A Numerical Approach to Space-Time Finite Elements for the Wave Equation

TL;DR

This paper develops a space-time finite element method for the wave equation, employing continuous Galerkin discretization and advanced preconditioning techniques to enhance computational efficiency across multiple dimensions.

Contribution

It introduces a novel time decomposition preconditioning strategy that significantly improves solver performance for space-time finite element discretizations of the wave equation.

Findings

Preconditioning with time decomposition accelerates convergence.

The method is effective in 1+1, 2+1, and 3+1 dimensions.

Krylov solvers with additive Schwarz preconditioning outperform unpreconditioned approaches.

Abstract

We study a space-time finite element approach for the nonhomogeneous wave equation using a continuous time Galerkin method. We present fully implicit examples in 1+1, 2+1, and 3+1 dimensions using linear quadrilateral, hexahedral, and tesseractic elements. Krylov solvers with additive Schwarz preconditioning are used for solving the linear system. We introduce a time decomposition strategy in preconditioning which significantly improves performance when compared with unpreconditioned cases.