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.
翻译:在Trefftz不连续的 Galerkin 方法中,部分差异方程式是使用不连续的形状函数分解的,这些函数被选择在相应的差别操作员的内核中成为元素。 我们提出一个新的变体, 嵌入的特雷夫茨不连续的加勒金方法, 即Galerkin在Trefftz- 类型的子空间上对一种潜在的不连续的加列尔金方法的预测。 子空间可以用非常笼统的方式描述, 并且为了获得它, 没有特雷夫茨函数需要明确计算, 而不是构建相应的嵌入操作员。 在最简单的例子中, 方法恢复了特雷夫茨不连续的加勒金方法。 但是, 这种方法可以方便地推广到普通案例, 包括无热源和非一致的系数差操作员。 我们引入了这种方法, 讨论执行方面, 并探索其在一套标准的 PDE 问题上的潜力。 与标准的不连续的加勒金方法相比, 我们观察到在所有考虑的案例中, 全球相伴不连续的未知的未知的功能会大大减少相应的计算时间 。 此外, 我们甚至观察到了以Tretz- Galtraz- Galtraffer平流的平流方法为基础的精确度。