论文标题
嵌入的Trefftz不连续的Galerkin方法
Embedded Trefftz discontinuous Galerkin methods
论文作者
论文摘要
在Trefftz不连续的Galerkin方法中，使用不连续的形状函数离散了部分微分方程，这些函数被选择为相应差分操作员的内核中的元素。我们提出了一种新的变体，即嵌入的Trefftz不连续的Galerkin方法，该方法是基础不连续的Galerkin方法的Galerkin投影到Trefftz-type的子空间上。可以以非常一般的方式描述子空间，并且为其获得无需明确计算的trefftz函数，而是构建相应的嵌入操作员。在最简单的情况下，该方法恢复已建立的Trefftz不连续的Galerkin方法。但是该方法可以方便地扩展到一般情况，包括不均匀的来源和非恒定系数差分运算符。我们介绍该方法，讨论实施方面并探索其在一组标准PDE问题上的潜力。与标准不连续的盖尔金方法相比，我们观察到在所有被考虑的情况下，全球耦合未知数的严重降低，从而大大减少了相应的计算时间。此外，对于Helmholtz问题，我们甚至观察到基于平面波的不连续的Galerkin方法的提高精度。
In Trefftz discontinuous Galerkin methods a partial differential equation is discretized using discontinuous shape functions that are chosen to be elementwise in the kernel of the corresponding differential operator. We propose a new variant, the embedded Trefftz discontinuous Galerkin method, which is the Galerkin projection of an underlying discontinuous Galerkin method onto a subspace of Trefftz-type. The subspace can be described in a very general way and to obtain it no Trefftz functions have to be calculated explicitly, instead the corresponding embedding operator is constructed. In the simplest cases the method recovers established Trefftz discontinuous Galerkin methods. But the approach allows to conveniently extend to general cases, including inhomogeneous sources and non-constant coefficient differential operators. We introduce the method, discuss implementational aspects and explore its potential on a set of standard PDE problems. Compared to standard discontinuous Galerkin methods we observe a severe reduction of the globally coupled unknowns in all considered cases, reducing the corresponding computing time significantly. Moreover, for the Helmholtz problem we even observe an improved accuracy similar to Trefftz discontinuous Galerkin methods based on plane waves.
