On Error Estimates for Discretization Operators for the Solution of the Poisson Equation

Abstract

A discretization operator for the solution of the Poisson equation with the right-hand side from the Korobov class is constructed and its error is estimated in the \(L^{p} \)-metric, \(2\leq p\leq \infty \). It is proved that for \(p=2 \) the resulting error estimate for the discretization operator is order sharp on the power scale. An error in calculating the trigonometric Fourier coefficients used when constructing the discretization operator is also found. It should be noted that the obtained estimate in one case is better than previously known estimates of the errors of discretization operators constructed from the values of the right-hand side of the equation at the nodes of the modified Korobov grid and the Smolyak grid, and in the other case it coincides with them up to constants.