Superfast Multifrontal Method for Structured Linear Systems of Equations

In this paper we develop a fast direct solver for discretized linear systems using the multifrontal method together with low-rank approximations. For linear systems arising from certain partial differential equations such as elliptic equations we discover that during the Gaussian elimination of the matrices with proper ordering, the fill-in has a low-rank property: all off-diagonal blocks have small numerical ranks with proper definition of off-diagonal blocks. Matrices with this low-rank property can be efficiently approximated with some semiseparable structures, called hierarchically semiseparable (HSS) representations. We reveal the above low-rank property by ordering the variables with nested dissection and eliminating them with the multifrontal method. All matrix operations in the multifrontal method are performed in HSS forms. We present efficient ways to build compact HSS structures along the elimination. Necessary HSS matrix operations for the structured multifrontal method are proposed. By taking advantage of the low-rank property the multifrontal method with HSS structures can be shown to have linear complexity and to require only linear storage. We therefore call this structured method a superfast multifrontal method. It is especially suitable for large problems, and also has natural adaptability to parallel computations and great potential to provide effective preconditioners. Numerical results demonstrate the efficiency.