An Approximate Grid Solution of a Nonlocal Boundary Value Problem with Integral Boundary Condition for Laplace\'s Equation

A new method for the solution of a nonlocal boundary value problem with integral boundary condition for Laplace's equation on a rectangular domain is proposed and justified. The solution of the given problem is defined as a solution of the Dirichlet problem by constructing the approximate value of the unknown boundary function on the side of the rectangle where the integral boundary condition was given. Further, the five point approximation of the Laplace operator is used on the way of finding the uniform estimation of the error of the solution which is order of OO h2 where hi s the mesh size. Numerical experiments are given to support the theoretical analysis made.


Introduction
In [1]- [5], a new constructive method that reduces the multilevel nonlocal problem for Poisson's equation to local Dirichlet problem, was given.
In this paper, the idea of the method in [5] is applied for the approximate solution of the Laplace's equation with integral boundary condition. By applying trapeizodal rule for the integral boundary condition, the problem is approximated by the multilevel nonlocal boundary value problem which is solved as the sum of two classical 5-point finite-difference Dirichlet problems. Finally, the numerical experiments are illustrated to support to obtain theoretical results.
On the convergence of difference schemes for problems with integral boundary conditions without reducing to the local problems see [6], [7] and references therein.

Nonlocal Boundary Value Problem
Let (1) be an open rectangle be its sides including the endpoints, numbered in the clockwise direction, begining with the side lying on the -axis and let be the boundary of . Then the following inequality holds where ₀ is defined in (4).
It is obivious that, Relying on (8), (9) and (10), the funtion satisfies relation On the basis of (11), by analogy with the proof of Theorem 1 and 2 in [4] the following Theorem is proved.
There is a unique function ∈ ⁰ for which relation (13) holds.

Finite Difference Approximation
We say that ∈ if has -th derivatives on satisfying the Hölder condition with exponent λ. We assume that ∈ ∈ in (2) and (3), respectively.
We define a square mesh with the mesh size ℎ is an integer, constructed with the lines ℎ ℎ Let ℎ be the set of nodes of this square grid, ℎ ∩ ℎ where is the rectangle (1), and where ℎ ∈ ℎ is an arbitrary function.
Let ℎ be a linear operator from ℎ to ℎ defined by ℎ ℎ ℎ The following inequality holds where ℎ is the trace of μ on ℎ and ℎ ∈ ℎ are the solution of the system of the equations ℎ ℎ ℎ ℎ ℎ We investigate the solution of system (17) by the the following fixed point iteration where ℎ is the -th iteration of (18).

Theorem 2.
The following inequality is true.
where ℎ is the trace of ℎ ₀ is a number defined by (4), − , is a number given in (12), is a constant independent of ℎ.
Consider the actual finite difference problem for the approximate solution of problem (7) and (8), where ℎ is computed function which approximates to . On the basis of (7) and Theorem 2, the following Theorem is proved:

Numerical Experiments
The following problem is solved: where is the exact solution, − and and By using the present method, the approximate solutions found and showed on line according to the decreasing mesh sizes ℎ. This shows that the error of approximation has order of ℎ² To attain this accuracy just 4 iterations in (19) are applied. The comparisons of the CPU times of the proposed method and the method without reducing to the local problem show the efficiency of our approach.