Investigation of a Mixed Boundary Value Problem for the Poisson Equation

The mixed boundary value problem for the Poisson equation is considered. The zero extension of the problem to be solved is applied across the boundary with the Dirichlet condition. Problems are formulated on plane, bounded domains. The indicated problems are given in variational forms. The considered problem is extended in a variational form. In order to solve the extended problem, a modified method of fictitious components is used. For this iterative process, estimates of the convergence of analytical approximations to the exact solution are indicated. Discretization of the extended problem by the finite element method is carried out. An iterative process for solving the extended problem on a finite-dimensional subspace is indicated. Convergence estimates for the iterative method on a finite-dimensional subspace are given. After discretization of the extended problem, we formulate the problem at the matrix level. In order to solve the problem at the matrix level, the method of fictitious components is used. Convergence estimates for the method of fictitious components at the matrix level are obtained. Unlike relative error estimates, absolute error estimates are independent on sampling parameters. Also, when continuing through the boundary with the Dirichlet condition, the method of fictitious components is asymptotically optimal for obtaining a solution with a permissible absolute error. We formulate the programmable algorithm for numerically solving the computer problem in question.