[Paper Review] Trefftz discontinuous Galerkin methods on unstructured meshes for the wave equation

Andrea Moiola|arXiv (Cornell University)|May 1, 2015

Advanced Numerical Methods in Computational Mathematics9 references3 citations

TL;DR

This paper presents a space–time Trefftz discontinuous Galerkin method for the wave equation on unstructured meshes with arbitrary face orientations. It proves well-posedness, stability, and optimal a priori error bounds in both DG and $L^2$ norms, with convergence rates enhanced by Trefftz basis functions that satisfy the wave equation elementwise, enabling high-order accuracy and efficient local solution computation via a semi-explicit scheme.

ABSTRACT

We describe and analyse a space-time Trefftz discontinuous Galerkin method for the wave equation. The method is defined for unstructured meshes whose internal faces need not be aligned to the space-time axes. We show that the scheme is well-posed and dissipative, and we prove a priori error bounds for general Trefftz discrete spaces. A concrete discretisation can be obtained using piecewise polynomials that satisfy the wave equation elementwise.

Motivation & Objective

• To develop a stable and convergent space–time discontinuous Galerkin method for the wave equation on unstructured meshes with non-aligned space–time faces.
• To extend existing Trefftz DG methods to higher dimensions and more general meshes, including those with space-like and time-like faces.
• To prove a priori error bounds in both DG and $L^2$ norms for general Trefftz discrete spaces.
• To demonstrate that the method can be computed as a semi-explicit scheme using the tent-pitching algorithm, enabling local, sequential solution updates.

Proposed method

• The method uses a first-order system formulation of the wave equation with variables $v$ (time derivative of displacement) and ${\boldsymbol{\sigma}}$ (gradient of displacement).
• A Trefftz discrete space $\mathbf{V}(\mathcal{T}_h)$ is defined such that all functions satisfy the wave equation locally in each element $K \in \mathcal{T}_h$.
• The weak formulation employs numerical fluxes on mesh faces, with stabilization via penalty terms on Dirichlet, Neumann, and Robin boundaries.
• The scheme is shown to be well-posed and dissipative by energy estimates and duality arguments, ensuring stability and convergence.
• For a class of meshes satisfying the space-like face condition, the method admits a semi-explicit, element-by-element solution strategy using the tent-pitching algorithm.
• Polynomial Trefftz spaces are constructed using polynomial waves of the form $P_{\ell,j}({\mathbf{x}},t) = ({\mathbf{x}} \cdot {\mathbf{d}}_j - ct)^\ell$, ensuring local satisfaction of the wave equation.

Experimental results

Research questions

• RQ1 Can a Trefftz discontinuous Galerkin method be formulated and analyzed for the wave equation on unstructured space–time meshes with arbitrary face orientations?
• RQ2 Is the proposed Trefftz-DG scheme well-posed and stable under general mesh conditions, including space-like and time-like faces?
• RQ3 What a priori error bounds can be derived in the DG and $L^2$ norms for general Trefftz discrete spaces?
• RQ4 Under what mesh conditions can the Trefftz-DG method be computed as a semi-explicit scheme with local, sequential updates?
• RQ5 How do the convergence rates of the Trefftz-DG method compare to standard DG methods in terms of degrees of freedom and polynomial degree?

Key findings

• The Trefftz-DG method is well-posed and dissipative for any Trefftz discrete space, with stability proven via energy and duality arguments.
• A priori error bounds are established in the DG norm, with the bound depending on the approximation properties of the Trefftz space and mesh regularity.
• For meshes with only space-like internal faces (e.g., constructed via the tent-pitching algorithm), the method admits a semi-explicit, locally computable scheme with sequential element-by-element solution updates.
• The $L^2$ norm of the error is bounded by a constant times the $L^2$ norms of the data and residual, under a sufficient condition verified for tent-pitched meshes.
• Polynomial Trefftz spaces of degree $p$ have dimension $\mathcal{O}(p^n)$, significantly lower than full polynomial spaces ($\mathcal{O}(p^{n+1})$), enabling higher efficiency.
• For analytic solutions, exponential convergence in $p$ is expected, and for $n=1$, optimal algebraic convergence rates in $h_K$ and $p$ are proven, with higher efficiency than standard DG methods.