Solution of high order compact discretized 3D elliptic partial differential equations by an accelerated multigrid method

Abstract

A robust and efficient numerical solver for three-dimensional linear elliptic partial differential equations without cross derivative terms, but variable coefficients is presented. The proposed algorithm is based on a general meshsize fourth-order compact finite-difference scheme and an accelerated geometric multigrid solver. The former not only increases the accuracy of the discretization, but also does not affect the bandwidth of the common second-order finite-difference scheme, which is relevant for distributed memory parallel implementations of the multigrid preconditioner. The acceleration technique, on the other hand, allows for a dramatic reduction of the number of cycles and smoothing steps required to achieve convergence. Furthermore, it is found that depending on the coefficients of the elliptic partial differential equation, the multigrid preconditioned solver can be the only way to obtain a reliable numerical solution or attain convergence.