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A new constructive method for the finite-difference solution of the Laplace equation with the integral boundary condition is proposed and justified. In this method, the approximate solution of the given problem is defined as a sequence of 9-point solutions of the local Dirichlet problems. It is proved that when the exact solution \(u(x,y)\) belongs to the Holder calsses \(C^{4,\lambda }\), \(0<\lambda <1\), on the closed solution domain, the uniform estimate of the error of the approximate solution is of order \(O(h^{4})\), where h is the mesh step. Numerical experiments are given to support analysis made.

A fourth-order accurate difference Dirichlet problem for the approximate solution of Laplace’s equation with integral boundary condition