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Embedded Trefftz DG method for the Helmholtz equation

Paul Stocker, Igor Voulis

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.

Embedded Trefftz DG method for the Helmholtz equation

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 -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.
Paper Structure (23 sections, 18 theorems, 166 equations, 4 figures)

This paper contains 23 sections, 18 theorems, 166 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Outline.
  4. Preliminaries
  5. Abstract framework
  6. Helmholtz equation
  7. Assumptions and inequalities
  8. Embedded Trefftz DG formulation
  9. Analysis of the local problem
  10. Prototype operator
  11. Properties of the local operator AK
  12. Analysis of the global problem
  13. Continuity of ah
  14. T-coercivity of ah
  15. Quasi-best approximation
  16. ...and 8 more sections

Key Result

Theorem 2.1

Consider $V_{\!\star \space h} := V_h + V$ with two norms $\left\|\cdot\right\|_{V_h}$ and ${\Vert\cdot\Vert}_{V_{\!\star \space h}}$ such that $\left\|\cdot\right\|_{V_h}\leq {\Vert\cdot\Vert}_{V_{\!\star \space h}}$ on $V_{\!\star \space h}$ and such that $\left\|\cdot\right\|_{V_h}$ and ${\Vert\c Assume that each $v_h\in V_h$ admits a fixed (linear) splitting $v_h=v_\mathbb{L}+v_\mathbb{T}$ wit

Figures (4)

  • Figure 1: Convergence of the embedded Trefftz DG method and the standard DG method for the problem with exact solution \ref{['eq:hankel']}. Theoretical rates are indicated by the lines, i.e., $\mathcal{O}(h^{p+1})$ for the $L^2$-error and $\mathcal{O}(h^{p})$ for the DG-error.
  • Figure 2: Results for the problem with exact solution \ref{['eq:sinsin']} for the embedded Trefftz DG method and the standard DG method. Left: DG-error against polynomial degree $p$. Right: DG-error against number of degrees of freedom.
  • Figure 3: Error against number of degrees of freedom per wavelength for different wave numbers $\omega$ and polynomial degrees $p$ for the problem with exact solution \ref{['eq:exp']}.
  • Figure 4: Results for the problem with exact solution \ref{['eq:varo']} with smoothly varying wave number. The straight lines indicate the expected convergence rates of order $\mathcal{O}(h^{p+1})$ for the $L^2$-error and $\mathcal{O}(h^{p})$ for the DG-error.

Theorems & Definitions (29)

  • Theorem 2.1: LLSV_ARXIV_2024
  • Lemma 2.2: see LLSV_ARXIV_2024
  • Theorem 2.3
  • Lemma 2.4: Scaled Poincaré--Friedrichs
  • Lemma 2.5: Polynomial inverse trace
  • Lemma 2.6: Polynomial inverse inequality
  • Lemma 2.7: Continuous trace
  • Lemma 2.8: Polynomial norming-set inequality on an inscribed ball
  • Lemma 3.1
  • proof
  • ...and 19 more