Maximal using of specifics of some boundary problems in potential theory after their numerical analysis

Abstract

Some typical problems in the numerical analysis of certain types of boundary value problems of the potential theory in substantially spatial formulation are considered. On the basis of the integral equation method (IE) an approximate scheme of solving one model example is built and investigated. It is also considered that the doubly connected open surface where boundary conditions are set obtains the Abelian group of symmetry of the eighth order. This article shows how using the apparatus of the group theory it is possible to solve an initial problem by the help of the sequence of the eight independent IEs, where the integration is realized only on one of the congruent constituents of the surface. It creates the conditions for two parallel processes of problem solution in general. The collocation method for obtaining approximate values of needed “density of charge distribution” in the particular two-dimensional integral equations is used. To take into account the singular way of solving the problem in the circuit of the open surface the a posteriori method of error evaluation is created and the procedure of integrating clarification of solving the task in the mesh node is implemented. To prove the reliability and estimation of the technique efficiency the number of numerical experiments is carried out including the use of so called “plane” approximation of the examined spatial problem.