Seminar in Numerical Analysis: Markus Melenk (TU Wien)
We consider the Helmholtz equation with piecewise analytic coefficients at large wavenumber k > 0. The interface where the coefficients jump is assumed to be analytic. We develop a k-explicit regularity theory for the solution that takes the form of a decomposition 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 [MS10, MS11, EM11, MSP12], which considered the Helmholtz equation with constant coefficients, to the case of (piecewise) analytic coefficients. This regularity theory allows to show for high order Galerkin discretizations (hp-FEM) of the Helmholtz equation that quasi-optimality is reached 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. This extends the results of [MS10, MS11, EM11, MSP12] about the onset of quasi-optimality of hp-FEM for the Helmholtz equation to the case of the heterogeneous Helmholtz equation.
Joint work with: Maximilian Bernkopf (TU Wien), Théophile Chaumont-Frelet (Inria).
[EM11] S. Esterhazy and J.M. Melenk, On stability of discretizations of the Helmholtz equation, in: Numerical Analysis of Multiscale Problems, Graham et al., eds, Springer 2012
[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 finite element discretizations of the Helmholtz equation, SIAM J. Numer. Anal., 49:1210–1243, 2011
[MSP12] J.M. Melenk, S. Sauter, A. Parsania, Generalized DG-methods for highly indefinite Helmholtz problems, J. Sci. Comp. 57:536–581, 2013
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