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

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.

Figures (6)

• Figure 1: Indices ${{\bm{k}}}$ in the $(k_1,k_2)$-plane in the case $d=2$, $m=2$ and $p=6$. Each black dot $\bullet$ corresponds to the coefficient $a_{k _1, k _2}$ of the monomial expansion \ref{['lincombmon']} of $v$. The indices ${{\bm{k}}}$ with $k _1\in\{0,1\}$ are highlighted in the shaded yellow area; the corresponding coefficients are determined by the Cauchy data $\psi_0,\psi_1$ of $v$. Left panel: the indices highlighted in the shaded blue area correspond to the coefficients appearing in formula \ref{['recursiveformula']} for computing $a_{{{\bm{i}}}+2{{\bm{e}}}_1}$, with ${{\bm{i}}}=(2,2)$, identified by the dot surrounded by the red circle . Right panel: illustration of the index ordering in Algorithm \ref{['Algo:general']}. All coefficients with indices located in the non-shaded region are computed with formula \ref{['recursiveformula']} in a double loop: first across diagonals $\nearrow$, and then along each diagonal $\searrow$. The ordering is shown by the magenta arrows .
• Figure 2: Error norms for the non-homogeneous problem in the unit cube, with the right-hand side and coefficients chosen to manufacture the solution given in \ref{['eq:ex2']}. We compare the quasi-Trefftz method (${\mathbb{Q\!T}}^p$) to the standard DG method using the full polynomial spaces ($\mathbb{P}^p$) for polynomial degrees $p=2,3,4$ on the same mesh sequence. Reference lines for the optimal convergence rates $\mathcal{O}(h^{p+1})$ and $\mathcal{O}(h^p)$ are shown in full and dashed lines, respectively.
• Figure 3: Left: $p$-convergence comparison between quasi-Trefftz and full polynomials DG in terms of degrees of freedom and computational time for the problem with coefficients \ref{['ex2']} using $h = 0.1$. Right: Condition numbers of the quasi-Trefftz DG (solid lines) and the standard DG (dashed lines) matrices for the Dirichlet problem on the unit square stated in \ref{['sec:cond']}; the numbers in the yellow markers show the algebraic rate in $h$ of the corresponding segment.
• Figure 4: Numerical result for the advection-dominated problem of section \ref{['layer']}. The first row shows results for the full polynomial space and the the second for the quasi-Trefftz space. From the first to the last column we vary the diffusion coefficient $\nu=10^{-j}$ for $j=1,2,3,4$.
• Figure 5: Error dependence on the diffusion parameter $\nu$ for the advection-dominated problem of section \ref{['layer']}. The numbers in the yellow boxes are the empirical rates.

Theorems & Definitions (24)

• Definition 2.1: Quasi-Trefftz space
• Remark 2.2: Non nested spaces
• Remark 2.3: Constant-coefficients: Trefftz and quasi-Trefftz spaces
• Theorem 2.4
• proof
• Proposition 2.5
• proof
• Remark 2.6: Computational cost
• Proposition 2.7
• proof