Error analysis when solving Laplace's equation numerically by iteration

摘要

Numerical solution methods of Laplace's equation /spl Delta/V=0 when boundary values of potential V are specified abound, and many computer programs employing relaxation techniques, finite-element techniques, etc. have been discussed in the literature. The finite-mesh relaxation method of numerical iteration is discussed in many physics and electrical engineering texts, but little attention is given to the error analysis, which, moreover, is incorrect more often than not in these texts. The authors show that the error in the iterated solution can be found by a relatively simple analysis, and discuss its implications. The authors illustrate the problem by using a very simple PC program that solves the two-dimensional Laplace's equation with Dirichlet conditions on a rectangular boundary.