Highly heterogeneous, anisotropic coefficients, e.g. in the simulation of carbon-fibre composite components, can lead to extremely challenging finite element systems. Direct solvers for the resulting large and sparse linear systems suffer from severe memory requirements and limited parallel scalability, while iterative solvers in general lack robustness. Two-level spectral domain decomposition methods can provide such robustness for symmetric positive definite linear systems, by using coarse spaces based on independent generalized eigenproblems in the subdomains. Rigorous condition number bounds are independent of mesh size, number of subdomains, as well as coefficient contrast. However, their parallel scalability is still limited by the fact that (in order to guarantee robustness) the coarse problem is solved via a direct method. In this paper, we introduce a multilevel variant in the context of subspace correction methods and provide a general convergence theory for its robust convergence for abstract, elliptic variational problems. Assumptions of the theory are verified for conforming, as well as for discontinuous Galerkin methods applied to a scalar diffusion problem. Numerical results illustrate the performance of the method for two- and three-dimensional problems and for various discretization schemes, in the context of scalar diffusion and linear elasticity.
翻译:在模拟碳纤维复合元件时,高异性、厌异性系数,例如模拟碳纤维复合元件时,可能导致极具挑战性的有限元素系统。因此产生的大型和稀疏线性系统的直接溶剂具有严重的内存要求和有限的平行伸缩性,而迭代溶剂一般缺乏稳健性。两种水平的光谱域分解方法可以提供对正直线系统如此强健的分解性,方法是使用基于独立普遍普遍分布的分解质的粗微空间。严格条件限与网状大小、子焦素数量和系数对比无关。然而,由于(为了保证稳健,)通过直接方法解决了共性问题,因此它们的平行伸缩性仍然受到限制。在本文件中,我们从子空间校正方法的角度引入了一种多层次的变异体,并为抽象的、顺流性变异性问题提供了一种总体趋同理论。理论的假设是经过核实的,以符合网状、不连续的加热质和子变异性的方法,以及用于不同直径系统传播结果的两种直径性方法。