An asymptotically optimal Schur complement reduction for the Stokes equation

Abstract

In this paper we develop an efficient Schur complement method for solving the 2D Stokes equation. As a basic algorithm, we apply a decomposition approach with respect to the trace of the pressure. The alternative stream function-vorticity reduction is also discussed. The original problem is reduced to solving the equivalent boundary (interface) equation with symmetric and positive definite operator in the appropriate trace space. We apply a mixed finite element approximation to the interface operator by $P_1$ iso $P_2/P_1$ triangular elements and prove the optimal error estimates in the presence of stabilizing bubble functions. The norm equivalences for the corresponding discrete operators are established. Then we propose an asymptotically optimal compression technique for the related stiffness matrix (in the absence of bubble functions) providing a sparse factorized approximation to the Schur complement. In this case, the algorithm is shown to have an optimal complexity of the order $O(N \log^q N)$ , q = 2 or q = 3, depending on the geometry, where N is the number of degrees of freedom on the interface. In the presence of bubble functions, our method has the complexity $O(N^2 \log N)$ arithmetical operations. The Schur complement interface equation is resolved by the PCG iterations with an optimal preconditioner.