In this article, we discuss the efficient implementation of powerful domain decomposition smoothers for multigrid methods for high order discontinuous Galerkin (DG) finite element methods. In particular, we study the inversion of matrices associated to mesh cells and to the patches around a vertex, respectively, in order to obtain fast local solvers for additive and multiplicative subspace correction methods. The effort of inverting local matrices for tensor product polynomials of degree $k$ is reduced from $\mathcal O(k^{3d})$ to $\mathcal O(dk^{d+1})$ by exploiting the separability of the differential operator and resulting low rank representation of its inverse as a prototype for more general low rank representations.

中文翻译：

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用于高阶不连续伽辽金方法的快速张量积 Schwarz 平滑器

在本文中，我们讨论了用于高阶不连续伽辽金 (DG) 有限元方法的多重网格方法的强大域分解平滑器的有效实现。特别是，我们分别研究了与网格单元和顶点周围的块相关的矩阵的求逆，以便为加法和乘法子空间校正方法获得快速的局部求解器。通过利用以下的可分性，将阶数为 $k$ 的张量积多项式的局部矩阵求逆的工作量从 $\mathcal O(k^{3d})$ 减少到 $\mathcal O(dk^{d+1})$微分算子及其逆的低秩表示作为更一般的低秩表示的原型。