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

Abstract

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 Hölder calsses C 4,λ , 0 < λ < 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.

Nonlinear Elliptic Equation With Nonlocal Integral Boundary Condition Depending on Two Parameters

“…The values of solution of two-dimensional problem (1.1)-(1.3) in one coordinate direction are conected by nonlocal condition (1.3). This is quite often and characteristic formulation of nonlocal condition for elliptic equation in two-or multi-dimensional case [1,2,3,4,5,13,14,18].…”

“…Difference method of fourth order of accuracy for two-dimensional Laplace equation with nonlocal condition u(x, 0) = α b ξ u(x, y)dy + µ(x), 0 < x < a, 0 < y < b has been considered in [14].…”

New solvability condition of 2-d nonlocal boundary value problem for Poisson’s operator on rectangle

“…Before finishing this introduction, we note that for the NLBVP which we consider in our present paper, the most relevant references [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] from [16, p. 51-52] are included in the bibliography.…”