A highly accurate difference method for solving the Dirichlet problem of~the~Laplace equation on a rectangular parallelepiped with~boundary~values~in~$C^{k,1}$

A. A. Dosiyev

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 24 (2024) no. 2, pp. 162-172

Résumé

A three-stage difference method for solving the Dirichlet problem of Laplace's equation on a rectangular parallelepiped is proposed and justified. In the first stage, approximate values of the sum of the pure fourth derivatives of the solution are defined on a cubic grid by the $14$-point difference operator. In the second stage, approximate values of the sum of the pure sixth derivatives of the solution are defined on a cubic grid by the simplest $6$-point difference operator. In the third stage, the system of difference equations for the sought solution is constructed again by using the $6$-point difference operator with the correction by the quantities determined in the first and the second stages. It is proved that the proposed difference solution to the Dirichlet problem converges uniformly with the order $O(h^{6}(|\ln h|+1))$, when the boundary functions on the faces are from $C^{7,1}$ and on the edges their second, fourth, and sixth derivatives satisfy the compatibility conditions, which follows from the Laplace equation. A numerical experiment is illustrated to support the analysis made.