Preconditioning of Linear Systems Arising in Finite Element Discretizations of the Brinkman Equation

Abstract

In this work we present a preconditioner for the pressure Schur complement of the linear system, resulting from finite element discretizations of the Stokes-Brinkman equation. The work is motivated by the need to solve numerically the Stokes-Brinkman system. The particular focus are two specific applications: industrial filtration problems and vuggy subsurface flows. The first problem features complex filtering media, coupled to a free flow (Stokes) domain. In vuggy subsurface flows one has free flow inclusions of various connectivity, embedded in highly heterogeneous diffusive media. The Birnkman equation provides a new modeling path to both problems, which warrants the search for efficient methods of solving the resulting linear systems. We consider a block-preconditioning approach for the pressure Schur complement. The starting point is an Incomplete Cholesky factorization of the velocity block. Based on it, an approximate pressure Schur complement is constructed and applied using Preconditioned Conjugate Gradient (PCG). The key in this scheme is an efficient preconditioning of this approximate Schur complement. This is achieved by introducing a second approximation of the pressure Schur complement based on an incomplete back-substitution scheme, followed by a second IC factorization. Numerical examples, illustrating the efficiency of this approach are also presented.