Optimization of Bai-Parlett-Wang's iteration method for solving saddle-point linear systems with applications to 2D Stokes flow problems

Abstract

For the two-dimensional Stokes equations we derive a saddle-point linear system to computing the velocities and pressure on nodal points. The equivalent form of the splitting iterative algorithm is expressed in terms of descent vector and residual vector, which are two basic vectors often used in the iterative algorithm. The splitting iterative algorithm is proven to be absolute convergence, if the orthogonality condition is fulfilled. An orthogonalized iterative algorithm (OIA) can be derived by preceding a stabilization factor to the descent vector. For the OIA the Jordan structure correlates the (k + 1)th step residual vector to the kth step residual vector and descent vector is explored. The convergence is happened automatically because the OIA exhibits a pull-back mechanism. By using the orthogonality condition the non-stationary parameter with optimal value per iteration is derived explicitly in Bai-Parlett-Wang's iteration method, which is able to maximally reduce the residual per step. Three splitting iterative algorithms are tested by five examples including the Stokes flow problems. Highly accurate numerical solutions with the accuracy in the order 10-14 for velocities and 10-13 for pressure are obtained by the proposed optimal Bai-Parlett-Wang's iteration method.