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

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