A Trefftz Continuous Galerkin method for Helmholtz problems

TL;DR

This paper presents a new Trefftz Continuous Galerkin method for 2D Helmholtz problems using evanescent plane waves, achieving stable, spectrally accurate solutions with exponential error decay and linear degrees of freedom scaling.

Contribution

It introduces a globally conforming Trefftz space based on EPWs, enabling closed-form system assembly and improved approximation properties for Helmholtz problems.

Findings

Spectral accuracy achieved for analytic solutions.

Exponential decay of approximation error with discretization.

Linear scaling of degrees of freedom with wavenumber.

Abstract

This work introduces a novel Trefftz Continuous Galerkin (TCG) method for 2D Helmholtz problems based on evanescent plane waves (EPWs). We construct a new globally-conforming discrete space, departing from standard discontinuous Trefftz formulations, and investigate its approximation properties, providing wavenumber-explicit best-approximation error estimates. The mesh is defined by intersecting the domain with a Cartesian grid, and the basis functions are continuous in the whole computational domain, compactly supported, and can be expressed as simple linear combinations of EPWs within each element. This ensures they remain local solutions to the Helmholtz equation and allows the system matrix to be assembled in closed form for polygonal domains. The discrete space provides stable approximations with bounded coefficients and spectral accuracy for analytic Helmholtz solutions. The approximation error is proved to decay exponentially both at a fixed frequency, with respect to the discretization parameters, and along suitable sequences of increasing wavenumbers, with the number of degrees of freedom scaling linearly with the frequency. Numerical results confirm these theoretical estimates for the full Galerkin error.