An efficient iterative scheme for the highly constrained augmented Stokes problem for the numerical simulation of flows in porous media

Abstract

In this work, we present a new efficient iterative solution technique for large sparse matrix systems that are necessary in the mixed finite-element formulation for flow simulations of porous media with complex 3D architectures in a representative volume element. Augmented Stokes flow problems with the periodic boundary condition and the immersed solid body as constraints have been investigated, which fowl a class of highly constrained saddle point problems mathematically. By solving the generalized eigenvalue problem based on block reduction of the discrete systems, we investigate structures of the solution space and its subspaces and propose the exact form of the block preconditioner. The exact Schur complement using the fundamental solution has been proposed to implement the block-preconditioning problem with constraints. Additionally, the algebraic multigrid method and the diagonally scaled conjugate gradient method are applied to the preconditioning sub-block system and a Krylov subspace method (MINRES) is employed as an outer solver. We report the performance of the present solver through example problems in 2D and 3D, in comparison with the approximate Schur complement method. We show that the number of iterations to reach the convergence is independent of the problem size, which implies that the performance of the present iterative solver is close to O(N).