A spacetime DPG method for the wave equation in multiple dimensions

TL;DR

This paper introduces a spacetime discontinuous Petrov-Galerkin (DPG) method for solving the linear wave equation across multiple dimensions, providing theoretical analysis, error estimates, and adaptive refinement strategies.

Contribution

It develops a novel spacetime DPG formulation for the wave equation, proves its well-posedness, and demonstrates its effectiveness through numerical experiments and error estimation.

Findings

The method is well-posed and stable across various mesh types.

Error estimates are established for different mesh geometries.

Numerical results confirm the method's accuracy and adaptive capabilities.

Abstract

A spacetime discontinuous Petrov-Galerkin (DPG) method for the linear wave equation is presented. This method is based on a weak formulation that uses a broken graph space. The wellposedness of this formulation is established using a previously presented abstract framework. One of the main tasks in the verification of the conditions of this framework is proving a density result. This is done in detail for a simple domain in arbitrary dimensions. The DPG method based on the weak formulation is then studied theoretically and numerically. Error estimates and numerical results are presented for triangular, rectangular, tetrahedral, and hexahedral meshes of the spacetime domain. The potential for using the built-in error estimator of the DPG method for an adaptivity mesh refinement strategy in two and three dimensions is also presented.