10. Robust and Efficient Multifrontal Solver for Large Discretized PDEs

This paper presents a robust structured multifrontal factorization method for large symmetric positive definite sparse matrices arising from the discretization of partial differential equations (PDEs). For PDEs such as 2D and 3D elliptic equations, the method costs roughly O(n) and O(n ^4/3) flops, respectively. The algorithm takes advantage of a low-rank property in the direct factorization of some discretized matrices. We organize the factorization with a supernodal multifrontal method after the nested dissection ordering of the matrix. Dense intermediate matrices in the factorization are approximately factorized into hierarchically semiseparable (HSS) forms, so that a data-sparse Cholesky factor is computed and is guaranteed to exist, regardless of the accuracy of the approximation. We also use an idea of rank relaxation for HSS methods so as to achieve similar performance with flexible structures in broader types of PDE. Due to the structures and the rank relaxation, the performance of the method is relatively insensitive to parameters such as frequencies and sizes of discontinuities. Our method is also much simpler than similar structured multifrontal methods, and is more generally applicable (to PDEs on irregular meshes and to general sparse matrices as a black-box direct solver). The method also has the potential to work as a robust and effective preconditioner even if the low-rank property is insignificant. We demonstrate the efficiency and effectiveness of the method with several important PDEs. Various comparisons with other similar methods are given.