Numerical Analysis Seminar

Date: May 4, 2022

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Dr J.M. Melenk, TU Wien

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

Abstract: We consider the Helmholtz equation with variable coefficients at large wavenumber k. In order to understand how k affects 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 first component is a piecewise analytic, but highly oscillatory function and the second one has infinite regularity but features wavenumber-independent bounds. This decomposition generalizes earlier decompositions of Melenk and Sauter (2010 & 2022) which considered the Helmholtz equation with constant coefficients, to the case of piecewise analytic coefficients. This regularity theory for the Helmholtz equation with variable coefficients 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 sufficiently small. This scale resolution condition ensures quasi-optimality for a variety of time-harmonic wave propagation problems including FEM-BEM coupling and Maxwell problems.