Zurich colloquium in applied and computational mathematics

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16 March 2022
16:30-18:00

Prof. Dr. Michael Feischl
TU Wien

Event Details

Title	Towards optimal adaptivity for time-dependent problems
Speaker, Affiliation	Prof. Dr. Michael Feischl, TU Wien
Date, Time	16 March 2022, 16:30-18:00
Location	Y27 H 35/36 Zoom Meeting
Abstract	We prove new optimality results for adaptive mesh refinement algorithms for non-symmetric, indefinite, and time-dependent problems by proposing a generalization of quasi-orthogonality which follows directly from the inf-sup stability of the underlying problem. This completely removes a central technical difficulty in modern proofs of optimal convergence of adaptive mesh refinement algorithms. The main technical tools are new stability bounds for the LU-factorization of matrices together with a recently established connection between quasi-orthogonality and matrix factorization.

Towards optimal adaptivity for time-dependent problemsread_more

Y27 H 35/36
Zoom Meeting

23 March 2022
16:30-18:00

Prof. Dr. Walter Boscheri
Department of Mathematics and Computer Science, University of Ferrara

Event Details

Title	On the construction of conservative semi-Lagrangian IMEX advection schemes for multiscale time dependent PDE
Speaker, Affiliation	Prof. Dr. Walter Boscheri, Department of Mathematics and Computer Science, University of Ferrara
Date, Time	23 March 2022, 16:30-18:00
Location	Y27 H 35/36 Zoom Meeting

On the construction of conservative semi-Lagrangian IMEX advection schemes for multiscale time dependent PDE

Y27 H 35/36
Zoom Meeting

6 April 2022
16:30-18:00

Dr. Martin Licht
EPF Lausanne

Event Details

Title	Local finite element approximation of Sobolev differential forms
Speaker, Affiliation	Dr. Martin Licht, EPF Lausanne
Date, Time	6 April 2022, 16:30-18:00
Location	Y27 H 35/36 Zoom Meeting
Abstract	We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Clément interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert Lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over simplicial triangulations.

Local finite element approximation of Sobolev differential formsread_more

Y27 H 35/36
Zoom Meeting

11 May 2022
16:15-17:15

Prof. Dr. Barbara Verfürth
Department of Mathematics, KIT

Event Details

Title	Numerical homogenization for nonlinear multiscale problems
Speaker, Affiliation	Prof. Dr. Barbara Verfürth, Department of Mathematics, KIT
Date, Time	11 May 2022, 16:15-17:15
Location	Y27 H 35/36 Zoom Meeting
Abstract	Many applications, such as geophysical flow problems or scattering from Kerr-type media, require the combination of nonlinear material laws and multiscale features, which together pose a huge computational challenge. In this talk, we discuss how to construct a problem-adapted multiscale basis in a linearized and localized fashion for nonlinear problems such as the quasilinear diffusion equation or the nonlinear Helmholtz equation. For this, we will adapt two different perspectives: (a) determining a fixed multiscale space for the nonlinear problem or (b) adaptively and iteratively updating the multiscale space during an iteration scheme for the nonlinear problem. We prove optimal error estimates for the corresponding generalized finite element methods. In particular, neither higher regularity of the exact solution nor structural properties of the coefficients such as scale separation or periodicity need to be assumed. Numerical examples show very promising results illustrating the theoretical convergence rates.

Numerical homogenization for nonlinear multiscale problemsread_more

Y27 H 35/36
Zoom Meeting

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