Adaptive space-time finite element methods for
the wave equation on unbounded domains
Lonny L. Thompson and Dantong He
Department of Mechanical Engineering and Engineering Mechanics
Clemson University
Clemson, South Carolina 29634-0921
Computer Methods in Applied Mechanics and Engineering
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.