Schur preconditioning of the Stokes equations in channel-dominated domains

Abstract

A novel precondition operator for the Schur complement of the stationary Stokes equations is proposed. Numerical experiments demonstrate that its discrete version is superior to established precondition operators if the computational domain is a thin channel or contains such (e.g. porous media or filters). Discrete diffusion is added to the established precondition operator, which is heuristically motivated by Darcy’s law—a homogenization result of Stokes’ equations on perforated domains. As proof of concept, we embed the operator into a block-diagonal precondition matrix for MINRES and solve the stationary Stokes saddle-point system. For domains with complex geometries, the number of iterations required to meet a certain tolerance is significantly reduced. The proposed operator can easily be incorporated into an already existing numerical solver, does not change the sparsity pattern of the established precondition operator for finite element discretizations, and is in particular suitable for distributed parallel implementations.