Overlapping Schwarz methods with GenEO coarse spaces for indefinite and nonself-adjoint problems
Overlapping Schwarz methods with GenEO coarse spaces for indefinite and nonself-adjoint problems
复制标题
将 Schwarz 方法与 GenEO 粗空间重叠，解决不定和非自伴随问题
DOI：
10.1093/imanum/drac036
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发表时间：
2023
影响因子：
2.1
通讯作者：
Bootland N
中科院分区：
文献类型：
--
作者：
Bootland N
重叠上的广义特征值问题（GenEO）是一种计算算子相关谱粗空间的方法，该方法与子域上的局部解相结合，形成椭圆偏微分方程的鲁棒并行域分解预条件。以前已经证明，在自伴随正定情况下，该方法作为共轭梯度的前置条件时，所得到的迭代数与偏微分算子的系数场的非均质性完全无关。我们将这一理论推广到非自伴不定的对流-扩散-反应问题，这些问题的离散化可以用预条件GMRES求解。本文利用基于自伴随正定子问题的广义特征值问题定义了GenEO粗空间。我们证明了对GMRES迭代计数的估计与算子中扩散项系数的变化无关，并且仅非常轻微地依赖于其他系数的变化。这些都是在子域直径足够小，构造粗糙空间的特征值容差足够大的假设下证明的。随着算子的非自伴随性和不确定性的增加，迭代估计的次数也会增加，但实际测试表明，迭代估计的退化程度要轻得多。因此，我们得到了一个并行有效的迭代求解器，对大范围的对流-扩散-反应问题非常有效。
Generalized eigenvalue problems on the overlap(GenEO) is a method for computing an operator-dependent spectral coarse space to be combined with local solves on subdomains to form a robust parallel domain decomposition preconditioner for elliptic PDEs. It has previously been proved, in the self-adjoint and positive-definite case, that this method, when used as a preconditioner for conjugate gradients, yields iteration numbers that are completely independent of the heterogeneity of the coefficient field of the partial differential operator. We extend this theory to the case of convection–diffusion–reaction problems, which may be nonself-adjoint and indefinite, and whose discretizations are solved with preconditioned GMRES. The GenEO coarse space is defined here using a generalized eigenvalue problem based on a self-adjoint and positive-definite subproblem. We prove estimates on GMRES iteration counts that are independent of the variation of the coefficient of the diffusion term in the operator and depend only very mildly on variations of the other coefficients. These are proved under the assumption that the subdomain diameter is sufficiently small and the eigenvalue tolerance for building the coarse space is sufficiently large. While the iteration number estimates do grow as the nonself-adjointness and indefiniteness of the operator increases, practical tests indicate the deterioration is much milder. Thus, we obtain an iterative solver that is efficient in parallel and very effective for a wide range of convection–diffusion–reaction problems.