论文标题
优化对反应扩散方程DG离散的两级方法
Optimization of two-level methods for DG discretizations of reaction-diffusion equations
论文作者
论文摘要
我们分析并优化应用于对称内部惩罚不连续的盖尔金有限元元素离散化的两级方法，对奇异的反应 - 反应扩散方程式。先前对此类方法的分析已通过Hemker等人数值进行。 al。对于泊松问题。我们的主要创新是，我们为1D中的两级方法的最佳弛豫参数获得了明确的公式，并且在所有制度中，在反应扩散案例中，在反应扩散案例中最佳选择的非常准确的闭合形式近似公式。我们在矩阵级别进行的本地傅立叶分析使线性代数社区更容易访问它，这表明，对于在实践中使用的DG惩罚参数值，最好使用Schwarz类型的细胞块 - 雅各布人Smoothers，与早期的结果相反，这表明单独的基于平滑的分析，这表明Point Block-Jacobi Smoothorss偏爱Point Block-Jacobi Smoothers。我们的分析还揭示了迭代求解器的性能如何取决于DG惩罚参数，以及应选择哪些值以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提供了新的直接联系。我们通过数值实验和更高维度和不同几何形状的比较来说明我们的分析。
We analyze and optimize two-level methods applied to a symmetric interior penalty discontinuous Galerkin finite element discretization of a singularly perturbed reaction-diffusion equation. Previous analyses of such methods have been performed numerically by Hemker et. al. for the Poisson problem. Our main innovation is that we obtain explicit formulas for the optimal relaxation parameter of the two-level method for the Poisson problem in 1D, and very accurate closed form approximation formulas for the optimal choice in the reaction-diffusion case in all regimes. Our Local Fourier Analysis, which we perform at the matrix level to make it more accessible to the linear algebra community, shows 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. Our 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. We illustrate our analysis with numerical experiments and comparisons in higher dimensions and different geometries.
