On the Approximate Solution of the Dirichlet Problem for Poisson's Equation Using A-Priori Pointwise Bounds and Continuous Approximating Functions

摘要

Pointwise bounds on the solution of the Dirichlet problem for Poisson's equations are developed. The bounds are a-priori in nature and thus are completely determined by the Dirichlet data and the geometry of the region. To obtain a solution at a point one first formally substitutes the difference between the solution function and an unspecified continuous approximating function into the original bound expression. The resultant expression represents a bound on the error between the two functions at the point. This error bound is then minimized by proper choice of the approximating function as determined by the Rayleigh-Ritz procedure. In the limit the value of the approximating function equals the value of the solution at the point in question. Error bounds, calculated by the method of Rayleigh-Ritz, as well as actual errors are determined for a number of specified test problems. In this paper this approach is evaluated on the basis of both the effort required to obtain results and the quality of the results themselves in order to ascertain its value as a general method of solution. (Author)