## 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.