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M. Bernkopf:
"Finite element analysis of the heterogeneous Helmholtz equation and least squares methods";
Supervisor, Reviewer: J. Melenk, S. Nicaise, S. Sauter; Institut für Analysis und Scientific Computing, 2021; oral examination: 2021-07-22.

The present thesis is concerned with three main topics. The first on beeing at least squares finite element approach for numerical discretizations of the homogeneous Helmholtz equation. We perform a wave number-explicit convergence theory for this method. Secondly, we prove optimality for a first order system least squares finite element methodd applied to second order partial differential equations focusing on minimal regularity assumptions on the data. Finally, we consider a class of time-harmonic wave propagation problems in piecewise smooth media. For these problems, a wavenumber-explicit regularity theory is performed. This in turn allows for a complete and wavenumber-explicit convergence analysis of a Galerkin method applied to our model class.

Helmholtz equation; hp-version of finite element method; least squares method

http://dx.doi.org/10.34726/hss.2021.90831