优化反应扩散方程 DG 离散的两级方法

M. Gander, José Pablo Lucero Lorca
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引用次数: 0

摘要

本手稿分析了应用于奇异扰动反应扩散方程的对称内部惩罚非连续 Galerkin 有限元离散化的两级方法。 Hemker 等人曾针对泊松问题对此类方法进行过数值分析。 这项工作的主要创新之处在于,获得了一维泊松问题两级方法最优松弛参数的明确公式,以及反应扩散情况下所有状态下最优选择的非常精确的闭式近似公式。 利用局部傅里叶分析(在矩阵级上进行,使线性代数界更容易理解),结果表明,对于实际中使用的 DG 惩罚参数值,最好使用施瓦茨类型的单元块-贾科比平滑器,而与之相反的是,早先的结果表明,仅基于平滑分析,点块-贾科比平滑器更可取。 分析还揭示了迭代求解器的性能如何取决于 DG 惩罚参数,以及应该选择什么值才能获得最快的迭代求解器,从而在 DG 离散化和迭代求解器性能之间建立了新的直接联系。 数值实验和比较表明,所获得的表达式适用于更高维度和更一般的几何形状。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Optimization of two-level methods for DG discretizations of reaction-diffusion equations
In this manuscript, two-level methods applied to a symmetric   interior penalty discontinuous Galerkin finite element discretization   of a singularly perturbed reaction-diffusion equation are analyzed.   Previous analyses of such methods have been performed numerically by   Hemker et al. for the Poisson problem.   The main innovation in this work is that explicit formulas for the   optimal relaxation parameter of the two-level method for the Poisson   problem in 1D are obtained, as well as very accurate closed form   approximation formulas for the optimal choice in the   reaction-diffusion case in all regimes.   Using Local Fourier Analysis, performed at the matrix level to make   it more accessible to the linear algebra community, it is shown that   for DG penalization parameter values used in practice, it is better to   use cell block-Jacobi smoothers of Schwarz type, in contrast to   earlier results suggesting that point block-Jacobi smoothers   are preferable, based on a smoothing analysis alone.   The analysis also reveals how the performance of the iterative   solver depends on the DG penalization parameter, and what value should   be chosen to get the fastest iterative solver, providing a new, direct   link between DG discretization and iterative solver performance.   Numerical experiments and comparisons show the applicability of the   expressions obtained in higher dimensions and more general geometries.
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