Adaptive Coarse Spaces for the Overlapping Schwarz Method and Multiscale Elliptic Problems

In science and engineering, many problems exhibit multiscale properties, making the development of efficient algorithms to compute accurate solutions often challenging. We consider finite element discretizations of linear, second-order, elliptic partial differential equations with highly heterogeneous coefficient functions. The condition numbers of the associated linear systems of equations strongly depend on the mesh resolution and the heterogeneity of the coefficient function. To obtain scalable solvers, we employ the conjugate gradient method and additive overlapping Schwarz domain decomposition preconditioners with suitable coarse spaces. To ensure that the convergence is independent of large variations in the coefficient function, we construct adaptive coarse spaces: The domain decomposition interface is partitioned into small components, on which we solve local generalized eigenvalue problems to incorporate adaptivity into the coarse spaces. A tolerance is specified a priori to select the most effective eigenfunctions, which are subsequently extended energy-minimally to the subdomains to construct coarse functions. To obtain small coarse problems, we present several techniques that affect the generalized eigenvalue problems: the incorporation of an energy-minimizing extension, the construction of specific interface partitions, and the enforcement of additional Dirichlet boundary conditions in the energy-minimizing extensions. Additionally, we present approaches to reduce the computational cost and facilitate a parallel implementation. For all adaptive coarse spaces, we prove condition number bounds that only depend on a user-prescribed tolerance and on a constant that is independent of the typical mesh parameters and of the coefficient function. We provide supporting numerical results for diffusion and linear elasticity problems