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Two-level domain decomposition preconditioners lead to fast convergence andscalability of iterative solvers. However, for highly heterogeneous problems,where the coefficient function is varying rapidly on several possiblynon-separated scales, the condition number of the preconditioned systemgenerally depends on the contrast of the coefficient function leading to adeterioration of convergence. Enhancing the methods by coarse spacesconstructed from suitable local eigenvalue problems, also denoted as adaptiveor spectral coarse spaces, restores robust, contrast-independent convergence.However, these eigenvalue problems typically rely on non-algebraic information,such that the adaptive coarse spaces cannot be constructed from the fullyassembled system matrix. In this paper, a novel algebraic adaptive coarsespace, which relies on the a-orthogonal decomposition of (local) finite element(FE) spaces into functions that solve the partial differential equation (PDE)with some trace and FE functions that are zero on the boundary, is proposed. Inparticular, the basis is constructed from eigenmodes of two types of localeigenvalue problems associated with the edges of the domain decomposition. Toapproximate functions that solve the PDE locally, we employ a transfereigenvalue problem, which has originally been proposed for the construction ofoptimal local approximation spaces for multiscale methods. In addition, we makeuse of a Dirichlet eigenvalue problem that is a slight modification of theNeumann eigenvalue problem used in the adaptive generalized Dryja-Smith-Widlund(AGDSW) coarse space. Both eigenvalue problems rely solely on local Dirichletmatrices, which can be extracted from the fully assembled system matrix. Bycombining arguments from multiscale and domain decomposition methods we derivea contrast-independent upper bound for the condition number.

A fully algebraic and robust two-level Schwarz method based on optimal local approximation spaces