Embedded Trefftz discontinuous Galerkin methods

TL;DR

The paper introduces an embedded Trefftz discontinuous Galerkin method that generalizes existing approaches, reduces computational complexity, and improves accuracy for PDE problems, including inhomogeneous sources and variable coefficients.

Contribution

It proposes a novel embedded Trefftz DG method that simplifies implementation and extends applicability without explicitly calculating Trefftz functions.

Findings

Significant reduction in globally coupled unknowns.

Reduced computational time compared to standard DG methods.

Improved accuracy for Helmholtz problems.

Abstract

In Trefftz discontinuous Galerkin methods a partial differential equation is discretized using discontinuous shape functions that are chosen to be elementwise in the kernel of the corresponding differential operator. We propose a new variant, the embedded Trefftz discontinuous Galerkin method, which is the Galerkin projection of an underlying discontinuous Galerkin method onto a subspace of Trefftz-type. The subspace can be described in a very general way and to obtain it no Trefftz functions have to be calculated explicitly, instead the corresponding embedding operator is constructed. In the simplest cases the method recovers established Trefftz discontinuous Galerkin methods. But the approach allows to conveniently extend to general cases, including inhomogeneous sources and non-constant coefficient differential operators. We introduce the method, discuss implementational aspects and explore its potential on a set of standard PDE problems. Compared to standard discontinuous Galerkin methods we observe a severe reduction of the globally coupled unknowns in all considered cases, reducing the corresponding computing time significantly. Moreover, for the Helmholtz problem we even observe an improved accuracy similar to Trefftz discontinuous Galerkin methods based on plane waves.