An algebraic multiscale solver with local Robin boundary value problems for flows in high-contrast media

Abstract

The algebraic and semi-algebraic formulations of a new multiscale elliptic solver based on a non-overlapping domain decomposition method are presented. By algebraic we mean that all information needed to implement the proposed procedure can be extracted from the underlying fine grid finite volume linear system. In addition to the entries of this linear system, the proposed semi-algebraic procedure will also use the coefficients of the elliptic equation at subdomain boundaries. Initially we construct multiscale basis functions (or local solutions) subject to Robin boundary conditions. Although the implementation of Robin conditions in the formulation of the multiscale method uses the coefficients of the elliptic equation of interest, we modify it such that only entries of the finite volume linear system appear in the calculation of local solutions. A linear combination of local solutions gives a (discontinuous at subdomain boundaries) solution that is refined in a smoothing step. To this end we propose an algebraic scheme to remove discontinuities that needs a staggered (or dual) coarse grid. By iterating with the defect correction scheme, we construct an effective two-stage preconditioner that combines both local solutions and the smoothing step; the final result is obtained in a small number of iterations. Our focus in this work is the presentation of the algebraic and semi-algebraic formulations of the multiscale elliptic solver, which are carefully explained, and in verifying their accuracy and efficiency in the solution of challenging problems. We consider two- and three-dimensional problems with coefficients exhibiting high-contrast, anisotropic, and channelized structures that are solved to reveal the good properties of the proposed schemes.