Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems

TL;DR

This paper introduces polynomial quasi-Trefftz discontinuous Galerkin methods for variable-coefficient elliptic PDEs, achieving high-order accuracy by using elementwise approximate solutions, and demonstrates their stability and effectiveness through numerical experiments.

Contribution

It develops a new polynomial quasi-Trefftz DG framework for elliptic PDEs with smooth coefficients, including basis construction and stability analysis, enhancing accuracy over standard methods.

Findings

High-order convergence demonstrated in numerical tests.

Method outperforms standard DG schemes in accuracy for similar degrees of freedom.

Effective in both diffusion- and advection-dominated regimes.

Abstract

Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying PDE. Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous, and have piecewise-constant coefficients. However, if the equation has variable coefficients, exact solutions are generally unavailable. Quasi-Trefftz methods overcome this limitation relying on elementwise "approximate solutions" of the PDE, in the sense of Taylor polynomials. We define polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients and source term, describe their approximation properties and, under a non-degeneracy condition, provide a simple algorithm to compute a basis. We then focus on a quasi-Trefftz DG method for variable-coefficient elliptic diffusion-advection-reaction problems, showing stability and high-order convergence of the scheme. The main advantage over standard DG schemes is the higher accuracy for comparable numbers of degrees of freedom. For non-homogeneous problems with piecewise-smooth source term we propose to construct a local quasi-Trefftz particular solution and then solve for the difference. Numerical experiments in 2 and 3 space dimensions show the excellent properties of the method both in diffusion-dominated and advection-dominated problems.