Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions
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Abstract
In the paper the convergence of a finite difference scheme for twodimensional nonlinear elliptic equation in the rectangular domain with the integral boundary condition is considered. The majorant is constructed for the error of the solution of the system of difference equations, and the estimation of this error is obtained. With this aim, the idea of application of the Mmatrices for the theoretical investigation of the system of difference equations was developed. Main results for the convergence of the difference schemes are obtained considering the structure of the spectrum and properties of the Mmatrices for a wider class of boundary value problems for nonlinear equations with nonlocal conditions. The main advantage of the suggested method is that the error of approximate solution is estimated in the maximum norm.
Keywords
Elliptic equation Integral boundary conditions Convergence of finitedifference method Eigenvalue problem MmatricesMSC
65M06 65M12 65N251 Introduction
Many physical phenomena have been formulated as a mathematical model with nonlocal boundary conditions. A short overview of these models is presented in many papers (see, for example, works [1, 2]). Particulary, many problems in thermoelasticity can be formulated as nonlocal problems (see [3, 4, 5] and the references therein). Two of the latest new mathematical models in the biotechnology are presented in [6, 7]. A separate class of such nonlocal models is boundary value problem for elliptic equation with nonlocal boundary conditions [8].
Numerical methods for boundary value problem of linear and nonlinear elliptic equations with various types of nonlocal conditions have been intensively investigated during past decades. Finite difference methods for linear elliptic equations with Bicadze–Samarski or multipoint nonlocal conditions were analyzed in works [8, 9]. In the papers [10, 11, 12], the main goal was the investigation of the existence and uniqueness of the solution of difference problem with integral conditions, as well as estimation of the error in certain norms.
Various iterative methods for the systems of linear difference equations, derived from elliptic equations with nonlocal conditions, and proofs of convergence of these methods can be found in [13, 14, 15]. In the articles [16, 17, 18, 19], the iterative methods are generalized for the systems of nonlinear difference equations with nonlocal conditions. Some other difference methods for elliptic equations with nonlocal conditions were described in [13, 20, 21, 22].
In most cases of elliptic equations with nonlocal conditions, the matrix of the discrete problem is characterized by the properties appropriate for Mmatrices [23, 24, 25]. This was used to prove the convergence of iterative methods for linear and nonlinear elliptic equations with nonlocal conditions [17, 26].
Application of the Mmatrices for the elliptic and parabolic differential equations with Dirichlet boundary conditions has been described by Varga [25] (for some new research in this field, see the works [27, 28, 29]).
In the present paper, the idea of application of Mmatrices for theoretical investigation of difference methods with nonlocal conditions is further developed. Namely, Mmatrices are used to prove the convergence of difference schemes. It is well known that the property of diagonal dominance of the matrix of discrete problem is necessary for applicability of the maximum principle. However, matrices with nonlocal conditions are not diagonally dominant. We can overcome this problem by applying the methodology of Mmatrices.
We will use that in the spectrum of the matrix there are no eigenvalues with negative real part. This became apparent after investigating the structure of the spectrum of twodimensional differential and appropriate difference operators with nonlocal conditions (see the works [14, 30, 31, 32, 33, 34, 35, 36] and the references therein). As far as authors know, the idea of application of Mmatrices for the convergence of the difference schemes in the case of nonlocal boundary conditions is applied for the first time. We note that in the case of nonlocal conditions, the matrix of system of finite difference equations is neither diagonally dominated nor symmetric.
Using the structure of the spectrum and properties of Mmatrices, we succeed in proving the convergence of difference schemes for a wider class of nonlinear equations with nonlocal conditions than it was proved before. This is the main result of the research.
The structure of the paper is the following. In Sect. 2, the differential problem is formulated and its discrete approximation is provided. The difference problem for the error of the solution is investigated in Sect. 3. In Sect. 4, the main properties of the matrix of difference problem are described, the case of matrix being as Mmatrix is analyzed. The auxiliary lemmas on the evaluation of the solution of the system of the equations with an Mmatrix are provided in Sect. 5. The majorant for the error is constructed in Sect. 6. The results of numerical experiments are presented in Sect. 7. In Sect. 8, we conclude and generalize some results.
2 Problem formulation
 H1.

\(\partial f/\partial u \ge 0\) for all the values \((x, y) \in D\) and u;
 H2.

γ is a given real number and \(0\le \gamma \le 2 \delta \), \(0<\delta \le 2\).
The solution of the system of difference equations (4)–(6) by the iterative methods was investigated in [17]. It follows from the results there that under hypotheses H1 and H2, the unique solution of the system of difference equations (4)–(6) exists. In the present paper, the error estimation for this solution and the convergence of difference scheme (4)–(6) are considered.
3 The investigation of the difference problem for the error
Furthermore, all three forms of system (12)–(14) will be used: the equivalent system (18), (19), (16), (14) with the order \(N(N1)\); partial system (18), (19), (14) with the order \((N1)^{2}\); the matrix form (20) of system (18), (19), (14).
4 Properties of the matrix A
Now we will use a few properties of Mmatrices. We reformulate some of these properties, applied for the system of difference equations (20) as new lemmas.
Definition 1
A real square matrix \(\mathbf{A}=\{a_{kl}\}\), \(k, l=\overline{1, n}\) with \(a_{kl}\le 0\) for all \(k\neq l\) is called an Mmatrix if A is nonsingular and \(\mathbf{A}^{1}\) is nonnegative.
It follows from the definition that \(a_{kk}>0\). Throughout the rest of this paper, we will denote \(\mathbf{A}>0 (\mathbf{A}\ge 0)\) if \(a_{kl}>0\) (\(a_{kl}\ge 0\)) for all k, l. Additionally, \(\mathbf{A}< \mathbf{B}\) if \(a_{kl}< b_{kl}\). Similar notation for vectors is also used. The following property of Mmatrices is correct [23, 24, 25].
Lemma 1
 (i)
\(\mathbf{A}^{1}\)exists and\(\mathbf{A}^{1}\ge 0\);
 (ii)
The real parts of all the eigenvalues of the matrixAare positive: \(\operatorname{Re}\lambda (\mathbf{A})>0\).
The main properties of matrix A for the system of equations (20) as lemmas are formulated below.
Lemma 2
The diagonal elements of the matrixAof system (20) are positive.
Lemma 3
The nondiagonal elements of matrixAof system (20) are nonpositive.
The statement of lemma follows from the fact that nondiagonal elements of matrix Λ are \(h^{2}\) or 0, nondiagonal elements of matrix C are \(h^{2}\alpha \) or 0, and matrix D is diagonal.
Lemma 4
All eigenvalues of matrix\(\boldsymbol{\varLambda }\mathbf{C}\)are positive.
Proof
Corollary 1
Matrices\(\mathbf{A}_{1}=\boldsymbol{\varLambda }\mathbf{C}\)and\(\mathbf{A}=\boldsymbol{\varLambda }\mathbf{C}+\mathbf{D}\)are Mmatrices, so\((\boldsymbol{\varLambda }\mathbf{C})^{1}\ge 0\), \(\mathbf{A}^{1} \ge 0\).
5 Comparison theorem
Lemma 5
If the matrixAin system (29) is an Mmatrix and\(\mathbf{f}\ge 0\), then\(\mathbf{u}\ge 0\).
The statement of lemma follows from \(\mathbf{u}= \mathbf{A}^{1} \mathbf{f} \) and \(\mathbf{A}^{1}\ge 0\), \(\mathbf{f}\ge 0\).
Let us denote the elements of vector u as \(\{u_{k}\}\), i.e., \(\mathbf{u}=\{u_{k}\}\). The vector with elements \(\vert u_{k}\vert \) can be denoted as \(\vert \mathbf{u}\vert \), i.e., \(\vert \mathbf{u}\vert =\{\vert u_{k}\vert \}\).
Lemma 6
(Comparison theorem)
Proof
Remark 2
Lemma 7
Proof
Remark 3
6 Construction of majorant and the main theorem
Now the main result of the research on the estimate of the error of the difference method can be formulated and convergence of the method can be proved.
Theorem 8
Proof
7 Numerical results
To justify the theoretical results and investigate the efficiency of numerical schemes in the case of nonlocal boundary conditions, we consider a model problem where the analytical solution is explicitly known. The theoretical results presented in the previous sections do not depend upon the numerical method that is used to solve nonlinear system of finite difference equations (4)–(6). This system can be solved using one of the iterative methods designed for problems with nonlocal conditions [16, 17, 18]. We can also use some classical method used for solving nonlinear systems, for example, [38]. Here we have used a generalization of the alternatingdirection implicit (ADI) method for problems with nonlocal boundary conditions [18] defined on twodimensional uniform grids on \(D=[0, 1] \times [0, 1]\). Meshes of different sizes with \(h=h_{x}=h_{y}\) and variation of parameter γ were used in simulations.
Test 1
Problem with nonlocal BC. The first numerical example is a simple test case for validating the error estimates and demonstrating the performance of the finitedifference method for several values of parameter of nonlocality γ.
The errors for different γ in the case of \(h=0.01\)
γ  \(E_{h}\)  \(\epsilon _{h}\) 

0.0  1.89729⋅10^{−6}  1.26332⋅10^{−5} 
0.3  2.01499⋅10^{−6}  1.49908⋅10^{−5} 
1.0  5.77622⋅10^{−5}  5.03429⋅10^{−5} 
1.95  2.13652⋅10^{−5}  8.20321⋅10^{−5} 
2.0  2.54840⋅10^{−5}  8.00939⋅10^{−5} 
3.0  1.03093⋅10^{−3}  1.78941⋅10^{−3} 
3.2  4.17644⋅10^{−3}  8.49890⋅10^{−3} 
3.22  4.88902⋅10^{−3}  7.93334⋅10^{−3} 
3.224  8.65916⋅10^{−3}  9.80129⋅10^{−2} 
The errors for different stepsizes h and γ
γ  h  \(E_{h}\)  \(\epsilon _{h}\)  Order p 

0.0  0.25  1.08749⋅10^{−3}  2.62507⋅10^{−3}  
0.125  2.90550⋅10^{−4}  6.83678⋅10^{−4}  1.9042  
0.0625  7.35630⋅10^{−5}  1.73016⋅10^{−5}  1.9817  
0.03125  1.85130⋅10^{−5}  4.38179⋅10^{−5}  1.9904  
1.0  0.25  1.44959⋅10^{−3}  3.00694⋅10^{−3}  
0.125  3.72072⋅10^{−4}  7.81139⋅10^{−4}  1.9620  
0.06250  9.47515⋅10^{−5}  1.97871⋅10^{−4}  1.9734  
0.03125  2.38487⋅10^{−5}  5.84610⋅10^{−5}  1.9902 
We determined from the numerical results in Table 1 that the error of the solution slightly increases as γ grows, when \(0 \leq \gamma < 2\). This agrees well with the fact that approximation error \(R_{j}(h)\) of nonlocal conditions (2) is linearly dependent on γ (see (11)).
From the results in Table 2, we see that error which theoretically is \(\mathcal{O}(h^{2})\) (Theorem 8) matches theory rather well both in the case of Dirichlet condition (\(\gamma =0\)) and in the case of nonlocal condition (\(\gamma =1\)).
We also note that Test 1 was also successfully solved with \(\gamma \in [2; 3.224]\). When \(\gamma > 2\), the matrix corresponding to difference operator \(\delta _{x}^{2}\) with nonlocal condition (5) has negative eigenvalue and ADI might not converge (for more details, see [14, 15, 16]).
Test 2
Let us explain the role of additional term \(cx^{2}\) in (49). Recall that approximation error for problem (8) depends on constants \(M_{2}\) and \(M_{4}\) from (11). Varying \(c>0\) lets us investigate the influence of the approximation error of nonlocal condition (which is bounded by constant \(M_{2}\)) without changing approximation error of differential equation (which is bounded by constant \(M_{4}\)).
The errors for different γ and c in the case of \(h=0.01\)
γ  c  \(E_{h}\)  \(\epsilon _{h}\) 

0.0  0.0  1.89729⋅10^{−6}  1.26332⋅10^{−5} 
0.5  1.84946⋅10^{−6}  2.79954⋅10^{−6}  
1.0  1.80728⋅10^{−6}  2.38339⋅10^{−6}  
5.0  3.70598⋅10^{−5}  8.10686⋅10^{−6}  
1.0  0.0  5.77622⋅10^{−5}  5.03429⋅10^{−5} 
0.5  6.89338⋅10^{−6}  6.53304⋅10^{−5}  
1.0  7.80954⋅10^{−6}  7.91123⋅10^{−5}  
5.0  2.09098⋅10^{−5}  1.85548⋅10^{−4}  
2.0  0.0  2.54840⋅10^{−5}  8.00939⋅10^{−5} 
0.5  3.73614⋅10^{−5}  5.85915⋅10^{−5}  
1.0  4.96933⋅10^{−5}  5.29029⋅10^{−5}  
5.0  7.63759⋅10^{−3}  3.08093⋅10^{−3} 
The errors for different γ, h, and c in the case of the RHS function (50)
γ  c  h  \(E_{h}\)  \(\epsilon _{h}\)  Order p 

0.0  1.0  0.25  1.01492⋅10^{−3}  
0.125  2.77342⋅10^{−4}  3.68367⋅10^{−4}  1.8716  
0.0625  6.98376⋅10^{−5}  9.25603⋅10^{−5}  1.9896  
0.03125  1.75949⋅10^{−5}  2.32000⋅10^{−5}  1.9888  
1.0  1.0  0.25  1.36950⋅10^{−3}  2.06912⋅10^{−3}  
0.125  3.53840⋅10^{−4}  5.20942⋅10^{−4}  1.9525  
0.06250  8.96690⋅10^{−5}  1.31948⋅10^{−4}  1.9804  
0.03125  2.26954⋅10^{−5}  6.66756⋅10^{−5}  1.9822 
We may observe that the actual convergence follows very closely the expected theoretical error. Numerical tests reinforce the theoretical convergence results.
8 Remarks and generalizations
Majorant \(w(x, y)\) is constructed according to formula (31) in the case \(0\le \gamma \le 2\delta \), \(\delta >0\). But the matrix A of system (20) is an Mmatrix with the values for \(\gamma \in [0, \gamma _{0})\), \(\gamma _{0}\approx 3.42\) [26]. It is not clear how the majorant can be defined for \(\gamma \ge 2\).
The theory of Mmatrices was first used in the comparison theorem instead of the maximum principle for the estimation of the error of approximate solution. This idea let us estimate error in the maximum norm in the case \(\gamma > 1\).
The numerical results presented in Sect. 7 confirm theoretical results about the estimate of the error. Method’s error is second order, independent of the type of boundary conditions (Dirichlet with \(\gamma = 0\) or nonlocal with \(\gamma > 0\)). Moreover, the numerical experiment provides additional information about quantitative dependence of error of the solution on γ and \(M_{2}\).
We prove the convergence of the difference scheme for one concrete case of nonlocal condition. Furthermore, this methodology may also be applied in another case when, for the operator with nonlocal conditions, all the eigenvalues are positive only.
Notes
Acknowledgements
The authors would like to thank the referees for helpful suggestions.
Availability of data and materials
Not applicable.
Authors’ contributions
All authors equally have made contributions. All authors read and approved the final manuscript.
Funding
Not applicable.
Competing interests
There are no competing interests regarding this research work.
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