Sparse Approximate Inverses in a Finite Element Framework
My thesis describes a method for constructing sparse approximate inverses (SPAI) of blocks of matrices arising in the numerical solution of partial differential equations (PDEs), discretized using the finite element method (FEM). The matrices have a 2 by 2 block form, which in this case is obtained via refinement of the discretization mesh and classification of the nodes as coarse and fine. The SPAI matrix is assembled from small matrices, obtained by manipulation of the local stiffness matrices; a procedure that is fully parallelizable.
The framework can be used recursively on a sequence of meshes and is suitable for constructing of algebraic Multi-Level preconditioners of block-multiplicative type.
As an illustration, numerical results of using this method to solve the Poisson problem are presented, both serial and parallel. The serial test results show that this preconditioning method is robust with respect to problem size and to discontinuity of the coefficients of the PDEs, when the areas corresponding to different coefficients are aligned with the coarse mesh. The method is not fully robust to anisotropy. Parallel tests show that the performance of the method is independent of the number of processors used to solve the problem, and that the method is highly scalable with excellent parallelization properties.