On the convergence rate of a difference solution of the Poisson equation with fully nonlocal constraints

Abstract. We consider the Poisson equation in a rectangular domain. Instead of the classical specification of boundary data, we impose an integral constraints on the inner stripe adjacent to boundary having the width ξ. The corresponding finite-difference scheme is constructed on a mesh, which selection does not depend on the value ξ. It is proved the unique solvability of the scheme. An a priori estimate of the discretization error is obtained with the help of energy inequality method. It is proved that the scheme is convergent with the convergence rate of order s − 1, when the exact solution belongs to the fractional Sobolev space of order s (1 < s 6 3).


Introduction
Nonlocal boundary-value problems naturally arise in the mathematical modeling of many problems of ecology, physics, and engineering, when it is impossible to determine the boundary values of the unknown function (see, e.g., [1][2][3][4][5] and the references therein).At the same time, they are a very interesting generalization of classical boundary-value problems (see, e.g., [6]).The investigation of boundary-value problems with integral conditions goes back to Cannon [7].The systematic investigation of a certain class of spatial nonlocal problems was carried out by Bitsadze and Samarskii [8].Later, for elliptic equations, were posed and analyzed nonlocal boundary-value problems of various types (see, e.g., [9][10][11][12][13][14]).
In [15], we considered the nonlocal problem for the Poisson equation, when the Dirichlet-Neumann conditions are posed on a pair of adjacent sides of a rectangle, and integral constraints l k 0 u(x) dx k = 0, k = 1, 2, were given instead of classical boundary conditions on the other pair.It is proved that corresponding difference scheme converges in the energy norm at the rate O(|h| s−1 ), when the desired solution belongs to the Sobolev c Vilnius University, 2014 space W s 2 (1 < s 3).The proof bases on procedure of obtaining convergence estimate (compatible with smoothness of the exact solution) developed by Samarskii et al. [16] (see, also [17,18]).
In this paper, we study the case, when the classical boundary conditions are completely replaced by nonlocal ones: where Ω = {(x 1 , x 2 ): 0 < x k < l k , k = 1, 2} be the rectangle; l = max{l 1 , l 2 }.We assume that the solution u of the nonlocal boundary-value problem (1), ( 2) belongs to the fractional-order Sobolev space W s 2 (Ω), s > 1.For the corresponding difference scheme, estimate of convergence similar to [15], is obtained.Besides the fact that the operator of the difference scheme is not positive definite, basic difficulties comparing with [15] are as follows: • It is not required that points with coordinates ξ k or l k − ξ k belong to the mesh, which complicates investigation; • Full disregard of classical boundary conditions complicates obtaining a priori estimates.

Finite-difference scheme and main results
Consider the following grid domains on . For the values of net function in several points, we apply the notation y ij = y(ih 1 , jh 2 ).When it does not lead to ambiguity, for simplicity, we use the notations y i = y(ih 1 , x 2 ), y j = y(x 1 , jh 2 ).
We define the finite-difference operators where r k is the unit vector on the x k axis.Let where m k is positive integer.By H we denote the set of all discrete functions v = v(x), defined on the grid ω and satisfying conditions where We need the following averaging operators for functions defined on Ω: We approximate the problem (1), (2) by the difference scheme Theorem 1.A solution of difference scheme (4) exists and is unique.
Indeed, according to the Lemma 7, the homogeneous problem Λy = 0 has only trivial solution y = 0. Therefore, the nonhomogeneous problem is uniquely solvable.
Theorem 2. Let a solution u(x) of the problem (1), (2) belong to the space W s 2 (Ω), s > 1.Then the convergence rate of the difference scheme (4) in the discrete weighted W 1 2 -norm is determined by the estimate Proof.Indeed, Hence using the relation which follows from the equality we obtain Here By applying nonlocal conditions we see that From here and (6) follows the first identity of Lemma 1.The proof of the last part leads analogously.
We define the weight functions In H, we introduce the inner product and norm as Let, in addition, Lemma 2. Let grid functions v(x), y(x) be defined on ω, and y(x) satisfy the conditions P (y) = 0, P (y) = 0, x 2 ∈ ω2 or P (y) = 0, P (y) = 0, x 1 ∈ ω1 . Then Proof.Using the summation by parts, we obtain Adding the equalities ( 7)-( 9) and applying following from the nonlocal conditions identities we verify the validity of the Lemma 2 in the case k = 1.The case k = 2 may be proved analogously.
Lemma 3. If a grid function y(x), defined on ω, satisfies the conditions P (y) = 0, P (y) = 0, x 2 ∈ ω2 or P (y) = 0, P (y) = 0, x 1 ∈ ω1 , Proof.It may be showed that the identity holds, where Let us note that, according to nonlocal conditions, In addition, the inequality follows from We can obtain analogously that Adding inequalities (11), (12) and replacing in the right-hand side From this inequality and (10) follows the validity of Lemma 3 in the case k = 1.We can consider the case k = 2 analogously.
Lemma 4. If a grid function y(x), defined on ω, satisfies the conditions P (y) = 0, P (y) = 0, x 2 ∈ ω2 or P (y) = 0, P (y) = 0, x 1 ∈ ω1 , Proof.For arbitrary y(x), defined on ω, the identity is true, where Let us estimate this sum.If Applying this inequality to (13), we have y 2 (1) From the nonlocal condition follows and, therefore, Based on the nonlocal condition, we have as well From ( 14) with the help of ( 15), ( 16) we obtain y 2 (1) If we increase the first and second sums by multiplication on the quantity l 1 /(2ξ 1 ) > 1 and apply the inequality in the summands with the indices i = m 1 + 1, n 1 − m 1 , we will be sure that the Lemma 4 is true.

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To determine the convergence rate of the finite-difference scheme (4), we apply the following lemma.
Lemma 7. Assume that the linear functional η(u) is bounded in W s 2 (E), where s = s+ , s is an integer, 0 < 1, and η(P ) = 0 for every polynomial P of degree s in two variables.Then there exists a constant c, independent of u, such that |η(u)| c u W s 2 (E) .This lemma is a particular case of Dupont-Scott approximation theorem [19] and it represents a generalization of the Bramble-Hilbert lemma [20] (see also [16, p. 29]).

The problem for the error
Let us define on the particular subintervals the components of approximation errors for the integral conditions (2): Lemma 8. Let u be the solution of the problem (1), ( 2) and y be the solution of the finite-difference scheme (4).Then discretization error z = y − u satisfies the following problem: Indeed, ( 22) can be obtained from substituting y = z + u into (4) and taking into account Further, in view of the conditions (2), (3), we have We can verify other equalities of (23) analogously.
As we see, the nonlocal conditions for the error problem, unlike the difference scheme, are not homogeneous.Therefore, in order to use the results obtained in the Section 3, we pass to the new unknown function.
First of all, let us define the functions For them, the following hold: We can verify straightforward that P (w) = 0 and P (w) = 0.
According to (25), we have If we apply the first inequality of Lemma 6 in the left-hand side of this identity, and in the right-hand side the Lemmas 2 and 5, we obtain The second inequality of the Lemma 6 together with (2), ( 27) gives an a priori estimate for the problem (22) For the estimation of J 1 , notice that the summands ζ , ζ , as linear functionals with respect to u(x), vanish on the polynomials of first order and are bounded on W s 2 , s > 1.Consequently, using Lemma 7, we have J 1 c|h| s u W s 2 (Ω) , 1 < s 2, from which J 1 c|h| s−1 u W s 2 (Ω) , 1 < s 3.For the estimation of J 2 , notice that the summands ζ x1 , ζ x2 , as linear functionals with respect to u(x), vanish on the polynomials of second order and are bounded on W s 2 , s > 1.Consequently, using Lemma 7, we receive J 2 c|h| s−1 u W s 2 (Ω) , 1 < s 3.For the estimation of J 3 , we represent its summands in the expanded form, for example, This may be estimated analogously to J 1 .
As a result from (28) it follows the validity of Theorem 2.

Conclusion
A nonlocal problem posed for Poisson equation is considered-classical boundary conditions are fully replaced with integral conditions on the inner stripe adjacent to boundary having the width ξ.The corresponding difference scheme is constructed for which convergence with rate s − 1 is proved when the exact solution belongs to Sobolev space W s 2 , 1 < s 3, with fractional exponent.
The obtained results may be expanded: for a case when the width of the stripe defined by integral conditions is different at all sides of the rectangle; for a system of statical theory of elasticity with constant coefficients, also for three dimensional case.