Vorträge in der Woche 13.02.2023 bis 19.02.2023

Freitag, 17.02.2023: Stability and convergence of Galerkin discretizations of the Helmholtz equation in piecewise smooth media

Jens Markus Melenk (TU Wien)

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 finite regularity but features wavenumber-independent bounds. This decomposition generalizes earlier decompositions of [MS11] 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 if (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. Besides impedance boundary conditions, the technique presented is applicable to other boundary conditions, in particular, boundary conditions suitable for treating full-space problems such as PML and FEM-BEM coupling. Furthermore, the technique is applicable to other time-harmonic wave propagation problems such as Maxwell's equations. References [MS11] J.M. Melenk and S. Sauter, Wavenumber explicit convergence analysis for finite element discretizations of the Helmholtz equation, SIAM J. Numer. Anal., 49:1210-1243, 2011 [BCFM22] M. Bernkopf, T. Chaumont-Frelet, J.M. Melenk, Wavenumber-explicit stability and convergence analysis of hp finite element discretizations of Helmholtz problems in piecewise smooth media. arXiv 2209.03601