Program
The presentations take place in the seminar room 2.417 (RUD 25).
Schedule on August 08
| Start | End | Speaker | Title |
|---|---|---|---|
| 10:00 | 10:45 | Ngoc Tien Tran (Augsburg) | Lower eigenvalue bounds with hybrid high-order methods |
| 10:45 | 11:30 | Dietmar Gallistl (Jena) | A posteriori error control in the max norm for the Monge-Ampère equation |
| 11:30 | 12:00 | tea break | (30 minutes) |
| 12:00 | 12:45 | Dirk Praetorius (Wien) | Functional a-posteriori error estimates for BEM |
| 12:45 | 14:00 | lunch break | (75 minutes) |
| 14:00 | 14:45 | Joscha Gedicke (Bonn) | A symmetric interior penalty method for an elliptic distributed optimal control problem with pointwise state constraints |
| 14:45 | 15:30 | Rekha Khot (Paris) | Virtual element methods for the Biot-Kirchhoff poroelasticity |
| 15:30 | 16:00 | tea break | (30 minutes) |
| 16:00 | 16:45 | Andreas Schröder (Salzburg) | A posteriori error estimates for variational inequalities |
| 16:45 | 17:30 | Philipp Bringmann (Wien) | On full linear convergence and optimal complexity of adaptive FEM with inexact solver |
Abstracts
| Speaker: | Ngoc Tien Tran (Uni Augsburg) |
|---|---|
| Title: | Lower eigenvalue bounds with hybrid high-order methods |
| This talk proposes hybrid high-order eigensolvers for the computation of guaranteed lower eigenvalue bounds. These bounds display higher order convergence rates and are accessible to adaptive mesh-refining algorithms. The involved constants arise from local embeddings and are available for all polynomial degrees. A wide range of applications is possible including the linear elasticity and Steklov eigenvalue problem. | |
| Speaker: | Dirk Praetorius (TU Wien) |
| Title: | Functional a-posteriori error estimates for BEM |
| We consider the Poisson model problem with inhomogeneous Dirichlet boundary conditions. For the numerical solution, we employ the boundary element method (BEM). Unlike existing work, we aim for a-posteriori error control of the potential error (in the domain) instead of the error of the approximated integral density (on the boundary). To this end, we employ the well-known technique of functional error estimates. One key feature is that the derived error estimates are independent of the BEM discretization and provide guaranteed lower and upper bounds for the unknown error. In particular, the analysis covers Galerkin BEM and the collocation method, what makes the approach of particular interest for scientific computations and engineering applications. Numerical experiments for the Laplace problem confirm the theoretical results. | |
| Speaker: | Joscha Gedicke (Uni Bonn) |
| Title: | A symmetric interior penalty method for an elliptic distributed optimal control problem with pointwise state constraints |
| We construct a symmetric interior penalty method for an elliptic distributed optimal control problem with pointwise state constraints on general polygonal domains. The resulting discrete problems are quadratic programs with simple box constraints that can be solved efficiently by a primal-dual active set algorithm. Both theoretical analysis and corroborating numerical results are presented. | |
| Speaker: | Andreas Schröder (Uni Salzburg) |
| Title: | A posteriori error estimates for variational inequalities |
| The talk presents a posteriori error estimates for variational inequalities with linear constraints. For this purpose, an abstract framework is used in which the error contributions representing non-penetration, non-conformity and complementarity conditions form a weighted functional. The use of the minimizer of this functional enables the derivation of reliable and efficient a posteriori error estimates. The abstract findings are applied to model contact problems and an optimal control problem with control constraints. Several numerical experiments are presented to discuss the properties of the error estimates and their applicability in adaptive schemes. | |
| Speaker: | Philipp Bringmann (TU Wien) |
| Title: | On full linear convergence and optimal complexity of adaptive FEM with inexact solver |
|
The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to compute an approximation of user-prescribed accuracy at quasi-minimal computational time. To this end, the algorithmic realization of a standard adaptive finite element method (AFEM) integrates an inexact solver and nested iterations with discerning stopping criteria to balance the different error components. The analysis ensuring optimal convergence order of AFEM with respect to the overall computational cost critically hinges on the concept of R-linear convergence of a suitable quasi-error quantity. This talk presents recent advances in the analysis of AFEMs to overcome several shortcomings of previous approaches. First, a new proof strategy with a summability criterion for R-linear convergence allows removing typical restrictions on the stopping parameters of the nested adaptive algorithm. Second, the usual assumption of a (quasi-)Pythagorean identity is replaced by the generalized notion of quasi-orthogonality from [Feischl, Math. Comp., 91 (2022)]. Importantly, this paves the way towards extending the analysis to general inf-sup stable problems beyond the energy minimization setting. Numerical experiments investigate the choice of the adaptivity and stopping parameters.
(This talk presents joint work with Michael Feischl, Ani Miraçi, Dirk Praetorius, Julian Streitberger.) |
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