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Quasi-Trefftz spaces for a first-order formulation of the Helmholtz equation

Lise-Marie Imbert-Gérard, Andréa Lagardère, Guillaume Sylvand, Sébastien Tordeux

TL;DR

The paper tackles high-frequency wave approximation by developing quasi-Trefftz spaces for a first-order Helmholtz system, avoiding reliance on an auxiliary scalar equation. It introduces two explicit algorithms and two SVD-based approaches to construct polynomial quasi-Trefftz bases, analyzes their computational complexity, and demonstrates (via numerical tests) high-order best-approximation properties for pressure and velocity. The decoupled-pressure formulation proves particularly effective, yielding exact first-two quasi-Trefftz properties and robust convergence for the third property, while explicit methods offer favorable computational efficiency. This work lays a foundation for efficient high-order discretizations of first-order wave systems and points toward extensions to convected Helmholtz and GPW-type spaces in broader Friedrichs-system contexts.

Abstract

This work is concerned with the development of quasi-Trefftz methods for first-order differential systems. It focuses on discrete quasi-Trefftz spaces, starting from their definition and including the construction of corresponding bases together with their computational aspect. This is the first attempt at constructing quasi-Trefftz bases for a problem governed by a first-order system without relying on an auxiliary scalar equation. A decoupling approach, with a second order scalar equation for the one unknown, is proposed here simply as a point of comparison to this new approach.

Quasi-Trefftz spaces for a first-order formulation of the Helmholtz equation

TL;DR

The paper tackles high-frequency wave approximation by developing quasi-Trefftz spaces for a first-order Helmholtz system, avoiding reliance on an auxiliary scalar equation. It introduces two explicit algorithms and two SVD-based approaches to construct polynomial quasi-Trefftz bases, analyzes their computational complexity, and demonstrates (via numerical tests) high-order best-approximation properties for pressure and velocity. The decoupled-pressure formulation proves particularly effective, yielding exact first-two quasi-Trefftz properties and robust convergence for the third property, while explicit methods offer favorable computational efficiency. This work lays a foundation for efficient high-order discretizations of first-order wave systems and points toward extensions to convected Helmholtz and GPW-type spaces in broader Friedrichs-system contexts.

Abstract

This work is concerned with the development of quasi-Trefftz methods for first-order differential systems. It focuses on discrete quasi-Trefftz spaces, starting from their definition and including the construction of corresponding bases together with their computational aspect. This is the first attempt at constructing quasi-Trefftz bases for a problem governed by a first-order system without relying on an auxiliary scalar equation. A decoupling approach, with a second order scalar equation for the one unknown, is proposed here simply as a point of comparison to this new approach.

Paper Structure

This paper contains 24 sections, 3 theorems, 61 equations, 11 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. The PDE problem and corresponding notation
  3. Polynomial spaces and corresponding notation
  4. Quasi-Trefftz properties and corresponding notation
  5. Quasi-Trefftz spaces
  6. Pressure-velocity definition
  7. Decoupled pressure
  8. Best approximation properties
  9. Two explicit algorithms
  10. From the coupled definition
  11. From the decoupled pressure
  12. Extension to 3D
  13. Quasi-Trefftz bases
  14. Constructing individual basis elements
  15. Computational complexity
  16. ...and 9 more sections

Key Result

Proposition 2.2.1

Given a point $\bm{x}_0\in\mathbb R^2$, a polynomial degree $d\in\mathbb N$ with $d\geq 2$, as well as the parameters of the Helmholtz partial differential operator $\mathcal{L}$ introduce in Section ssec:PDEnot, namely $\omega\in\mathbb R$, $\rho\in\mathbb R_+$, and $c\in\mathcal{C}^{d-2}$ at $\bm{

Figures (11)

  • Figure 1: Values of the number $s=\min\{m,\ell\}-\max\{0,m+\ell-M\}+1$ of terms in the inside sum in \ref{['eq:dsum']}, as a function of $(m,\ell)$ for an odd value and an even value of $M$.
  • Figure 2: Structure of a matrix representing the linear operator $Q^F_d$ for $d=8$.
  • Figure 3: Structure of matrices representing the linear operators $Q^S_d$ and $G_d$ for $d=8$.
  • Figure 4: Number of floating-point operations required to construct one quasi-Trefftz basis of ${\mathbb S}_d$, for degrees $d$ from 2 to 10, using the four methods (EXPL1, EXPL2, ALGE1, and ALGE2).
  • Figure 5: Representation of the computational domains, their meshes, and the corresponding variable coefficients $\omega^2/c^2$ for the two test cases.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Remark 1.3.1
  • Definition 2.1.1
  • Remark 2.1.2
  • Proposition 2.2.1
  • proof
  • Proposition 2.3.1
  • Remark 2.3.2
  • proof
  • Proposition 2.3.3
  • proof
  • ...and 2 more