A Fast Direct Solver for a Class of 3-D Elliptic Partial Differential Equation with Variable Coefficient

Abstract

We propose a direct solver for the three-dimensional Poisson equation with a variable coefficient, and an algorithm to directly solve the associated sparse linear systems that exploits the sparsity pattern of the coefficient matrix. Introducing some appropriate finite difference operators, we derive a second-order scheme for the solver, and then two suitable high-order compact schemes are also discussed. For a cube containing N nodes, the solver requires O(N^3/2log²N) arithmetic operations and O(NlogN) memory to store the necessary information. Its efficiency is illustrated with examples, and the numerical results are analysed.