Scaling up through domain decomposition

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Abstract
In this article, we discuss domain decomposition parallel iterative solvers for highly heterogeneous problems of flow and transport in porous media. We are particularly interested in highly unstructured coefficient variation where standard periodic or stochastic homogenization theory is not applicable. When the smallest scale at which the coefficient varies is very small, it is often necessary to scale up the equation to a coarser grid to make the problem computationally feasible. Standard upscaling or multiscale techniques require the solution of local problems in each coarse element, leading to a computational complexity that is at least linear in the global number N of unknowns on the subgrid. Moreover, except for the periodic and the isotropic random case, a theoretical analysis of the accuracy of the upscaled solution is not yet available. Multilevel iterative methods for the original problem on the subgrid, such as multigrid or domain decomposition, lead to similar computational complexity (i.e. 𝒪(N)) and are therefore a viable alternative. However, previously no theory was available guaranteeing the robustness of these methods to large coefficient variation. We review a sequence of recent papers where simple variants of domain decomposition methods, such as overlapping Schwarz and one-level FETI, are proposed that are robust to strong coefficient variation. Moreover, we also extend the theoretical results, for the first time, to the dual-primal variant of FETI.