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Trefftz Discontinuous Galerkin discretization for the Stokes problem

Philip L. Lederer, Christoph Lehrenfeld, Paul Stocker

TL;DR

This work develops a Trefftz-DG discretization for the Stokes problem that enforces the Stokes equations inside each mesh element, producing elementwise divergence-free velocity fields and a significant reduction in degrees of freedom compared with standard DG methods. By employing an embedded Trefftz-DG framework, the method accommodates inhomogeneous body forces and provides a rigorous a priori error analysis, including a saddle-point decomposition, LBB stability, and quasi-best approximation results in energy and $L^2$ norms. Numerical experiments with exact solutions and Moffatt eddies confirm optimal convergence rates and demonstrate substantial computational savings relative to standard DG, while remaining competitive with HDG and Taylor–Hood approaches at higher orders. The framework offers a generic pathway to extend Trefftz-DG reductions to linearizations of Oseen or Navier–Stokes problems and to other PDEs with inhomogeneous sources.

Abstract

We introduce a new discretization based on the Trefftz-DG method for solving the Stokes equations. Discrete solutions of a corresponding method fulfill the Stokes equation pointwise within each element and yield element-wise divergence-free solutions. Compared to standard DG methods, a strong reduction of the degrees of freedom is achieved, especially for higher order polynomial degrees. In addition, in contrast to many other Trefftz-DG methods, our approach allows to easily incorporate inhomogeneous right hand sides (driving forces) by using the concept of the embedded Trefftz-DG method. On top of a detailed a priori error analysis, we further compare our approach to standard discontinuous Galerkin Stokes discretizations and present numerical examples.

Trefftz Discontinuous Galerkin discretization for the Stokes problem

TL;DR

This work develops a Trefftz-DG discretization for the Stokes problem that enforces the Stokes equations inside each mesh element, producing elementwise divergence-free velocity fields and a significant reduction in degrees of freedom compared with standard DG methods. By employing an embedded Trefftz-DG framework, the method accommodates inhomogeneous body forces and provides a rigorous a priori error analysis, including a saddle-point decomposition, LBB stability, and quasi-best approximation results in energy and norms. Numerical experiments with exact solutions and Moffatt eddies confirm optimal convergence rates and demonstrate substantial computational savings relative to standard DG, while remaining competitive with HDG and Taylor–Hood approaches at higher orders. The framework offers a generic pathway to extend Trefftz-DG reductions to linearizations of Oseen or Navier–Stokes problems and to other PDEs with inhomogeneous sources.

Abstract

We introduce a new discretization based on the Trefftz-DG method for solving the Stokes equations. Discrete solutions of a corresponding method fulfill the Stokes equation pointwise within each element and yield element-wise divergence-free solutions. Compared to standard DG methods, a strong reduction of the degrees of freedom is achieved, especially for higher order polynomial degrees. In addition, in contrast to many other Trefftz-DG methods, our approach allows to easily incorporate inhomogeneous right hand sides (driving forces) by using the concept of the embedded Trefftz-DG method. On top of a detailed a priori error analysis, we further compare our approach to standard discontinuous Galerkin Stokes discretizations and present numerical examples.
Paper Structure (22 sections, 12 theorems, 60 equations, 5 figures, 7 tables)

This paper contains 22 sections, 12 theorems, 60 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Trefftz-DG for Stokes
  4. The underlying DG Stokes discretization
  5. The Trefftz-DG Stokes discretization
  6. Trefftz constraints and the dimension of T
  7. Implementation via embedded Trefftz method
  8. A-priori error analysis
  9. Saddle-point structure and space decomposition
  10. Continuity and kernel-coercivity
  11. LBB-stability
  12. Inf-Sup-Stability
  13. Quasi-best approximation and Aubin-Nitsche
  14. Approximation and a-priori error bounds
  15. Numerical examples
  16. ...and 7 more sections

Key Result

Lemma 1

The pointwise Stokes operator $\mathcal{L} : [\mathcal{P}^{k}(T)]^d \times \mathcal{P}^{k-1}(T) \to [\mathcal{P}^{k-2}(T)]^d \times \mathcal{P}^{k-1}(T)$, $(v,q) \mapsto (-\Delta v + \nabla p, - \mathop{\mathrm{\mathrm{div}}}\nolimits v)$ is surjective and the local Trefftz space on an element $T\in

Figures (5)

  • Figure 1: Example basis functions of a Stokes-Trefftz space for $k=2$ on a triangle. The coloring corresponds to the pressure value while the arrows indicate the velocity. The pressure scaling is different between the first seven and the last three basis functions. Note, that these basis functions are obtained from the generic approach of the embedded Trefftz-DG method 2201.07041, cf. \ref{['sec:impl']}, and hence do not offer a complete insight into an available structure of the Trefftz space such as a clean decomposition into lower and higher order basis functions. Nevertheless, we observe that velocity and pressure functions are coupled for most basis functions except for the three lowest order Trefftz basis functions: The first two basis functions (in the upper row) have a zero pressure and are constant and linear, respectively, and divergence-free and the last (in the lower row) basis function corresponds to a zero velocity and a constant pressure.
  • Figure 2: Numerical results for the DG and Trefftz-DG method for the two dimensional example on the top row and three dimensional example on the bottom. The gray (solid and dashed) lines indicate the expected convergence rates.
  • Figure 3: Moffatt eddies with $k=10$ on the left, including a zoom on the bottom eddies on the bottom left. On the right we show the computational mesh.
  • Figure 4: Sketch of fourth order FE discretisations with different types of unknowns for velocity and pressure: unknowns that can be removed beforehand if a suitable basis is used (red), local unknowns that can be eliminated by static condensation (green) and the remaining global unknowns (blue). Note that every arrow corresponds to one scalar dof.
  • Figure 5: Comparison of ndof, ncdof and nnze for a selection of methods in \ref{['sec:numbercrunching']} for the 2D and 3D Stokes problem on the displayed mesh.

Theorems & Definitions (27)

  • Lemma 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3: Lowest order subspaces
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 17 more