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 in [11] 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. The dependence on the distribution of the plane wave directions has been studied in [1] based on a least squares method. However, the emphasis in [1] has been on the h-version of the PWDG method, i.e., decreasing the mesh width h for a fixed number p of plane wave directions. In this contribution, we are interested in the p-version, i.e., increasing p for a fixed mesh-width h. We formulate the choice of the plane wave directions as a control constrained optimal control problem with a continuously differentiable objective functional and the variational formulation of the PWDG method as a further 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.