Convergence Analysis of an Adaptive Interior Penalty Discontinuous Galerkin Method for the Helmholtz Equation

In this thesis, we are mainly concerned with the numerical solution of the two dimensional Helmholtz equation by an adaptive Interior Penalty Discontinuous Galerkin (IPDG) method based on adaptively refined simplicial triangulations of the computational domain. The a posteriori error analysis involves a residual type error estimator consisting of element and edge residuals and a consistency error which, however, can be controlled by the estimator. The refinement is taken care of by the standard bulk criterion (Dörfler marking) known from the convergence analysis of adaptive finite element methods for linear second-order elliptic PDEs. The main result is a contraction property for a weighted sum of the energy norm of the error and the estimator which yields convergence of the adaptive IPDG approach. Numerical results are given that illustrate the quasi-optimality of the method.