Numerical Modeling of Solutions of the Applied Problems for the Poisson’s Equation

Abstract

Many problems of continuum physics, applied physics, electrostatics, heat engineering, and structural mechanics are described by boundary value problems for the Poisson equation. The article is devoted to the construction of a numerical algorithm for solving plane boundary value problems for the Poisson equation. The algorithm is based on the transition to a polyharmonic equation, which is solved by the method of linear boundary elements. We consider two boundary value problems with different boundary conditions: Dirichlet boundary conditions and Neumann boundary conditions. To construct a numerical solution using the linear boundary element method, the boundary of the area is replaced by an inscribed polygon. The boundary conditions are satisfied at the middle (control) points of the elements. A polyharmonic equation is reduced to a system of linear integral equations, which can be reduced to a system of linear algebraic equations. Test numerical examples confirming the effectiveness of the proposed algorithm are given.