| October 19, 2011 | 09:15 | Axel Målqvist (Uppsala University) |
|---|---|---|
| »A Priori Error Analysis of a Multiscale Method« (Abstract) | ||
| October 26, 2011 | 09:15 | Kersten Schmidt (TU Berlin) |
| »Asymptotische Modellierung von Akkustik in viskosen Gasen« | ||
| November 2, 2011 | 09:15 | Carsten Carstensen |
| »Elements of Mathematical Paper Writing« | ||
| November 9, 2011 | 09:15 | Joscha Gedicke |
| »Lower Bounds for Eigenvalues« | ||
| November 14, 2011 | 16:00 | Ricardo Duran |
| »Poincare and Related Inequalities: A New Approach and Some Applications« | ||
| November 23, 2011 | 09:30 | Charles Elliott (University of Warwick) |
| »Computational Surface PDEs« (Abstract) | ||
| November 30, 2011 | 15:15 | Dietrich Braess (Ruhr-Universität Bochum) |
| »A Posteriori Fehlerschätzung für Hindernisprobleme mittels der Hyperkreismethode« (Abstract) | ||
| December 7, 2011 | 09:15 | Markus Schmuck (Imperial College London) |
| »Derivation of Effective Equations for Elektrokinetic Transport in Porous Media« | ||
| December 14, 2011 | 10:00 | Paulo Rafael Bösing (Federal University of Santa Catarina) |
| »Discontinuous Galerkin method for the linear Poisson-Boltzmann equation« (Abstract) | ||
| January 4, 2012 | 09:15 | Martin Brokate (TU München) |
| »Optimal control of a rate independent variational inequality« | ||
| January 11, 2012 | 09:15 | Peter Gottschling (TU Dresden) |
| »Wissenschaftliches Hochleistungsrechnen mit MTL4« | ||
| January 18, 2012 | 09:15 | Alexander Eckert |
| »Arnold-Winther MFEM für lineare Elastizitätstheorie und a posteriori Fehlerschätzung« | ||
| January 25, 2012 | 09:15 | Mira Schedensack |
| »Comparison Results of First-Order FEMs« | ||
| 10:00 | Dietmar Gallistl | |
| »Quasi Optimal Adaptive Pseudostress Approximation of the Stokes Equations« | ||
| February 1, 2012 | 09:15 | Daniel Elfverson (Uppsala University) |
| »An Adaptive Discontinuous Galerkin Multiscale Method for Elliptic Problems based on Energy norm a posteriori error estimates« (Abstract) | ||
| February 8, 2012 | 09:15 | Philipp Morgenstern |
| TBA | ||
We derive an a priori error estimate and thereby prove convergence for the multiscale method presented by Larson and Malqvist in "Adaptive variational multiscale methods based on a posteriori error estimation" [CMAME, 196, 2007, 2313-2324]. The proof strongly relies on the local behavior of the elliptic differential operator on fine scales. We use iterative techniques to track down the decay rate of the fine scale basis functions for arbitrary positive bounded diffusion coefficient. The decay rate is the key result which leads to an a priori bound of the error in the multiscale approximation. We present five numerical test cases in order to illustrate the theoretical results of the paper.
I will motivate the computational solution of surface PDES with some examples from biology (cell motility and phase separation on biomembranes) and material science (surface dissolution and the formation of nanoporosity) which couple PDEs on surfaces to the evolution of the surfaces. I will formulate the evolving surface finite element method and describe the finite element error analysis which yields optimal order bounds for piece-wise linear elements in the semi-discrete and the fully discrete backward Euler schemes. This error analysis is joint work with G. Dziuk. I will describe the phase field approach to interface and surface problems.
Hindernisprobleme gehören zu den einfachen Beispielen für Variationsungleichungen. Trotzdem haben die meisten klassischen a posteriori Schätzer den Nachteil, dass die Fehler in den Lagrangeschen Multiplikatoren die Abschätzungen der eigentlichen Zielfunktion verschlechtern. Die Hyperkreismethode liefert Schranken ohne generische Konstanten im Hauptterm. Zur primal zulässigen Funktion, hier ist es die Finite-Element-Lösung, ist eine dual zulässige zu konstruieren. Das analytische und das numerische Vorgehen erfordern hier geeignete Erweiterungen.
In this seminar we are going to discuss the linearized Poisson-Boltzmann equation with singular source term. As a consequence of the discontinuous coefficient, the linear Poisson-Boltzmann solution has low regularity, which prevents the use of discontinuous Galerkin method as we need the existence of the normal derivatives at the interfaces. To get around this, we introduced a regularization method, and the regularized equation is solved by the Galerkin discontinuous method with interior penalization. As the regularization method consists in approximate the discontinuous coefficient by a continuous coefficient that depends on a parameter $\epsilon$, we prove a elliptic regularity result for the solution that depends on this parameter. We use this to prove that the numerical solution of regularized equation converges to the weak solution of the linear Poisson-Boltzmann equation as $\epsilon$ (and $h$)tend to zero. At the end, some related numerical results are shown.
An adaptive discontinuous Galerkin multiscale method based on an energy norm a posteriori error bound is proposed. The method is based on splitting the problem into a coarse and a fine scale. Localized fine scale constituent problems are solved on patches of the the domain and are used to obtain a modified coarse scale equation. The coarse scale equation has considerably less degrees of freedom than the original problem. The a posteriori error estimate is to be used within an adaptive algorithm to tune the critical paramets i.e the refinement level and the size of the different patches where the fine scale constituent problems are solved. The fine scale computations are completely parallelizable, since no communication between different processors is required for solving the constituent fine scale problems. The convergence, the adaptive strategy and the computational effort are investigated through a series of numerical experiments.