Periodicity Effects on the Iterative Solution of Elliptic Difference Equations

The first part of this paper considers the effect on the solution of the difference equations from the general self-adjoint elliptic second order partial differential equation of a periodicity condition in the x-direction. There are different effects according to whether n, the number of mesh lengths in this direction, is odd or even, but for Laplace’s equation the asymptotic rates of convergence of the iteration are independent of its actual value. Hence attention can be focused on the number of mesh lengths in the other direction. Results are given for small numbers of these as well as large.The periodicity condition can either be regarded as due to a repeating pattern in a plane or due to the region's being the surface of a solid of revolution. Hence the region can be regarded either as a rectangular region with a periodicity condition, or as a region with the error function vanishing on the boundary to which any theorems can be applied which do not depend on the region being simply connected. In contr...