Abstract
In this study, the BitsadzeSamarskii type nonlocal boundaryvalue problem with integral condition for an elliptic differential equation in a Hilbert space H with selfadjoint positive definite operator A is considered. The second order of the accuracy difference scheme for the approximate solutions of this nonlocal boundaryvalue problem is presented. The wellposedness of this difference scheme in Hölder spaces with a weight is proved. The theoretical statements for the solution of this difference scheme are supported by the results of numerical example.
Keywords:
wellposedness; difference scheme; elliptic equation1 Introduction
In 1969 Bitsadze and Samarskii [1] stated and studied a new problem in which a nonlocal condition is related to the values of the solution on parts of the boundary and on an interior curve for a uniformly elliptic equation. Furthermore, in [216] the BitsadzeSamarskii type nonlocal boundaryvalue problems were investigated for the various differential and difference equations of elliptic type. The role played by coercive inequalities in the study of local boundaryvalue problems for elliptic differential equations is well known [17]. Methods of solutions of elliptic differential and difference equations have been studied extensively by many researchers (see [1827] and the references therein). In the present paper we consider the BitsadzeSamarskii type nonlocal boundaryvalue problem with integral condition,
for the differential equation of elliptic type in a Hilbert space H with the selfadjoint positive definite operator A with a closed domain . Here, let be a given abstract continuous function defined on with values in H, φ, and ψ are elements of and is a scalar function. A function is called a solution of problem (1) if the following conditions are satisfied:
i. is a twice continuously differentiable on the segment .
ii. The element belongs to for all , and the function is continuous on the segment .
iii. satisfies the equation and nonlocal boundary conditions (1).
The paper is organized as follows. In Section 2 the second order of the accuracy difference scheme for the approximate solution (1) is presented. The stability, the almost coercive stability, and the coercive stability estimates for the solution of the difference scheme for an approximate solution of the nonlocal boundaryvalue problem with integral condition for elliptic equations are obtained. Section 3 contains the applications of Section 2. The final section is devoted to the numerical result. Theoretical statements for the solution of the second order of the accuracy difference scheme is supported by a numerical experiment.
2 The second order of the accuracy difference scheme
Let us associate the nonlocal boundaryvalue problem (1) with the corresponding difference problem,
We will study the problem (2) under the following assumption:
It is well known [28] that for a selfadjoint positive definite operator A it follows that is selfadjoint positive definite and , which is defined on the whole space H is a bounded operator. Here, I is the unit operator. Furthermore, we have
In this paper, positive constants, which can differ in time (hence they are not a subject of precision considerations) will be indicated with M. On the other hand is used to focus on the fact that the constant depends only on .
Lemma 1The operator
has an inverse
and the following estimate is satisfied:
whereMdoes not depend onτ.
The proof of the estimate (5) is based on the estimate
Here
The estimate (6) follows from the spectral representation of A and the Cauchy inequality.
Theorem 2For any, , the solution of the problem (2) exists and the following formula holds:
Proof
has a solution and the following formula holds [29]:
Applying formula (9) and the nonlocal boundary condition
we obtain
Since the operator
has an inverse , it follows that
Theorem 2 is proved. □
Let be the linear space of the mesh functions with values in the Hilbert space H. We denote by and , , Banach spaces with the norms
The nonlocal boundaryvalue problem (2) is said to be stable in if we have the inequality
Theorem 3The solutions of the difference scheme (2) under the assumption (3) satisfy the stability estimate
Proof By [29],
is proved for the solution of difference scheme (8). Then the proof of (10) is based on (11) and on the estimate
Using the formula (7) and the estimates (4), (5), we get
Theorem 3 is proved. □
Theorem 4The solutions of the difference problem (2) inunder the assumption (3) obey the almost coercive inequality
Proof By [29],
is proved for the solution of the boundaryvalue problem (8). Using the estimates (4), (5) and the formula (7), we obtain
for the solution of difference scheme (2). Applying formula (7) and , we get
where
To this end it suffices to show that
and
The estimate (15) follows from formula (13) and the estimates (4), (5). Using formula (14) and the estimates (4), (5), we obtain
From the last estimate and the estimate (15) follows the estimate (12). Theorem 4 is proved. □
Theorem 5The difference problem (2) is well posed in the Hölder spacesunder the assumption (3) and the following coercivity inequality holds:
Proof By [29],
is proved for the solution of difference scheme (8). Then the proof of (17) is based on (18) and on the estimate
Applying the triangle inequality, formula (7) and the estimate (15), we get
To this end it suffices to show that
Applying formula (14), we get
where
Second, let us estimate for any separately. We start with . Using estimates (4), (5) and the definition of the norm of the space , we get
From (3) it follows that
Now, let us estimate . Using estimates (4), (5) and the definition of the norm of the space , we obtain
The sum
is the lower Darboux integral sum for the integral
It follows that
By the lower Darboux integral sum for the integral it concludes that
For , applying (4), (5) and the definition of the norm of the space , we get
The sum
is the lower Darboux integral sum for the integral
Since
it follows that
By the lower Darboux integral sum for the integral it follows that
From (3) it follows that
Next, let us estimate . Using the estimates (4), (5), and the definition of the norm space , we obtain
The sum
is the lower Darboux integral sum for the integral
Since
it follows that
By the lower Darboux integral sum for the integral it follows that
Finally, let us estimate . Using the estimates (4), (5), and the definition of the norm space , we get
The sum
is the lower Darboux integral sum for the integral
Thus, we show that
By the lower Darboux integral sum for the integral it follows that
From (3) it follows that
Combining estimates for , we get the estimate (19). Theorem 5 is proved. □
3 Applications
Now, the application of Theorems 35 will be given. First, we consider the mixed boundaryvalue problem for elliptic equation
where , , and are given sufficiently smooth functions and , , . The discretization of problem (21) is carried out in two steps. In the first step, let us define the grid space
We introduce the Hilbert space of the grid functions defined on , equipped with the norms
To the differential operator A generated by the problem (21) we assign the difference operator by the formula
acting in the space of the grid functions satisfying the conditions , . It is know that is a selfadjoint positive definite operator in . With the help of , we arrive at the nonlocal boundaryvalue problem
for an infinite system of ordinary differential equations. Therefore, in the second step, equation (23) is replaced by the difference scheme (2), and we get the following difference scheme:
for the numerical solution of (21).
Theorem 6Letτandhbe sufficiently small positive numbers. Then under the assumption (3), the solution of the difference scheme (24) satisfies the following stability and almost coercivity estimates:
The proof of Theorem 6 is based on Theorems 3 and 4, on the estimate
and on the symmetry properties of the difference operator defined by the formula (22) in .
Theorem 7Letτandbe sufficiently small positive numbers. Then under the assumption (3), the solution of the difference scheme (24) satisfies the following coercivity estimate:
The proof of Theorem 7 is based on Theorem 5 and the symmetry properties of the difference operator defined by formula (22).
Second, let Ω be the unit open cube in with boundary S, . In , the DirichletBitsadzeSamarskii type mixed boundaryvalue problem for the multidimensional elliptic equation
is considered. We will study the problem (26) under the assumption (3). Here, (), , () and (, ) are smooth functions and . The discretization of problem (26) is carried out in two steps. In the first step let us define the grid sets
We introduce the Hilbert space of the grid functions defined on , equipped with the norms
To the differential operator A generated by the problem (26), we assign the difference operator by the formula
acting in the space of the grid functions , satisfying the conditions for all . It is known that is a selfadjoint positive definite operator in . With the help of , we arrive at the nonlocal boundaryvalue problem for an infinite system of ordinary differential equations
In the second step, (28) is replaced by the difference scheme (2), and we get the following difference scheme:
for the numerical solution of (26).
Theorem 8Letτandbe sufficiently small positive numbers. Then under the assumption (3) the solution of the difference scheme (29) satisfies the following stability estimates:
The proof of Theorem 8 is based on Theorem 3 and the symmetry properties of the difference operator defined by (27) in .
Theorem 9Letτandbe sufficiently small positive numbers. Then under the assumption (3) the solution of the difference scheme (29) satisfies the following almost coercivity estimates:
The proof of Theorem 9 is based on Theorem 4, on the estimate (25), on the symmetry properties of the difference operator defined by (27) in , and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in .
Theorem 10For the solutions of the elliptic difference problem
the following coercivity inequality holds[30]:
Theorem 11Letτandbe sufficiently small positive numbers. Then under the assumption (3) the solution of the difference scheme (29) satisfies the following coercivity stability estimate:
The proof of Theorem 11 is based on Theorem 5, on the symmetry properties of the difference operator defined by the formula (27), and on Theorem 10 on the coercivity inequality for the solution of the elliptic difference equation in .
4 Numerical results
We consider the Bitsadze Samarskii type nonlocal boundary problem for the elliptic equation
The exact solution of this problem is
In the present part for the approximate solutions of the BitsadzeSamarskii type nonlocal boundaryvalue problem (30), we will use the first and second orders of the accuracy difference schemes with grid intervals , for t and x, respectively. For the approximate solution of the nonlocal boundary BitsadzeSamarskii type problem (30), we consider the set of a family of grid points depending on the small parameters τ and h,
Applying the first order of the accuracy difference scheme from [31] and the second order of the accuracy difference scheme (2) for the approximate solutions of the problem, we have the secondorder difference equations with respect to n with matrix coefficients. To solve these difference equations, we have applied the procedure of a modified Gauss elimination method for the difference equations with respect to n with matrix coefficients. To obtain the solution of (2), we use MATLAB programming. The errors are computed by
of numerical solutions for different values of M and N, where represents the exact solution and represents the numerical solution at . The results are shown in Table 1, respectively.
Table 1. The errors for first and secondorder difference scheme
5 Conclusion
In this paper, the second order of the accuracy difference scheme for the approximate solution of the BitsadzeSamarskii type nonlocal boundaryvalue problem with the integral condition for elliptic equations is presented. Theorems on the stability estimates, almost coercive stability estimates, and coercive stability estimates for the solution of difference scheme for elliptic equations are proved. The theoretical statements for the solution of this difference scheme are supported by the result of a numerical example. As can be seen from Table 1, the second order of the accuracy difference scheme is more accurate than the first order of the accuracy difference scheme.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
EO carried out the studies, participated in the sequence alignment and drafted the manuscript and AA carried out the studies, participated in the sequence alignment. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank Prof. Dr. PE Sobolevskii for his helpful suggestions on the improvement of this paper.
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