The block–grid method for the approximation of the pure second order derivatives for the solution of Laplace’s equation on a staircase polygon

摘要

The combined block–grid method is developed for the highly accurate approximation of the pure second order derivatives for the solution of Laplace’s equation on a staircase polygon. By approximating the pure derivatives with respect to one of the variables on an artificial boundary around the reentry vertices, the approximation problem of this derivative is reduced to the solution of a special Dirichlet finite difference problem on the “nonsingular” part of the polygon. For the error in the maximum norm O(hτ), τ=2,4,6 order of estimations are obtained, when the boundary functions on the boundary of the “nonsingular” part of the polygon are from the Hölder classes Cτ+2,λ,0<λ<1. Numerical experiments are illustrated to support the theoretical results.