Convergence analysis of Galerkin discretizations of the Helmholtz equation in piecewise smooth media

Maximilian Bernkopf

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 from the work of J. M. Melenk and S. A. Sauter, who considered the Helmholtz equation with constant coefficients, to the case of (piecewise) analytic coefficients. This regularity theory allows for the analysis of high order Galerkin discretizations of the Helmholtz equation that are explicit in the wavenumber k. This is joint work with T. Chaumont-Frelet (INRIA) and J. M. Melenk (TU Wien).