We describe and analyze a quasi-Trefftz DG method for solving boundary value problems for the homogeneous diffusion-advection-reaction equation with piecewise-smooth coefficients. Trefftz schemes are high-order Galerkin methods whose discrete functions are elementwise exact solutions of the underlying PDE. Trefftz basis functions can be computed for many PDEs that are linear, homogeneous and with piecewise-constant coefficients. However, if the equation has varying coefficients, in general, exact solutions are unavailable, hence the construction of discrete Trefftz spaces is impossible. Quasi-Trefftz methods have been introduced to overcome this limitation, relying on discrete spaces of functions that are elementwise "approximate solutions" of the PDE. A space-time quasi-Trefftz DG method for the acoustic wave equation with smoothly varying coefficients has recently been studied; since it has shown excellent results, we propose a related method that can be applied to second-order elliptic equations. The DG weak formulation is derived using an interior penalty parameter and the upwind numerical fluxes. We choose polynomial quasi-Trefftz basis functions, whose coefficients can be computed with a simple algorithm based on the Taylor expansion of the PDE's coefficients. The main advantage of Trefftz and quasi-Trefftz schemes over more classical ones is the higher accuracy for comparable numbers of degrees of freedom. We prove that the dimension of the quasi-Trefftz space is smaller than the dimension of the full polynomial space of the same degree and that yields the same optimal convergence rates. The quasi-Trefftz DG method is well-posed, consistent and stable and we prove its high-order convergence. We present some numerical experiments in two dimensions that show excellent properties in terms of approximation and convergence rate.

翻译：本文描述并分析了一种用于求解具有分段光滑系数的均匀扩散-对流-反应方程边值问题的准Trefftz DG方法。Trefftz格式是一类高阶Galerkin方法，其离散函数在单元上精确满足底层偏微分方程。对于许多线性、均匀且具有分段常数系数的偏微分方程，可以计算Trefftz基函数。然而，若方程具有变系数，通常无法获得精确解，因此无法构建离散Trefftz空间。为克服此局限性，研究者引入了准Trefftz方法，其依赖的函数离散空间在单元上满足偏微分方程的“近似解”。近期已有研究针对光滑变系数声波方程提出时空准Trefftz DG方法；鉴于其优异表现，我们提出一种可应用于二阶椭圆型方程的相关方法。DG弱形式通过内部罚参数和迎风数值通量导出。我们选择多项式准Trefftz基函数，其系数可通过基于偏微分方程系数泰勒展开的简单算法计算。与经典方法相比，Trefftz和准Trefftz格式的主要优势在于相同自由度数量下具有更高精度。我们证明了准Trefftz空间的维数小于同阶完全多项式空间的维数，且两者可达到相同的最优收敛速率。该准Trefftz DG方法具有适定性、相容性和稳定性，并证明了其高阶收敛性。我们展示了二维数值实验，在逼近精度和收敛速率方面均表现出优异特性。