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

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.

💡 Research Summary

This paper introduces a high‑order discontinuous Galerkin (DG) method for linear elliptic diffusion‑advection‑reaction equations with smooth, spatially varying coefficients. The key novelty is the definition of a polynomial quasi‑Trefftz space (Q_T^p_f(E)) on each mesh element (E). For a linear differential operator (M=\sum_{|j|\le m}\alpha_j D^j) of order (m) and a source term (f), the space consists of all polynomials of degree ≤ p whose residual (M v-f) and all its derivatives up to order (p-m) vanish at a chosen point (x_E\in E). Under the regularity assumption (\alpha_j,f\in C^{p-m}(E)) and (p\ge m), the authors prove (Theorem 2.4) that the Taylor polynomial of the exact solution belongs to (Q_T^p_f(E)) and that the best‑approximation error in any (C^q) seminorm decays like (h^{p+1-q}). Hence the quasi‑Trefftz space delivers the same convergence order as the full polynomial space (P_p(E)) while having far fewer degrees of freedom.

A non‑degeneracy condition (\alpha_{(m,0,\dots,0)}(x_E)\neq0) guarantees that the dimension of (Q_T^p_0(E)) is dramatically reduced (formula (14)). The authors present Algorithm 1, a recursive procedure that computes the monomial coefficients of a basis of (Q_T^p_f(E)) using only the values and derivatives of the coefficients (\alpha_j) and the source term at the point (x_E). This algorithm is embarrassingly parallel, incurs modest pre‑processing cost, and works for any polynomial degree.

The DG formulation uses the symmetric interior penalty for the diffusion term and an upwind penalty for the advection term. Because the trial space is affine when (f\neq0), the authors first compute an element‑wise particular quasi‑Trefftz polynomial (u_h^) (via Algorithm 1) and then solve for the correction (w_h = u_h-u_h^) in the homogeneous quasi‑Trefftz space. This retains linearity of the global system while exactly incorporating the source term.

Stability and optimal error estimates are obtained by adapting standard DG analysis to the quasi‑Trefftz setting. The consistency error vanishes for the particular polynomial, and the coercivity follows from the penalty terms. Consequently, for smooth solutions the method achieves (O(h^{p})) convergence in the energy norm and (O(h^{p+1})) in the (L^2) norm, with the same rates holding under (p)-refinement (spectral convergence) when the solution is analytic.

Numerical experiments in two and three dimensions confirm the theory. Using polygonal (polytopal) meshes, the authors compare the quasi‑Trefftz DG with a standard DG that employs the full polynomial space of the same degree. The quasi‑Trefftz version attains identical error levels while using 30–50 % fewer degrees of freedom, leading to lower assembly and solve times (Table 2). Both diffusion‑dominated and advection‑dominated regimes are tested; the method captures sharp layers and respects the directionality of the flow without loss of stability.

The implementation is built on the open‑source NGSolve library, and the code is made publicly available, facilitating reproducibility. The paper also notes that the presented framework extends naturally to other linear PDEs with smooth coefficients, and that the quasi‑Trefftz concept can be combined with alternative stable DG formulations.

In summary, the authors provide a rigorous and practical DG scheme that leverages polynomial quasi‑Trefftz spaces to reduce the number of unknowns while preserving high‑order accuracy for variable‑coefficient elliptic problems. The combination of a simple basis construction algorithm, solid theoretical guarantees, and demonstrated computational efficiency makes this approach a valuable addition to the toolbox of high‑order finite element methods for complex, heterogeneous media.