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arXiv 2201.07041math.NAcs.NA

嵌入式 Trefftz 间断 Galerkin 方法

Embedded Trefftz discontinuous Galerkin methods

Christoph Lehrenfeld, Paul Stocker

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25 pages, 14 figures, 1 table
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AI总结：

本文提出嵌入式 Trefftz 间断 Galerkin 方法，通过构造嵌入算子获得 Trefftz 型子空间而无需显式 Trefftz 函数，可处理非齐次源与变系数，并显著减少全局未知量、计算时间且提升 Helmholtz 问题精度。

摘要：

在 Trefftz 间断 Galerkin 方法中，偏微分方程采用间断形函数进行离散，这些形函数被选为逐单元位于相应微分算子的核中。我们提出一种新的变体，即嵌入式 Trefftz 间断 Galerkin 方法，它是底层间断 Galerkin 方法到 Trefftz 型子空间上的 Galerkin 投影。该子空间可以用非常一般的方式描述，并且为了得到它，不必显式计算任何 Trefftz 函数，而是构造相应的嵌入算子。在最简单的情形下，该方法恢复已有的 Trefftz 间断 Galerkin 方法。但该方法能够方便地推广到一般情形，包括非齐次源项和非常系数微分算子。我们介绍该方法，讨论实现方面，并在一组标准 PDE 问题上考察其潜力。与标准间断 Galerkin 方法相比，我们在所有考虑的情形中观察到全局耦合未知量大幅减少，从而显著缩短相应计算时间。此外，对于 Helmholtz 问题，我们甚至观察到与基于平面波的 Trefftz 间断 Galerkin 方法相似的精度提升。

英文摘要：

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.

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