On the High Order Convergence of the Difference Solution of Laplace’s Equation in a Rectangular Parallelepiped

Abstract

The boundary functions ϕ_{j} of the Dirichlet problem, on the faces Γ_{j}, j=1,2,...,6 of the parallelepiped R are supposed to have seventh derivatives satisfying the Hölder condition and on the edges their second‚ fourth and sixth order derivatives satisfy the compatibility conditions which result from the Laplace equation. For the error u_{h}-u of the approximate solution u_{h} at each grid point (x₁,x₂,x₃), a pointwise estmation O(ρh⁶) is obtained, where ρ=ρ(x₁,x₂,x₃) is the distance from the current grid point to the boundary of R; h is the grid step. The solution of difference problems constructed for the approximate values of the first and pure second derivatives converge with orders O(h⁶|ln h|) and O(h^{5+λ}), 0<λ<1, respectivly.