A unified multilevel framework of upscaling and domain decomposition

Abstract

We consider multiscale preconditioners for a class of mass-conservative domain-decomposition (MCDD) methods. For the application of reservoir simulation, we need to solve large linear systems, arising from finite-volume discretisations of elliptic PDEs with highly variable coefficients. We introduce an algebraic framework, based on probing, for constructing mass-conservative operators on a multiple of coarse scales. These operators may further be applied as coarse spaces for additive Schwarz preconditioners. By applying different local approximations to the Schur complement system based on a careful choice of probing vectors, we show how the MCDD preconditioners can be both efficient preconditioners for iterative methods or accurate upscaling techniques for the heterogeneous elliptic problem. Our results show that the probing technique yield better approximation properties compared with the reduced boundary condition commonly applied with multiscale methods.