Block Tridiagonal Matrix Is ​​incomplete Decomposition Preconditioner Method

The discretization of partial differential equations (PDEs) in 2D or 3D, by finite difference or finite element approximation, leads often to large sparse block-tridiagonal linear systems. Therefore, how to sovle this system fastly and efficiently is studied by many scientists. For this large linear system, direct solvers become prohibitively expensive because of the large amount of storage required. As an alternative, we ususlly consider the Krylov subspace method. In general, the convergence is not guaranted or may be extremely slow. Hence, a preconditioner is applied to this system to tansform it into a more tractable form. ILU factorization is considered a good precondition technology, but ILU factorizetion preconditioner is difficult to be computed in parallel. In [1,12,13,14], J. H. Yun has proposed new ILU factorization preconditioners for block tridiagonal H-matrices and M-matrices. These preconditioners can be computed in parallel, therefore the time to construct the preconditioners will decrease; meanwhile the number of the iterations of the Krylov subspace methods with this kind of preconditioners is smaller than with ILU(0). In this paper, to make an M-matrix as an example, this method is improved to make it need less expense but more effective. Furthermore, this method is proved that is suitable for block M-matrices and H-matrices. Theoretical propertieds of the preconditioners are compared with those of block ILU preconditioers apoposed by J. H. Yun, then some theorems are drawn. Lastly, the numerical results of the methods we have proposed are compared with the old methods to see the effectiveness of the proposed methods.