Local space-time adaptive finite element methods for the wave equation on unbounded domains

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. In the first part of this work, global temporal error estimate arising from the discontinuous jump between time steps of both the interior field solutions and auxiliary boundary functions are used to adjust the global time-step size. For accurate data transfer (projection) between meshes, a new superconvergent interpolation (SI) method based on the SPR gradients is developed. Numerical studies of transient radiation and scattering demonstrate the accuracy, reliability and efficiency gained from the adaptive strategy.; In the second part of this work, local space-time adaptive methods are developed including high-order accurate, local radiation boundary conditions (RBC). Novel space-time elements are formulated which enable local sub-time steps within global space-time slabs. Recovery based error estimates within each local space-time element are used to determine the number and size of local space-time elements within a global time step. The result is an efficient and reliable temporal h-adaptive strategy which distributes local space-time elements where needed to accurately track time-dependent waves over large distances and time. The resulting coupled interior and boundary equation system is solved using a two-level iterative GMRES method with a ILUT preconditioner. Numerical examples of time-dependent acoustic radiation and scattering problems are given which demonstrate the potential of this new local adaptive strategy.