Optimization of Plane Wave Directions in Plane Wave Discontinuous Galerkin Methods for the Helmholtz Equation

Recently, the use of special local test functions other than polynomials in Discontinuous Galerkin (DG) approaches has attracted a lot of attention and became known as DG-Trefftz methods. In particular, for the 2D Helmholtz equation, plane waves have been used to derive an Interior Penalty (IP) type Plane Wave DG (PWDG) method and to provide an a priori error analysis of its p-version with respect to equidistributed plane wave directions. However, the dependence on the distribution of the plane wave directions has not been studied. In this thesis, we study the dependence by formulating the choice of the directions as an optimal control problem with a tracking type objective functional and the variational formulation of the PWDG method as a constraint. The necessary optimality conditions are derived and numerically solved by a projected gradient method. Numerical results are given which illustrate the benefits of the approach.