Adaptive space–time finite element methods for the wave equation on 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 based on the Gauss–Seidel method are developed to solve the resulting fully-discrete system equations for the interior hyperbolic equations coupled with the first-order temporal equations associated with auxiliary functions in the NRBC. Due to the local nature of wave propagation, the iterative strategy requires only a few 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 explicit multi-corrector method. An h-adaptive space–time strategy is employed based on the Zienkiewicz–Zhu spatial error estimate using the superconvergent patch recovery (SPR) technique, together with a temporal error estimate arising from the discontinuous jump between time steps of both the interior field solutions and auxiliary boundary functions. For accurate data transfer between meshes, a new enhanced interpolation (EI) method is developed and compared to standard interpolation and projection. Numerical studies of transient radiation and scattering demonstrate the accuracy, reliability and efficiency gained from the adaptive strategy.

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