A Fast and Memory Efficient Sparse Solver with Applications to Finite-Element Matrices

In this article we introduce a fast iterative solver for sparse matrices arising from the finite element discretization of elliptic PDEs. The solver uses a fast direct multi-frontal solver as a preconditioner to a simple fixed point iterative scheme. This approach combines the advantages of direct and iterative schemes to arrive at a fast, robust and accurate solver. We will show that this solver is much faster and more memory efficient compared to a conventional multi-frontal solver. Furthermore, the solver can be applied to both structured and unstructured meshes in a similar manner. We build on our previous work [2] and utilize the fact that dense frontal matrices in the multi-frontal algorithm can be represented as hierarchically off-diagonal low-rank (HODLR) matrices. Using this idea, we replace all large dense matrix operations in the multi-frontal elimination process with HODLR operations to arrive at a faster and more memory efficient multi-frontal solver.