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In this article, we introduce a fast and memory efficient solver for sparsematrices arising from the finite element discretization of elliptic partialdifferential equations (PDEs). We use a fast direct (but approximate)multifrontal solver as a preconditioner, and use an iterative solver to achievea desired accuracy. This approach combines the advantages of direct anditerative schemes to arrive at a fast, robust and accurate solver. We will showthat this solver is faster ($\sim$ 2x) and more memory efficient ($\sim$ 2--3x)than a conventional direct multifrontal solver. Furthermore, we willdemonstrate that the solver is both a faster and more effective preconditionerthan other preconditioners such as the incomplete LU preconditioner. Specificspeed-ups depend on the matrix size and improve as the size of the matrixincreases. The solver can be applied to both structured and unstructured meshesin a similar manner. We build on our previous work and utilize the fact thatdense frontal and update matrices, in the multifrontal algorithm, can berepresented as hierarchically off-diagonal low-rank (HODLR) matrices. Usingthis idea, we replace all large dense matrix operations in the multifrontalelimination process with $O(N)$ HODLR operations to arrive at a faster and morememory efficient solver.

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