Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the Helmholtz equation

Abstract

We are concerned with a convergence analysis of an adaptive interior penalty discontinuous Galerkin (IPDG) method for the numerical solution of acoustic wave propagation problems as described by the Helmholtz equation. The mesh adaptivity relies on a residual-type a posteriori error estimator that not only controls the approximation error but also the consistency error caused by the nonconformity of the approach. As in the case of IPDG for standard second-order elliptic boundary-value problems, the convergence analysis is based on the reliability of the estimator, an estimator reduction property and a quasi-orthogonality result. However, in contrast to the standard case, special attention has to be paid to a proper treatment of the lower-order term in the equation containing the wave number, which is taken care of by an Aubin–Nitsche-type argument for the associated conforming finite element approximation. Numerical results are given for an interior Dirichlet problem and a screen problem, illustrating the performance of the adaptive IPDG method.