Embedded Trefftz DG method for the Helmholtz equation

TL;DR

This paper introduces an embedded Trefftz discontinuous Galerkin method for the Helmholtz equation that enforces Trefftz properties via local constraints, providing stable and convergent solutions with explicit wavenumber dependence.

Contribution

It presents a novel embedded Trefftz DG approach that avoids explicit Trefftz basis functions and proves stability and convergence under explicit mesh conditions.

Findings

Wavenumber-explicit stability established.

Quasi-optimality and convergence proven.

Method avoids explicit Trefftz basis construction.

Abstract

We study an embedded Trefftz discontinuous Galerkin method for the Helmholtz equation. The method starts from a polynomial DG space and enforces the Trefftz property through local constraints, avoiding an explicit construction of Trefftz basis functions. For the global coupling we use a simple symmetric interior penalty DG bilinear form. Since the resulting formulation is not coercive, stability is proved by a $T$-coercivity argument combined with a Schatz-type duality technique. This yields wavenumber-explicit stability, quasi-optimality, and convergence estimates in standard DG norms under an explicit mesh resolution condition.