Adaptive Time-Discontinuous Galerkin Methods for Acoustic Scattering in Unbounded Domains

Abstract

Comprehensive adaptive procedures with efficient solution algorithms for the time-discontinuous Galerkin space-time finite element method (DGFEM) including high-order accurate nonreflecting boundary conditions (NRBC) for unbounded wave problems are developed. Sparse multi-level iterative schemes are developed to solve the resulting system equations for the interior hyperbolic equations coupled with the first-order equations associated with auxiliary functions in the NRBC. The iterative strategy requires only a f ew iterations per time step to resolve the solution to high accuracy. Further cost savings are obtained by diagonalizing the mass and boundary damping matrices. In this case the algebraic structure decouples the diagonal block matrices giving rise to an unconditionally stable explicit iterative method. An h-adaptive space-time strategy is employed based on the superconvergent patch recovery (SPR) technique, together with a temporal error estimate arising from the discontinuous jump between time steps. For accurate data transfer (projection) between meshes, we develop a new superconvergent interpolation (SI) method. Numerical studies of transient scattering demonstrate the accuracy, reliability and efficiency gained from the adaptive strategy.