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A Trefftz Continuous Galerkin method for Helmholtz problems

Nicola Galante, Bruno Després, Emile Parolin

TL;DR

This work introduces a globally conforming Trefftz Continuous Galerkin method for 2D Helmholtz problems using evanescent plane waves on a Cartesian grid. By constructing edge- and node-based Trefftz spaces and combining them, the method achieves stability with κ-explicit best-approximation estimates and exponential convergence for analytic solutions, while enabling closed-form assembly on polygonal domains. The approach addresses stability and conditioning issues that plague propagative plane-wave methods by enriching the approximation space with EPWs and nodal functions, and it includes a regularized least-squares Petrov–Galerkin discretization to manage redundancy. Numerical experiments across propagative waves, corner singularities, and complex geometries confirm the theoretical predictions and demonstrate high accuracy with linear scaling of degrees of freedom with frequency. The framework offers a principled path toward efficient high-frequency Helmholtz solvers and lays groundwork for extensions to Neumann/3D problems and related wave equations.

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.

A Trefftz Continuous Galerkin method for Helmholtz problems

TL;DR

This work introduces a globally conforming Trefftz Continuous Galerkin method for 2D Helmholtz problems using evanescent plane waves on a Cartesian grid. By constructing edge- and node-based Trefftz spaces and combining them, the method achieves stability with κ-explicit best-approximation estimates and exponential convergence for analytic solutions, while enabling closed-form assembly on polygonal domains. The approach addresses stability and conditioning issues that plague propagative plane-wave methods by enriching the approximation space with EPWs and nodal functions, and it includes a regularized least-squares Petrov–Galerkin discretization to manage redundancy. Numerical experiments across propagative waves, corner singularities, and complex geometries confirm the theoretical predictions and demonstrate high accuracy with linear scaling of degrees of freedom with frequency. The framework offers a principled path toward efficient high-frequency Helmholtz solvers and lays groundwork for extensions to Neumann/3D problems and related wave equations.

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.

Paper Structure

This paper contains 50 sections, 24 theorems, 173 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Plane wave-based Trefftz methods.
  3. A new conforming EPW-based Trefftz space.
  4. Main approximation results.
  5. Outline of the paper.
  6. Local edge-based Trefftz space
  7. Single-edge Helmholtz mode
  8. Fundamental properties
  9. Local approximation results
  10. Pre-asymptotic regime.
  11. Degeneracy as $\widehat{h}_2 \to 0$.
  12. High-frequency regime: $\kappa \widehat{h}_1 \to +\infty$.
  13. Low-frequency regime: $\kappa \widehat{h}_1 \to 0$.
  14. Global edge-based Trefftz space
  15. Construction of the edge-based Trefftz space
  16. ...and 35 more sections

Key Result

Lemma 2.5

The two families are Hilbert bases of $L^2(\hat{\mathbf{s}})$. Besides, the former coincides (up to normalization) with $\{ \widehat{\varphi}_n|_{\hat{\mathbf{s}}}\}_{n \in \mathbb{N}^*}$.

Figures (11)

  • Figure 1: Basis functions $\widehat{\varphi}_n$ associated with the top edge $\hat{\mathbf{s}}$ of $\widehat{K}$; $\kappa=15$, $\widehat{h}_1=\widehat{h}_2=1$.
  • Figure 2: Basis functions $\phi_{\mathbf{s},n}$ associated with an horizontal edge $\mathbf{s} \in \Sigma_h$; $\kappa=15$, $h_1=h_2=1$.
  • Figure 3: $H_\kappa^1(\Omega)$-orthogonal projection of the plane wave $u(\mathbf{x}) = e^{\imath \kappa \mathbf{d}\cdot \mathbf{x}}$ with $\mathbf{d} = (1/\sqrt{2}, 1/\sqrt{2})$; $\Omega=(0,1)^2$, wavenumber $\kappa=30$. Left: plane wave real part $\Re u$. Center and right: real part of the plane wave $H_\kappa^1(\Omega)$-projection and absolute error for $N_{\mathbf{e}} = 12$.
  • Figure 4: Basis functions $\psi_{\mathbf{p},n}$ associated with a node $\mathbf{p} \in \mathcal{N}_h$; $\kappa=15$, $h_1=h_2=1$.
  • Figure 5: A star-shaped domain $\Omega$ covered by a rectangular mesh $\mathcal{T}_h$. The red square highlights a mesh element $K \in \mathcal{T}_h$, while the darker region represents the intersection $K \cap \Omega$.
  • ...and 6 more figures

Theorems & Definitions (68)

  • Remark 2.2
  • Definition 2.3: Single-edge Helmholtz mode
  • Definition 2.4: $\kappa$-dependent Sobolev norms
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Definition 2.8
  • Lemma 2.9
  • ...and 58 more