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

Donnerstag, 8.12.2005, 14:00 Uhr (ESH)

Prof. E. Burman   (Ecole Polytechnique Federale de Lausanne)
Interior penalty procedures for transport problems: Asymptotic limits and monotonicity

In this talk we will consider interior penalty type finite element methods for convection dominated elliptic problems and pure transport operators. In the first part we will recall some basic notions of a framework for stabilization mechanisms in discontinuous Galerkin methods recently proposed by Brezzi, Cockburn, Marini and Suli. We consider the problem of pure transport and, starting from a stabilized discontinuous Galerkin methods we use an asymptotic analysis to recover different continuous Galerkin methods are limits when a certain penalization parameter goes to infinity. We then show some numerical examples comparing discontinuous and continuous stabilized finite element methods verifying the theoretical predictions. In the second half of the talk we will address the question of monotonicity for continuous finite element approximation of convection-diffusion problems. In an abstract setting we give sufficient conditions for a discrete maximum principle (DMP) to be satisfied. We then give some examples of nonlinear residual based artificial dissipation terms leading to methods that satisfy the DMP.

Donnerstag, 24.11.2005, 14:00 Uhr (ESH)

Claire Chainais   (Universite Blaise Pascal, France)
Reconstruction of gradients in finite volume scheme. Application to the discretization of the Joule heating term

The aim of this talk is to present two different finite volume schemes for the energy-transport model of semiconductor device. This coupled system of parabolic and elliptic equations involves a Joule heating term in the equation of energy. This term looks like \|\nabla V\|^2. We propose two kinds of reconstruction of the gradients in order to discretize this Joule heating term. Finally, we compare the resulting schemes on the ballistic diode test case.

Donnerstag, 3. 11. 2005, 10:00 Uhr (Raum 406)

Prof. R. Eymard   (Université de Marne-la-Vallée)
Finite volume schemes for anisotropic diffusion operators

The use of a finite volume method for the discretization of an anisotropic operator leads to approximate the full gradient of the unknown at each interface, and not only the normal component of this gradient. It is then possible to cheaply compute such a gradient in every grid block, and use this gradient to compute the fluxes. This methods provides a finite volume scheme for which it is possible to prove convergence properties. We present the mathematical background of this study, as well as some consequences for the discretization of engineering problems.

Donnerstag, 20. 10. 2005, 14:00 Uhr (ESH)

Dr. Olaf Schenk   (Universität Basel, Fachbereich Informatik)
On large scale diagonalization techniques for the Anderson model of localization

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for the largest sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation of Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Arnoldi method and in the Jacobi-Davidson method. Our preconditioning approaches for the shift-and invert symmetric indefinite linear system are based on new maximum weighted matchings and algebraic multilevel incomplete $LDL^T$ factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that these recent algebraic multilevel preconditioning solvers can accelerative the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude. This is joint work with Matthias Bollhoefer (TU Berlin, Germany) and Rudo Roemer (University of Warwick, UK)

Donnerstag, 6. 10. 2005, 14:00 Uhr (ESH)

D. Kourounis   (University of Ioannin, Department of Material Sciences)
The FEM-FCT stabilization method for convection dominated problems

Among various stabilization methods proposed in the literature, algebraic flux correction tools are a very promissing alternative, since they provide a problem independent implementation. In this study the Flux Corrected Transport (FCT) method is used to stabilize nonlinear convection-diffusion(Burgers) equations discretized by the finite element method on unstructured triangular meshes. The algorithm is based the one proposed by Kuzmin and Turek. A modified version of the above algorithm is also presented which accelarates significantly the performance of the method while achieving qualitative similar results. Various test cases are considered to demonstrate the stabilizing effectiveness of the method even for the nonhomogeneous case.

Donnerstag, 8. 9. 2005, 14:00 Uhr (ESH)

Prof. Dr. Peter Arbenz   (ETH Zürich, Institut für Wissenschaftliches Rechnen)
On a parallel multilevel preconditioned Maxwell eigensolver

We report on a parallel implementation of the Jacobi-Davidson algorithm for computing a few eigenvalues and corresponding eigenvectors of a large real sym-metric generalized matrix eigenvalue problem Ax = \lambda Mx, C^T x = 0. The eigenvalue problem originates from the design of cavities of particle accel-erators. It is obtained by the finiteelement discretization of the time-harmonic Maxwell equation in weak form by a combination of N'ed'elec (edge) and La-grange (node) elements. We found the Jacobi-Davidson (JD) method to be a very effective solver pro-vided that a good preconditioner is available for the correction equations that have to be solved in each step of the JD iteration. The preconditioner of ourchoice is a combination of a hierarchical basis preconditioner and a smoothed aggregation AMG preconditioner. It is close to optimal regarding iterationcount. The parallel code makes extensive use of the Trilinos software framework. Wediscuss procedures how to distribute data. Our examples from accelerator physics deliver very satisfactory speedups and efficiencies.

Donnerstag, 4. 8. 2005, 14:00 Uhr (ESH)

Dr. Henning Struchtrup   (Department of Mechanical Engineering, University of Victoria, Canada)
A macroscopic transport model for PEM membranes based on the microscale behavior

Insight into the microscopic be structure and behavior of PEM membranes is used in the development of a general transport model for water and protons in PEMs based on the Binary Friction Model (BFM). In order to investigate the unknown parameters in the transport model, a simplified conductivity model (BFCM) is developed for perfluorosulfonic acid membranes. The model is first applied 1100 EW Nafion, and compared to other established membrane models, the new BFCM is shown to provide a more consistent fit to the data over the entire range of water contents and at different temperatures. The generality of the BFCM is also illustrated by applying the model to predict the conductivity of a Dow and Membrane C membrane using only rational and physically consistent changes in model parameters. The paper discusses possible experimental investigations and fundamental simulations to determine the model parameters required to apply the BCFM to other types of membranes. One advantage of the BFCM model and the associated transport model is that by fitting the BFCM to conductivity data we are able to gain insight into all the transport parameters, and this could eventually be used to develop a general membrane water transport model.
(H. Struchtrup, joint work with J. Fimrite, B. Cairns, N. Djilali)

Dienstag, 21. 6. 2005, 13:30 Uhr (ESH)

Dr. Jürgen Fuhrmann   (WIAS Berlin)
Ein Finite-Volumen-Verfahren auf der Basis der Lösung lokaler Dirichlet-Probleme