On the optimal discretization of the solution Poisson’s equation

Abstract

The paper studies the problem of discretizing the solution of the Poisson equation with the right hand side f  belonging to the multidimensional periodic Sobolev class. The research methodology is based on considering the problem of discretizing the solution of the Poisson equation as one of the concretizations of the general problem of optimal recovery of the operator Tf and using well known statements of approximation theory. Within the framework of this general optimal recovery problem, we first estimate from above the smallest discretization error N of the solution of the Poisson equation in the Hilbert metric using the discretization operator (l^(N) , N) constructed from a finite set of Fourier coefficients of the function f. A lower estimate, coinciding in order with the upper estimate, for the smallest error N was obtained by involving all linear functionals defined on the multidimensional Sobolev class. It should be noted that the optimal discretization operator (l^(N) , N) better approximates the solution under consideration in the Hilbert metric than any discretization operator constructed from values f at given points. Poisson’s equation is an elliptic partial differential equation and describes many physical phenomena such as electrostatic field, stationary temperature field, pressure field and velocity potential field in hydrodynamics. Therefore, the relevance of the research conducted here is beyond doubt.