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Embedded Trefftz DG framework for the analysis of discretizations with local-global decompositions

Philip L. Lederer, Christoph Lehrenfeld, Paul Stocker, Igor Voulis

TL;DR

<3-5 sentence high-level summary> The paper develops a general local-global framework to analyze discretizations that split problems into local subproblems and a global, Trefftz-like problem, enabling rigorous error analysis for embedded Trefftz DG and quasi-Trefftz DG methods across a broad class of second-order elliptic and advection–reaction PDEs. By deriving stability and error results from the approximation properties of the full discrete space rather than the Trefftz subset alone, the authors obtain optimal convergence in standard norms and, via Aubin–Nitsche arguments, optimal bounds in weaker norms. The framework unifies and extends existing approaches (classical Trefftz, embedded Trefftz, and quasi-Trefftz) and provides practical tools and a generic recipe (SVD-based space splitting) for constructing stable embedded Trefftz discretizations. Applications to advection-reaction, diffusion-advection-reaction, and quasi-Trefftz diffusion illustrate the method’s versatility, supported by numerical examples that corroborate the theoretical findings. This approach lays a foundation for broad extensions to more complex PDEs and local-global discretizations beyond Trefftz-type methods.

Abstract

This paper presents a framework for the analysis of discretization methods based on the decomposition into local and global problems. We apply the framework to provide a comprehensive error analysis for the embedded Trefftz discontinuous Galerkin method, for a wide range of second-order scalar elliptic partial differential equations and a scalar reaction-advection problem. We also analyze quasi-Trefftz methods with our framework, presenting the first optimal error bounds in weaker norms.

Embedded Trefftz DG framework for the analysis of discretizations with local-global decompositions

TL;DR

<3-5 sentence high-level summary> The paper develops a general local-global framework to analyze discretizations that split problems into local subproblems and a global, Trefftz-like problem, enabling rigorous error analysis for embedded Trefftz DG and quasi-Trefftz DG methods across a broad class of second-order elliptic and advection–reaction PDEs. By deriving stability and error results from the approximation properties of the full discrete space rather than the Trefftz subset alone, the authors obtain optimal convergence in standard norms and, via Aubin–Nitsche arguments, optimal bounds in weaker norms. The framework unifies and extends existing approaches (classical Trefftz, embedded Trefftz, and quasi-Trefftz) and provides practical tools and a generic recipe (SVD-based space splitting) for constructing stable embedded Trefftz discretizations. Applications to advection-reaction, diffusion-advection-reaction, and quasi-Trefftz diffusion illustrate the method’s versatility, supported by numerical examples that corroborate the theoretical findings. This approach lays a foundation for broad extensions to more complex PDEs and local-global discretizations beyond Trefftz-type methods.

Abstract

This paper presents a framework for the analysis of discretization methods based on the decomposition into local and global problems. We apply the framework to provide a comprehensive error analysis for the embedded Trefftz discontinuous Galerkin method, for a wide range of second-order scalar elliptic partial differential equations and a scalar reaction-advection problem. We also analyze quasi-Trefftz methods with our framework, presenting the first optimal error bounds in weaker norms.

Paper Structure

This paper contains 37 sections, 24 theorems, 149 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Outline.
  4. Examples of methods within the framework
  5. Motivation
  6. Classical polynomial Trefftz DG
  7. Embedded Trefftz DG
  8. Quasi-Trefftz DG
  9. A new perspective on static condensation
  10. Towards a local--global framework
  11. The framework for local-global discretizations
  12. Continuous problem
  13. Underlying discrete spaces and other notation
  14. Local subproblems
  15. Global problem
  16. ...and 22 more sections

Key Result

theorem 3.2

Assume that ass:localass:assumption_Thass:inexacttrefftz hold. Then, there exist maps $\mathop{\mathrm{T}}\nolimits_{{\mathcal{K}_h}}:$${\mathbb{V}_h} \to def:QTh$, and $\mathop{\mathrm{T}}\nolimits_{def:ITh}:$${\mathbb{V}_h} \to def:ITh$, so that $def:Bh(\cdot,\cdot)$ is $\mathop{\mathrm{T}}\nolimi where we made use of the unique decomposition $u_h = u_{\mathbb{T}} + u_\mathbb{L} \in def:ITh \opl

Figures (4)

  • Figure 1: Overview of the main components in the analysis of the considered framework.
  • Figure 2: Illustration of the flow fields $\beta$ and its approximation $\bar{\beta}$ in a triangle and the in-circle $B(K)$ as well as the hyperplane $\Gamma_{K,\beta}$ orthogonal to $\bar{\beta}$ (left). After translation, rotation and rescaling the configuration on an arbitrary element $K$ is mapped to a reference configuration (right).
  • Figure 3: Convergence of the Trefftz DG method for the problem \ref{['eq:arpde']} with exact solution \ref{['eq:exar']}. The left plot shows the $L^2$-error and the right plot the ${\mathbb{V}_h}$-error. We compare the Trefftz DG method with the standard DG method, plotted with dashed lines. The black lines indicate the expected convergence rates.
  • Figure 4: Convergence of the Trefftz DG method for the problem \ref{['eq:darpde']} with exact solution \ref{['eq:darex']}. The left plot shows the $L^2$-error and the right plot the ${\mathbb{V}_h}$-error. We compare the Trefftz DG method with the standard DG method, plotted with dashed lines. The black lines indicate the expected convergence rates.

Theorems & Definitions (52)

  • remark 2.1: Plane wave DG
  • remark 3.1
  • theorem 3.2
  • proof
  • corollary 3.3: Strang-type result
  • proof
  • remark 3.4
  • theorem 3.5
  • proof
  • lemma 3.6
  • ...and 42 more