On the Computational Aspects of Implementing Difference Diagrams for Solving a Dirichlet Problem for an Elliptic Mixed Derivation Equation

The most used in computational practice in solving elliptic boundary value problems are the finite difference method [1, 2]. This method is not only an effective numerical method for finding approximate solutions of various problems to partial differential equations, but also a very general and relatively simple method for evidence existence theorems and research the differential properties of solutions of these problems. As it known, the finite differences method essence consists of the following. The domain of continuous variation of arguments of the original problem is replaced by a discrete set of points (nodes), which is called a grid; derivatives in the differential equation and the boundary conditions are approximated by difference relations; herewith, the boundary value problem for the differential equation is replaced by a system of algebraic equations (a difference scheme). In this paper, we discuss computational schemes of explicit iterative methods for solving Dirichlet difference problems of various orders of accuracy for two-dimensional elliptic differential equations with mixed derivatives. The developed algorithms software implementation was performed, and a qualitative comparative analysis of the results was carried out on the example of solving specific problems. For the elliptic boundary value problem with mixed derivatives, various iterative methods of solving the corresponding finite-difference schemes are implemented. For this, a software product was developed in the C# programming language, which implements the above-mentioned iterative grid algorithms, to conduct capacious computational experiments, to study their practical convergence, and to carry out qualitative comparative analysis using the example of solving model problems.