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

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1 Nonlinear Analysis: Modelling and Control, 04, Vol. 9, No. 3, On the convergence rate of a difference solution of the Poisson equation with fully nonlocal constraints Givi Berikelashvili a,b, Nodar Khomeriki b a A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University Tamarashvili str. 6, Tbilisi 077, Georgia bergi@rmi.ge; berikela@yahoo.com b Department of Mathematics, Georgian Technical University Kostava str. 77, Tbilisi 075, Georgia n.khomeriki@gtu.ge Received: February 04 / Revised: 6 March 04 / Published online: 30 June 04 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, when the exact solution belongs to the fractional Sobolev space of order s < s 3. Keywords: integral conditions, energy inequalities, difference scheme, convergence rate. 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., [ 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 4]. In [5], 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 ux dx k = 0, k =,, 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, when the desired solution belongs to the Sobolev c Vilnius University, 04

2 368 G. Berikelashvili, N. Khomeriki space W s < s 3. The proof bases on procedure of obtaining convergence estimate compatible with smoothness of the exact solution developed by Samarskii et al. [6] see, also [7, 8]. In this paper, we study the case, when the classical boundary conditions are completely replaced by nonlocal ones: u x u x = fx, x Ω, ξ k 0 ux dx k = 0, k l k ξ k ux dx k = 0, 0 x 3 k l 3 k, k =,, where Ω = {x, x : 0 < x k < l k, k =, } be the rectangle; l = max{l, l }. We assume that the solution u of the nonlocal boundary-value problem, belongs to the fractional-order Sobolev space W s Ω, s >. For the corresponding difference scheme, estimate of convergence similar to [5], is obtained. Besides the fact that the operator of the difference scheme is not positive definite, basic difficulties comparing with [5] 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 Ω: ω = ω ω, ω = ω ω, where ω k = {x k,ik = i k h k : i k = 0,,..., n k, h k = l k /n k }, ω k = ω k 0, l k, ω k = ω k 0, l k ], k = h k for x k ω k, k = h k / for x k = 0, l k, h = h h /. For the values of net function in several points, we apply the notation y ij =yih, jh. When it does not lead to ambiguity, for simplicity, we use the notations y i = yih, x, y j = yx, jh. We define the finite-difference operators v xk = vx h kr k vx h k, v xk = vx vx h kr k h k, k =,, where r k is the unit vector on the x k axis. Let ξ k = m k θ k h k, 0 θ k <, h k ξ k l k /, k =,, where m k is positive integer. By H we denote the set of all discrete functions v = vx, defined on the grid ω and satisfying conditions ˇP v = 0, ˆP v = 0, x ω, ˇP v = 0, ˆP v = 0, x ω, 3

3 On the convergence rate of a difference solution of the Poisson equation 369 where m ˇP v j := i=0 m ˇP v i := ˆP v j := ˆP v i := j=0 h v ij h y 0j v mj θ h θ v mj θ v m,j, h v ij h v i0 v im θ h θ v im θ v i,m, n h v ij h v n m,j v nj i=n m θ h θ v n m,j θ v n m,j, n j=n m h v ij h v i,n m v in θ h θ v i,n m θ v i,n m. We need the following averaging operators for functions defined on Ω: T ux = h x h h x t ut, x dt, T ux = h x h x h h x t ux, t dt. x h We approximate the problem, by the difference scheme where Λy = ϕ, x ω ω, y H, 4 Λ := Λ Λ, Λ α y := y xαx α, ϕ := T T f. Theorem. 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. Let a solution ux of the problem, belong to the space W s Ω, s >. Then the convergence rate of the difference scheme 4 in the discrete weighted W -norm is determined by the estimate y u W ω,ρ c h s u W s Ω, < s 3. 5 Nonlinear Anal. Model. Control, 04, Vol. 9, No. 3,

4 370 G. Berikelashvili, N. Khomeriki 3 Auxiliary statements Lemma. For every discrete function v H, the following identities hold: Proof. Indeed, ˇP v 0 = ˇP v 0 ˇP v m = m j= θ h Hence using the relation j= ˇP v i = 0, ˆP v i = 0, i = 0, n. h v 0j v mj h v 00 v 0m v m0 v mm θ v 0m θ v 0,m θ v mm θ v m,m. m m v 0j v mj = v i0 v im m j= which follows from the equality m h j= m θ θ v mj θ v m,j, m ˇP v j = h θ θ v im θ v i,m ˇP v i, we obtain Here h ˇP v 0 = h ˇP v 0 ˇP v m Q = Q. 6 Q : = θ θ θ m θ θ θ m j= m v im v 0m v mm v i,m v 0,m v m,m m j= v mj v m 0 v mm v m,j v m,0 v m,m.

5 On the convergence rate of a difference solution of the Poisson equation 37 By applying nonlocal conditions we see that Q = θ θ θ θ v mm θ v m,m θ θ θ v m,m θ v m,m θ θ θ θ v mm θ v m,m θ θ θ v m,m θ v m,m = 0. From here and 6 follows the first identity of Lemma. The proof of the last part leads analogously. We define the weight functions i k h k ξ k, i k =,,..., m k,, i = m ρ k = ρ k k,..., n k m k, i k h k = n k i k h k ξ k, i k = n k m k,..., n k, θ k h k ξ k, i k = m k, i k = n k m k, h k 4ξ k, i k = 0, n k, i k h k ρ k = ρ k ξ i k h k = k, i k =,,..., m k,, i k = m k,..., n k m k, n k i k h k ξ k, i k = n k m k,..., n k. In H, we introduce the inner product and norm as Let, in addition, y = y, v = ω ρ ρ yv, y = y, y /. y, v ω = ω ω ω h ρ ρ y x h h yv, ω ω y W ρ,ω = y y, h ρ ρ y x, i ξ ih y ij h k=0 y kj h y ij y 0j, 0 i m, G y ij = y ij, m i n m, n ξ n ih y ij h k=i y kj h y ij y nj, n m i n, j ξ jh y ij h k=0 y ik h y ij y i0, 0 j m, G y ij = y ij, m i n m, n ξ n jh y ij h k=j y ik h y ij y in, n m j n. Nonlinear Anal. Model. Control, 04, Vol. 9, No. 3,

6 37 G. Berikelashvili, N. Khomeriki Lemma. Let grid functions vx, yx be defined on ω, and yx satisfy the conditions ˇP y = 0, ˆP y = 0, x ω or ˇP y = 0, ˆP y = 0, x ω. Then h k v xk x k G k y = h k ρ k v xk y xk ω k for k = or k =, respectively. Proof. Using the summation by parts, we obtain m n m i=m n ω k m h v xx i G y i = h ρ ih v x i y x i v x mg y m, 7 h v xx i G y i = i=n m h v xx i G y i = n m i=m h v x i y x i v x n m y n m v x m y m, 8 n i=n m h ρ ih v x i v x i v x n m G y n m. 9 Adding the equalities 7 9 and applying following from the nonlocal conditions identities G y m = y m θ h ξ y x m, G y n m = y n m θ h ξ y x n m, we verify the validity of the Lemma in the case k =. The case k = may be proved analogously. Lemma 3. If a grid function yx, defined on ω, satisfies the conditions then ˇP y = 0, ˆP y = 0, x ω or ˇP y = 0, ˆP y = 0, x ω, 7 8 k ρ k y h k yg k y k ρ k y ω k ω k for k = or k =, respectively. Proof. It may be showed that the identity n h y ij G y ij = h ξ m ω k n m iyij h i=m y ij h ξ n i=n m n iy ij h 8ξ y 0j y n j Š Ŝ 0

7 On the convergence rate of a difference solution of the Poisson equation 373 holds, where m Š := y kj y 0j, Ŝ := k=0 Let us note that, according to nonlocal conditions, In addition, the inequality follows from n Š = θ y mj θ y m,j. k=n m y kj y nj. Š θ y m j θ y m,j := A A Š θ θ A Š = θ θ y mj y m,j 0. We can obtain analogously that Ŝ θ y n m,j θ y n m,j. Adding inequalities, and replacing in the right-hand side θ m and θ ξ /h, we obtain h Š 8ξ Ŝ m h y 8ξ mj y h n m,j y 8 m,j yn m,j n ρ i y 8 ij. i=0 From this inequality and 0 follows the validity of Lemma 3 in the case k =. We can consider the case k = analogously. Lemma 4. If a grid function yx, defined on ω, satisfies the conditions then ˇP y = 0, ˆP y = 0, x ω or ˇP y = 0, ˆP y = 0, x ω, k y lk h k ρ k y x k ω k for k = or k =, respectively. Proof. For arbitrary yx, defined on ω, the identity ω k n y := yi = J l y m yn m i=0 3 Nonlinear Anal. Model. Control, 04, Vol. 9, No. 3,

8 374 G. Berikelashvili, N. Khomeriki is true, where m J = h i y n i yi m i=m n h n i y i y i. i=n m Let us estimate this sum. If then J m J := n i n m y x,i h 3 i=m h 3 n i y x,i, h 3 n i=n m n h i y i y i n i y x,i / h yi y i J / n h y i y i J 8 y J. Applying this inequality to 3, we have y 4J l y m y n m. 4 From the nonlocal condition follows m ξy m = h i y i y i θ h y m y m, and, therefore, y m m h i y x,i θ h y x,m. 5 Based on the nonlocal condition, we have as well yn m n h n i y x,i θ h y x,n m. 6 i=n m From 4 with the help of 5, 6 we obtain y 9l m h i y x,i 9l l n m i=m n i=n m h n i y x,i h y x,i l 8ξ θ h y x,m y x,n m.

9 On the convergence rate of a difference solution of the Poisson equation 375 If we increase the first and second sums by multiplication on the quantity l /ξ > and apply the inequality θ h < 9 θ h 8ξ 4 ξ in the summands with the indices i = m, n m, we will be sure that the Lemma 4 is true. Lemma 5. If a grid function yx, defined on ω, satisfies the conditions then ˇP y = 0, ˆP y = 0, x ω or ˇP y = 0, ˆP y = 0, x ω, for k = or k =, respectively. Proof. Let Š i := i k=0 h k G k y 0 k ρ k y, k =,, ω k ω k h y k h y i y 0, Ŝ i := According to the definition of the operator G, n We have from here n h G y i h G y i = m h ξ m m n k=i n h m ξ ih y i Ši n i=n m h ξ n m ih y i h ξ Ši i=m n h ξ i=n m It is not difficult to verify that m h Ši = m m h Šm h y k h y i y n. i=m. n iy i Ŝi n h y i h y i h ξ n i h yi i=n m Ŝi. 7 h i y i y i Ši Ši. We have from here m m h Ši 4m h Šm h h i y i y i. 8 Nonlinear Anal. Model. Control, 04, Vol. 9, No. 3,

10 376 G. Berikelashvili, N. Khomeriki Noting that we obtain from 8 Šm h y m ym, m m h Ši 8 h iξ yi 3h ξy m h 3 y0. 9 We can obtain analogously that n i=n m h Ŝi 8 n i=n m h n iξ y i 3h ξ y n m h 3 y n. 0 From the inequalities 7, 9, 0 follows validity of the Lemma 5 in the case k =. We can consider the case k = analogously. Lemma 6. For every y H, the following inequalities hold: 7 8 y Λy, G G y ω y, y W ρ,ω c Λy, G G y ω, c = 8 7 l. Proof. Based on the Lemma, Λy, G G y ω = h h ρ y x G y x ω ω ω ω h h ρ y x G y x. Taking into account in addition following from the Lemma 3 inequalities ω ω ω ω h ρ ρ y x h h ρ y x G y x h ρ ρ y x ω ω h h ρ y x G y x ω ω ω ω we ensure the validity for the first inequality of lemma. According to Lemma 4, From here and it follows Lemma 6. h ρ ρ y x, ω ω h ρ ρ y x, y c y, c = max l ; l.

11 On the convergence rate of a difference solution of the Poisson equation 377 To determine the convergence rate of the finite-difference scheme 4, we apply the following lemma. Lemma 7. Assume that the linear functional ηu is bounded in W s E, where s = sɛ, s is an integer, 0 < ɛ, and ηp = 0 for every polynomial P of degree s in two variables. Then there exists a constant c, independent of u, such that ηu c u W s E. This lemma is a particular case of Dupont Scott approximation theorem [9] and it represents a generalization of the Bramble Hilbert lemma [0] see also [6, p. 9]. 4 The problem for the error Let us define on the particular subintervals the components of approximation errors for the integral conditions : ζ ij := ζ i,m := ζ i,n m := ζ ij := ζ m,j := ζ n m,j := jh j h uih, t dt m θ h m h n m h uih, t dt n m θ h uih, t dt ih i h ut, jh dt m θ h m h n m h ut, jh dt n m θ h ut, jh dt h u i,j u ij, j m, n m, θ h θ u i,m θ u i,m, θ h θ u i,n m θ u i,n m, h u i,j u ij, i m, n m, θ h θ u m,j θ u m,j, θ h θ u n m,j θ u n m,j. Lemma 8. Let u be the solution of the problem, and y be the solution of the finite-difference scheme 4. Then discretization error z = y u satisfies the following problem: Λz = η x x η x x, x ω, ˇP z = ˇχ x, ˆP z = ˆχ x, x ω, 3 ˇP z = ˇχ x, ˆP z = ˆχ x, x ω, Nonlinear Anal. Model. Control, 04, Vol. 9, No. 3,

12 378 G. Berikelashvili, N. Khomeriki where η := T u u, η := T u u, m ˇχ i := ζ ij, ˆχ i := j= m ˇχ j := ζ ij, ˆχ j := n j=n m ζ ij, n i=n m ζ ij. Indeed, can be obtained from substituting y = z u into 4 and taking into account T k u/ x = u xk x k. Further, in view of the conditions, 3, we have ˇP z = ˇP y ˇP u = ξ 0 ut, x dt ˇP u = ˇχ x. We can verify other equalities of 3 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 ˇβ k x k = l k ξ k x k ξ k l k ξ k, ˆβk x k = x k ξ k, k =,. ξ k l k ξ k For them, the following hold: Let ˇP ˇβ =, ˇP ˇβ =, ˆP ˇβ = 0, ˆP ˇβ = 0, ˇP ˆβ = 0, ˇP ˆβ = 0, ˆP ˆβ =, ˆP ˆβ =. wx = zx ˇβ x ˇχ x ˆβ x ˆχ x ˇβ x ˇχ x ˇβ x ˇβ x ˇP ˇχ ˇβ x ˆβ x ˇP ˆχ ˆβ x ˆχ x ˆβ x ˇβ x ˆP ˇχ ˆβ x ˆβ x ˆP ˆχ. 4 We can verify straightforward that ˇP w = 0 and ˆP w = 0. For the verification of the conditions ˇP w = 0 and ˆP w = 0, we apply the consequences of 3, respectively, and ˇP ˇχ = ˇP ˇχ, ˇP ˆχ = ˆP ˇχ ˆP ˇχ = ˇP ˆχ, ˆP ˆχ = ˆP ˆχ.

13 On the convergence rate of a difference solution of the Poisson equation 379 It may be proved that the function wx represents a solution of the following problem: Λw = ψ, x ω, w H, 5 where ψ := Λk η k ˇβ 3 k Λ k ˇχ k ˆβ 3 k Λ k ˆχ k. k= 5 Proof of Theorem It follows from 4 where z w cj J J 3, 6 J : = ˇχ ω ˇχ ω ˆχ ω ˆχ ω, J : = ˇχ ω x ˇχ ω x ˆχ ω x ˆχ ω x, J 3 : = ˇχ ˇP ˆχ ˇP ˇχ ˆP ˆχ ˆP. According to 5, we have Λw, G G w ω = ψ, G G w ω. 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 and 5, we obtain w c η x ω ω η x ω ω J. 7 The second inequality of the Lemma 6 together with, 7 gives an a priori estimate for the problem z W ω,ρ c η x ω ω η x ω ω J J J 3. 8 For the estimation of J, notice that the summands ζ, ζ, as linear functionals with respect to ux, vanish on the polynomials of first order and are bounded on W s, s >. Consequently, using Lemma 7, we have J c h s u W s Ω, < s, from which J c h s u W s Ω, < s 3. For the estimation of J, notice that the summands ζ x, ζ x, as linear functionals with respect to ux, vanish on the polynomials of second order and are bounded on W s, s >. Consequently, using Lemma 7, we receive J c h s u W s Ω, < s 3. For the estimation of J 3, we represent its summands in the expanded form, for example, ˇP ˇχ = h m m j= ζ ij ζ i,j θ h This may be estimated analogously to J. m j= θ χ m j θ χ m,j. Nonlinear Anal. Model. Control, 04, Vol. 9, No. 3,

14 380 G. Berikelashvili, N. Khomeriki The norms of the functionals η k x k, k =,, are less than c h s u W s Ω, < s 3. The obtaining of these estimates are considered in detail, for example, in [6, pp.48 49]. As a result from 8 it follows the validity of Theorem. 6 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 is proved when the exact solution belongs to Sobolev space W s, < 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. The authors wish to thank the anonymous referee for many signifi- Acknowledgment. cant comments. References. J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge, New York, E. Obolashvili, Nonlocal problems for some partial differential equations, Appl. Anal., 45:69 80, V.V. Shelukhin, A non-local in time model for radionuclides propagation in Stokes fluid, Din. Splosh. Sredy, 07:80 93, A. Taflove, Computational Electrodynamics. The Finite-Difference Time-Domain Method, Artech House, Boston, MA, Z.G. Mansourati, L.L. Campbel, Non-classical diffusion equations related to the birth-death processes with two boundaries, Q. Appl. Math., 543:43 443, G.K.Berikelashvili, D.G. Gordeziani, On a nonlocal generalization of the biharmonic Dirichlet problem, Differ. Uravn., 463:38 35, 00 in Russian. English transl.: Differ. Equ., 463:3 38, J.R. Cannon, The solution of the heat equation subject to the specification of energy, Q. Appl. Math., :55 60, A.V. Bitsadze, A.A. Samarskii, On some simplest generalization of linear elliptic problems, Dokl. Akad. Nauk SSSR, 85: , 969 in Russian. 9. M.P. Sapagovas, R.J. Čiegis, The numerical solution of some nonlocal problems, Lit. Matem. Sb., 7: , 987 in Russian.

15 On the convergence rate of a difference solution of the Poisson equation V.A. Il in, E.I. Moiseev, Two-dimensional nonlocal boundary-value problem for the Poisson s operator in differential and difference interpretations, Math. Modelling, 8:39 56, 990 in Russian.. Y. Wang, Solutions to nonlinear elliptic equations with a nonlocal boundary conditions, Electron. J. Differ. Equ., 0005: 6, 00.. G. Berikelashvili, On a nonlocal boundary-value problem for two-dimensional elliptic equation, Comput. Methods Appl. Math., 3:35 44, 003. Dedicated to Raitcho Lazarov. 3. G. Avalishvili, M. Avalishvili, D. Gordeziani, On integral nonlocal boundary-value problem for some partial differential equations, Bull. Georgian Natl. Acad. Sci. N.S., 5:3 37, M.Sapagovas, A. Štikonas, and O.Štikonienė, Alternating direction method for the Poisson equation with variable weight coefficients in an integral condition, Differ. Uravn., 478:63 74, 0 in Russian. English transl.: Differ. Equ.,478:76 87, G. Berikelashvili, N. Khomeriki, On a numerical solution of one nonlocal boundary value problem with mixed Dirichlet-Neumann conditions, Lith. Math. J., 534: , A.A. Samarskii, R.D. Lazarov, V.L. Makarov, Difference Schemes for Differential Equations with Generalized Solutions, Vysshaya Shkola, Moscow, 987 in Russian. 7. G. Berikelashvili, Construction and analysis of difference schemes for some elliptic problems, and consistent estimates of the rate of convergence, Mem. Differ. Equ. Math. Phys., 38: 3, B.S. Jovanovi c, E.Süli, Analysis of Finite Difference Schemes. For Linear Partial Differential Equations with Generalized Solutions, Springer Ser. Comput. Math., Vol. 46, Springer, London, T. Dupont, R. Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comput., 34:44 463, J.H. Bramble, S.R. Hilbert, Bounds for a class of linear functionals with application to Hermite interpolation, Numer. Math., 6:36 369, 97. Nonlinear Anal. Model. Control, 04, Vol. 9, No. 3,