The network method for the external dirichlet problem

Abstract

A method is put forward below for the approximate solution on a polar network of the external Dirichlet problem for the two-dimensional Laplace equation. The use of a polar network is obviously the most natural for the solution of external boundary value problems and problems in unbounded regions. Although the external Dirichlet problem, as is well-known [1], reduces to an internal Dirichlet problem and in a number of cases we shall use this property, yet we shall construct difference equations for approximate solution directly on a network situated in a given unbounded region. Using a polar network with the number of nodes of the order h −2 ln h −1 fills a circle of radius of order h −1, where h is the dimensionless step of the network along the radius. The relation between h and the step in the angle is chosen so that as h → 0 the cells of the network approach squares and for any finite h the difference equations at inner nodes are written in exactly the same way as in a square network, i.e. the required unknown is equal to the mean arithmetical value of the unknowns at the four nearest nodes. On the assumption that the solution of the Laplace equation possesses bounded second derivatives on a closed region, the error of the approximate solution is of the order h 2(1 + ¦ln h¦) . In conclusion a method is given for the construction of an auxiliary system of difference equations with free terms, expressed in terms of the known quantities, whose solution majorizes the error of the approximate solution. A similar method of evaluating the error for the case of the internal Dirichlet problem is put forward in [2, 3]. We note that in [4] the question of the solution of the external Dirichlet problem by the network method is related to a number of problematical themes. The approximate solution of this problem by the method of potentials is put forward in [5].