An asymptotically optimal Schur complement reduction for the Stokes equation

In this paper we develop an efficient Schur complement method for solving t he 2D Stokes equation. As a basic algorithm, we apply a decomposition appro ach with respect to the trace of the pressure. The alternative stream funct ion-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 o f stabilizing bubble functions. The norm equivalences for the corresponding discrete operators are established. Then we propose an asymptotically opti mal compression technique for the related stiffness matrix tin the absence of bubble functions) providing a sparse factorized approximation to the Sch ur complement. In this case, the algorithm is shown to have an optimal comp lexity of the order O(N log(q) N), q = 2 or q = 3, depending on the geometr y, where N is the number of degrees of freedom on the interface. In the pre sence of bubble functions, our method has the complexity O(N-2 logN) arithm etical operations. The Schur complement interface equation is resolved by t he PCG iterations with an optimal preconditioner.