi finite volume gradient norm in $2d$ coincides with the gradient norm of continuous piecewise linear finite elements.

Donnerstag, 14. 06. 2012, 15:00 Uhr (ESH)

Dr. J. Mura   (Pontificia Universidad Catolica de Valparaiso, Chile)
On some applications of small amplitude homogenization in elasticity

The talk focuses on some applications of small amplitude homogeneization to two phases shape optimization problems. This approach is based on the assumption that the contrast on the values of the Lame elastic coefficients of each phase is not very large. This approach is promising because it is well adapted to cases when the two phases results from defect or weak inhomogeneities during the fabrication process. In particular, the methodology can be applied to optimize the distribution of reinforcements and to detect weak defects. We show different numerical examples, to demonstrate that the method is quite robust under noisy measurements and errors in the characterization of the defect.

Donnerstag, 07. 06. 2012, 14:00 Uhr (ESH)

Dr. H. Stephan   (WIAS Berlin)
Ein abstrakter Zugang zur Modellierung klassischer Transportprobleme

Lineare Transportphänomene wie Drift, Diffusion oder lineare Reaktionen (Teilchenumwandlungen) lassen sich für einfache Situationen (Zustandsraum ist beschränktes Gebiet, Gleichungsystem ist streng elliptisch und hat diagonalen Hauptterm) recht gut beschreiben. Unter anderem gilt Positivitätserhaltung und es läßt sich das asymptotische Verhalten durch die Existenz von Lyapunovfunktionen abschätzen - Eigenschaften, die aus physikalischen Gründen für jedes Transportproblem sinnvoll sind. In komplizierteren Situationen (Heterostrukturen, wechselnde Dimensionen) ist manchmal nicht einmal klar, wie der Zustandsraum zu wählen ist, was ein Wahrscheinlichkeitsmaß ist und welche Größe als Konzentration zu verstehen ist. Im Vortrag wird ein abstrakter Zugang für allgemeine Transportprobleme vorgestellt, mit dem sich auch kompliziertere Situationen behandeln lassen. Dabei wird unter `Transport' die zeitlichen Änderung des Zustands eines klassischen (d.h. nicht quantenmechanischen) physikalischen Systems verstanden. Damit läßt sich auf jeder Modellierungsstufe - einschließlich einiger numerischer Verfahren - ein Evolutionsproblem als Transportproblem für ein effektives System mit effektiven Zuständen verstehen. Trotz der Allgemeinheit dieses Zuganges lassen sich konkrete Aussagen treffen, die für alle derartigen Transportprobleme gelten. Das sind unter anderen: - Jedes solche Problem ist ursprünglich linear (nichtlineare Gleichungen sind abgeleitete Gleichungen). - Alle Systeme haben die gleichen universellen Lyapunovfunktionen (2. Hauptsatz der Thermodynamik, `Die Zeit läuft stets vorwärts'). - Positivität bleibt erhalten (Maximum-Prinzip) - Kanonisch ist die Zeit eine diskrete Größe Des weiteren lassen sich Aussagen zur kanonischen Struktur entsprechender Evolutionsgleichungen und zu optimalen Funktionenräumen für ihre Lösungen treffen.

Donnerstag, 24. 05. 2012, 14:00 Uhr (ESH)

Prof. T. Iliescu   (Virginia Tech, USA)
Reduced-order modeling of complex flows: Analysis and computation

The reduced-order models (ROMs) are frequently used in the simulation of complex flows to overcome the high computational cost of direct numerical simulations, especially for three-dimensional nonlinear problems. The proper orthogonal decomposition (POD), as one of the most commonly used tools to generate ROMs, has been utilized in many engineering and scientific applications. Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by ROMs and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed. In this talk, we put forth two new closure models for the POD-ROMs of structurally dominated turbulent flows: the dynamic subgrid-scale (DS) model and the variational multiscale (VMS) model. These models, which are considered state-of-the-art in large eddy simulation, are carefully derived and numerically investigated. We also propose a two-level method for an efficient and accurate numerical discretization of general nonlinear POD closure models. This method computes the nonlinear terms of the ROM on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method and the physical accuracy of the POD-ROMS in several computational settings, including a three-dimensional turbulent flow past a cylinder at Reynolds number Re = 1000. We also present a thorough numerical analysis of the finite element discretization of the VMS-POD-ROM for convection-dominated convection-diffusion-reaction equations. Numerical tests show the increased numerical accuracy over the standard POD-ROM and illustrate the theoretical convergence rates.
Finally, we discuss the use of the new POD-ROMs in realistic applications such as air flow simulation in energy efficient building design and control problems, as well as numerical simulation of large-scale ocean motions in climate modeling.

Donnerstag, 19. 04. 2012, 14:00 Uhr (ESH)

Prof. R. Eymard   (Universite Paris Est, France)
Finite volume schemes for flows in anisotropic heterogeneous porous medium

We first present a new finite volume scheme, whose advantage is to apply on general anisotropic and heterogeneous media. This method is based on the introduction of control volumes at the center of the grid blocks and at their vertices. We then consider the case of the simulation of the displacement of phases into a heterogeneous porous medium. This problem leadz to some difficulties, related to discontinuities of the capillary pressure function. We compare different approaches, and we show the interest of the introduction of an artificial dissolution.

und 15:00 Uhr (ESH)

M. A. Fernandez   (INRIA, France)
Time-splitting schemes for incompressible fluid-structure interaction

The numerical approximation of fluid-structure interaction problems involving an incompressible fluid and an elastic structure is very sensitive to the way the interface coupling conditions (kinematic and kinetic continuity) are treated at the discrete level. For instance, it is well known that the stability of explicit Dirichlet-Neumann coupling schemes (involving only one fluid and one solid resolution per time step) is dictated by the physics and geometry of the system, irrespectively of the discretization parameters. Examples in blood flows simulations are widespread. In this talk we will present an overview of some splitting alternatives circumventing these infamous instabilities.

Donnerstag, 16. 02. 2012, 14:00 Uhr (ESH)

Dr. S. V. Sobolev   (GFZ Potsdam)
Major challenges in computational geodynamics, Part I

Geological-scale processes in solid Earth are dominated by plate tectonics (PT). PT is a non-trivial (not existing at Venus or Mars) surface manifestation of the solid-state thermo-chemical convection in the Earth mantle. A key feature of the deformation processes in PT is extreme strain localization. The major challenge of computational geodynamics is to replicate main features of PT and its relation with mantle convection. I'll show examples of how that is done now and which computational problems arise.

und 14:30 Uhr (ESH)

B. Steinberger/E. Mulyukova   (GFZ Potsdam)
Major challenges in computational geodynamics, Part II

The dynamical model of the Earth's interior can be broadly described by the thermochemical convection of the mantle, comprising the approximately 3000 km thick outer shell of our planet. The main challenges of geodynamic modelling of the mantle flow arise due to the strong dependence of properties of the mantle-materials on temperature, depth and composition. Robust advection schemes and accurate interpolation techniques are some of the requirements for obtaining physically reliable results. We have developed a two-dimensional finite element method code for simulating thermochemical convection of a heterogeneous viscous fluid in an annular domain. We will present the preliminary results of our model, and the results of the benchmarking study of the numerical techniques involved.

Donnerstag, 09. 02. 2012, 14:00 Uhr (ESH)

Dr. R. Schneider   (Technische Universität Dresden)
With edge based refinement towards anisotropic adaptive refinement in FEM

We propose a new paradigm for adaptive mesh refinement. Instead of considering local mesh diameters and their adaption to solution features, we propose to evaluate the benefit of possible refinements in a direct fashion, and to select the most profitable refinements. We demonstrate that based on this approach a directional refinement of triangular elements can be achieved, allowing arbitrarily high aspect ratios. However, only with the help of edge swapping and/or node removal (directional un-refinement) near optimal performance can be achieved for strongly anisotropic solution features. With these ingredients even re-alignment of the mesh with arbitrary error directions is achieved. Numerical experiments demonstrate the utility of the proposed anisotropic refinement strategy.

Donnerstag, 19. 1. 2012, 14:00 Uhr (ESH)

Prof. M. Bause   (Helmut Schmidt Universität, Universität der Bundeswehr Hamburg)
Efficient and reliable numerical approximation of transport equations with small diffusion


Donnerstag, 12. 1. 2012, 14:00 Uhr (ESH)

G. Bauer   (Technische Universität München, Lehrstuhl für Numerische Mechanik )
A variational multiscale finite element method for the numerical simulation of electrochemical systems

The consideration of ion-transport due to convection, diffusion and (electro-)migration plays a fundamental role for the mathematical modeling of many electrochemical systems. For example, electrodeposition of metals is an important industrial application. In typical electro-plating baths rather complex, often turbulent flow conditions arise, directly influencing the plating process. Hence, the apparent coupling to (turbulent) flow needs to be taken into account in the computational model. In this presentation, first the governing equations for incompressible flow, multi-ion transport, electric field and electrochemical reactions at electrode surfaces will be summarized, and the respective computational challenges are highlighted. Afterwards, our novel variational multiscale finite element method for solving the coupled nonlinear problem in complex cell geometries will be presented, and results from various three-dimensional numerical examples will be shown, demonstrating that the proposed method is robust and provides accurate results. Finally, our recent progress towards the simulation of ionic mass transport under turbulent flow conditions will be addressed.