Higher order discretization of elliptic partial differential equations

Abstract

Higher order finite difference methods are discussed with respect to speed and accuracy when used in the solution of elliptic partial differential equations. Although fast direct methods for solving elliptic partial differential equations are currently often discussed in the literature, the methods usually lean towards using the conventional five-point differencing on a uniform rectangular mesh which gives rise to block tridiagonal and tridiagonal matrices of Toeplitz form. For the solution of large linear systems which result from the use of a finite difference formula involving more mesh-points, the matrix equation XA + AY = F is used instead of the usual composite matrix approach. Although the matrices involved become less sparse, the operation count remains 0(n[superscript 3] ) when using an n x n mesh. However, for a comparable accuracy, n is much smaller for a higher order finite difference formula than that required for a standard five-point formula.