Robust Structured Multifrontal Factorization and Preconditioning for Discretized Pdes

We present an approximate structured factorization method which is efficient, robust, and also relatively insensitive to ill conditioning, high frequencies, or wavenumbers for some discretized PDEs. Given a sparse symmetric positive definite discretized matrix A, we compute a structured approximate factorization A ≈ LLT with a desired accuracy, where L is lower triangular and data sparse. This can be used in direct solution or preconditioning for linear systems. The method uses the idea that during the direct factorization of some discretized matrices, certain dense intermediate matrices have a low-rank property, or, their off-diagonal blocks can be approximated by compact low-rank matrices. In this paper, we organize the factorization with a supernodal version multifrontal method using nested dissection ordering of the matrix. Each dense intermediate matrix is formed explicitly and then partially factorized so that the leading factor is a hierarchically semiseparable matrix and the Schur complement remains dense. The use of explicit dense matrices makes the method much simpler than existing structured factorization methods. The overall factor remains structured and can be used to solve systems in nearly linear complexity. The factorization algorithm costs O(rn log2 n), where n is the matrix size, and r is a parameter related to the tolerance and the problem. Schur complements are automatically compensated during the factorization so that LLT always exists for any accuracy and has enhanced positive definiteness. No extra stablization is needed. The method also works well as a preconditioner even if the low-rank property is not highly significant. We demonstrate the reliability and effectiveness of the method with various applications, including elliptic problems, linear elasticity equations, Helmholtz equations, Maxwell equations, etc.