ON SOLVING NON-SYMMETRIC SADDLE-POINT SYSTEMS ARISING FROM FICTITIOUS DOMAIN APPROACHES∗

where the diagonal block A is an (n × n) matrix, the off-diagonal blocks B1 and B2 are (m × n) matrices with full row-rank and vectors f , g are of order n, m, respectively. Our contribution is inspired by a class of saddle-point systems arising from fictitious domain formulations of PDEs [3, 4]. Therefore we will be interested especially in systems (1) with n large, A singular and B1, B2 sparse. Moreover, we will assume that m is much smaller than n and that the defect l of A, i.e. l = n − rank A, is much smaller than m. There are several basic approaches used for solving (1); see e.g. [1]. Due to the structure of our matrices, we pay our attention to the class of methods that are based on the Schur complement reduction. Their key idea consists in eliminating the first unknown u. This leads, in the case of non-singular A, to the reduced system for the second unknown λ. The matrix of this system is the (negative) Schur complement −S = B2A−1B 1 . If this system is solved by an iterative method, we do not need to form S explicitly since only the matrix-vector products with S are needed. The situation is not so easy if A is singular. In this case, the first unknown u can not be completely eliminated from (1). The Schur complement reduction leads now to another saddle-point system for λ and a new unknown, say α, that corresponds to the null-space of A. Fortunately after applying orthogonal projectors, we obtain an equation only in terms of λ. As our original saddle-point system (1) is non-symmetric, this equation can be solved by a projected Krylov method for non-symmetric matrices. In our numerical tests, we will use the projected variant of the BiCGSTAB algorithm. The presented method generalizes ideas used in the algebraic description of FETI domain decomposition methods [2], in which A is symmetric, positive semidefinite and B1 = B2. ∗Supported by the National Program of Research ”Information Society” under project 1ET400300415 and by the grant IAA1075402 of the Grant Agency of the Czech Academy of Sciences.