Solution of the mixed boundary problem for the Poisson equation on two-dimensional irregular domains

Abstract

Objectives. A finite-difference computational algorithm is proposed for solving a mixed boundary-value problem for the Poisson equation given in two-dimensional irregular domains.Methods. To solve the problem, generalized curvilinear coordinates are used. The physical domain is mapped to the computational domain (unit square) in the space of generalized coordinates. The original problem is written in curvilinear coordinates and approximated on a uniform grid in the computational domain.The obtained results are mapped on non-uniform boundary-fitted difference grid in the physical domain.Results. The second order approximations of mixed Neumann-Dirichlet boundary conditions for the Poisson equation in the space of generalized curvilinear coordinate are constructed. To increase the order of Neumann condition approximations, an approximation of the Poisson equation on the boundary of the domain is used.Conclusions. To solve a mixed boundary value problem for the Poisson equation in two-dimensional irregular domains, the computational algorithm of second-order accuracy is constructed. The generalized curvilinear coordinates are used. The results of numerical experiments, which confirm the second order accuracy of the computational algorithm, are presented.

Keywords

• elliptic operator
• mixed derivatives
• generalized curvilinear coordinates
• neumann – dirichlet boundary problem
• finite-difference methods
• difference schemes