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Stability and convergence of Galerkin discretizations of the Helmholtz equation

Tuesday, February 14, 3:30PM – 5PM
ACE 6.304

J.M. Melenk

We consider boundary value problems for the Helmholtz equation at large wave numbers k. In order to understand how the wave number 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 an analytic, but highly oscillatory function and the second one has finite regularity but features wavenumber-independent bounds.

This new understanding of the solution structure opens the door to the analysis of discretizations of the Helmholtz equation that are explicit in their dependence on the wavenumber k. As a first example, we show for a conforming high order finite element method 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 small.

(Joint work with Stephan Sauter)

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