Embedded Trefftz DG method for the Helmholtz equation
This work addresses numerical challenges in wave propagation problems, representing an incremental improvement in finite element methods for acoustics and electromagnetics.
The paper tackles the Helmholtz equation by developing an embedded Trefftz discontinuous Galerkin method that enforces the Trefftz property through local constraints, resulting in wavenumber-explicit stability and quasi-optimal convergence estimates under a mesh resolution condition.
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.