An efficient approach for solving saddle point problems using block structure

Abstract

This paper focuses on saddle point problems with a 2-by-2 block coefficient matrix. When the number of columns in the upper-right block and the number of rows in the lower-left block of the coefficient matrix is large, the convergence behavior of Krylov subspace methods for the saddle point problems tends to be poor even if the upper-left block is a well-conditioned matrix. In this paper, an efficient approach for solving the saddle point problems using block structure of the problems is proposed. The most time-consuming part of our proposed approach is the solution of a linear system with multiple right-hand sides. To solve the linear system with multiple right-hand sides efficiently, we propose to apply Block Krylov subspace methods to this linear system. Numerical experiments show that the proposed approach with Block Krylov subspace methods can solve the saddle point problems more efficiently than the conventional approach in terms of the number of iterations and the computation time.