Past Event: Oden Institute Seminar

Stability and convergence of Galerkin discretizations of the Helmholtz equation in piecewise smooth media

Jens Markus Melenk, Professor, Computational Mathematics, TU Wien

Abstract

**This seminar will be presented LIVE in POB 6.304 and on Zoom**

We consider the Helmholtz equation with variable coecients at large wavenumber k. In order to understand how k a ects the convergence properties of discretizations of such problems, we develop a regularity theory for the Helmholtz equation that is explicit in k. At the heart of our analysis is the decomposition of solutions into two components: the rst component is a piecewise analytic, but highly oscillatory function and the second one has nite regularity but features wavenumber-independent bounds. This decomposition generalizes earlier decompositions of [MS10, MS11] which considered the Helmholtz equation with constant coecients, to the case of piecewise analytic coecients. This regularity theory for the Helmholtz equation with variable coecients allows for the analysis of high order Galerkin discretizations of the Helmholtz equation that are explicit in the wavenumber k. We show that quasi-optimality is guaranteed under the following scale resolution condition: (a) the approximation order p is selected as p = O(log k) and (b) the mesh size h is such that kh=p is suciently small. This scale resolution condition ensures quasi-optimality for a variety of time-harmonic wave propagation problems including FEM-BEM coupling and Maxwell problems.

References
[MS10] J.M. Melenk and S. Sauter, Convergence Analysis for Finite Element Discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp. 79:1871{1914, 2010.
[MS11] J.M. Melenk and S. Sauter, Wavenumber explicit convergence analysis for nite element discretizations of the Helmholtz equation, SIAM J. Numer. Anal., 49:1210{1243, 2011.

Biography

J.M. Melenk is professor of computational mathematics at TU Wien. He was head of the Institute of Analysis and Scientific Computing of TU Wien from 2012 until 2020. He received his PhD from the University of Maryland in 1995 and his Habilitation from ETH Zurich in 2000. Before joining TU Wien in 2005, he held postdoc positions at ETH Zurich (CH) and the Max-Planck-Institute for Mathematics in the Sciences in Leipzig (DE) and was a lecturer of Applied Mathematics at the University of Reading (UK).

His current research interests include high order finite element methods, singular perturbations, wave propagation, fast methods for integral equations as well as numerical methods for fractional differential equations.