On a difference scheme of the second order of accuracy for elliptic-parabolic equations

The second order of accuracy difference scheme generated by Crank-Nicholson difference scheme for approximately solving multipoint nonlocal boundary value problem is considered. Well-posedness of this difference scheme in Hölder spaces is established. Furthermore, as applications, coercivity estimates in Hölder norms for approximate solutions of the multipoint nonlocal boundary value problems for mixed type equations are obtained. Moreover, the method is illustrated by numerical examples.

In recent years, more and more mathematicians have been studying nonlocal problems for ordinary differential equations and partial differential equations because of their existence in many applied problems included in applied sciences. Theory and numerical methods of solutions of the nonlocal boundary value problems for these partial differential equations were investigated by many researchers (see, e.g., [1-13] and the references therein). Several types of problems in fluid mechanics, other areas of physics, and mathematical biology led to partial differential equations of elliptic-parabolic type (see, [14-18]). The purpose of this paper is to study the second order of accuracy difference schemes of elliptic-parabolic problem with nonlocal boundary value problems.

In [19], we established the well-posedness of multipoint nonlocal boundary value problem

in a Hilbert space H with the self-adjoint positive definite operator A under assumption

The well-posedness of multipoint nonlocal boundary value problem (1) in Hölder spaces with a weight was established. In applications, coercivity inequalities for the solutions of nonlocal boundary value problems for elliptic-parabolic equations were obtained.

In [20], we studied the well-posedness of the first order of accuracy difference scheme for the approximate solution of boundary value problem (1) under assumption (2).

In the present paper, we consider the second order of accuracy difference scheme generated by Crank-Nicholson difference scheme

for the approximate solution of boundary value problem (1) under assumption (2).

The well-posedness of difference scheme (3) in Hölder spaces is established. In applications, the stability, almost coercivity stability, coercivity stability estimates for solutions of the second order of accuracy difference scheme for elliptic-parabolic equations are obtained. Furthermore, the theoretical statements for the solution of the first and second order of accuracy schemes for one-dimensional elliptic-parabolic differential equation are supported by the results of a numerical example.

Let us give some auxiliary lemmas we need below. Throughout the paper, H is a Hilbert space and we denote , where A is a self-adjoint positive definite operator. Then, it is clear that B is a self-adjoint positive definite operator and , where , and which is defined on the whole space H is a bounded operator. The following operators

and

exist and are bounded for a self-adjoint positive operator A. Here,

and I is the identity operator.

Lemma 2.1For any, and, , the solution of problem (3) exists and the following formulas hold:

Proof Clearly, the solution formula of the problem

is [22]:

for any and γ. Equation (9) and the fact that yield Equation (6).

The solution of the problem

satisfies the following formula [21]:

Equation (5) follows from Equations (9) and (11), initial condition , and

Finally, let us obtain formula (7). Combining (5), (6), and the condition

we get

From Equation (4), it follows that

This finishes the proof of Lemma 2.1. □

Here, we study well-posedness of problem (3). First, we give some necessary estimates for , and .

Lemma 2.2For a self-adjoint positive operatorAthe following estimates are satisfied[21,22,24]:

whereis independent ofτ.

From these estimates, it follows that

Now, we study well-posedness of problem (3). Let be the linear space of mesh functions defined on with values in the Hilbert space H. Next, on we denote , , , , , and , Banach spaces with the norms

Theorem 2.1Nonlocal boundary value problem (3) is stable innorm.

Proof By [21], we have

for the solution of boundary value problem (10).

By [22], we have

for the solution of inverse Cauchy difference problem (8).

Then, the proof of Theorem 2.1 is based on stability inequalities (14), (15), and on estimates

for the solution of boundary value problem (3). Estimates (16) and (17) follow from formula (7) and estimates (12), (13). Theorem 2.1 is proved. □

Theorem 2.2Assume thatand. Then, for the solution of difference problem (3), we have the following almost coercivity inequality:

whereis independent not only of, , φbut also ofτ.

Proof By [24], we have

for the solution of inverse Cauchy difference problem (8).

By [21], we have

for the solution of boundary value problem (10). Then, the proof of Theorem 2.2 is based on almost coercivity inequalities (18), (19), and on the estimates

for the solution of boundary value problem (3). The proof of these estimates follows the scheme of papers [21,24] and relies on both formula (7) and estimates (12), (13). This finalizes the proof of Theorem 2.2. □

Theorem 2.3Let the assumptions of Theorem 2.2 be satisfied. Then, boundary value problem (3) is well-posed in Hölder spaces, and the following coercivity inequalities hold:

whereMis independent of, , φ, τandα.

Proof By [24],

for the solution of inverse Cauchy difference problem (8).

By [21], we have

for the solution of boundary value problem (10). Then, the proof of Theorem 2.3 is based on coercivity inequalities (20)-(22) and estimates

for the solution of difference scheme (3). The proof of these estimates follows the scheme of the papers [21,24] and relies on both formula (7) and estimates (12), (13). This is the end of the proof of Theorem 2.3. □

Now, the application of the abstract result is considered. In , let us consider the boundary value problem for multi-dimensional elliptic-parabolic equation

where (), (, ), (, ), and (, ) are given smooth functions. Here, Ω is the unit open cube in the n-dimensional Euclidean space (, ) with boundary S, , and .

The discretization of problem (23) is carried out in two steps. In the first step, the grid sets

are defined. To the differential operator A generated by problem (23), we assign the difference operator by formula

acting in the space of grid functions , satisfying the conditions for all . With the help of , we arrive at the nonlocal boundary value problem

for an infinite system of ordinary differential equations (see [21]).

Secondly, problem (25) is replaced by difference scheme (3), so that the following second order of accuracy difference scheme

is obtained (see [21], [22]).

To formulate the results, we introduce the spaces , , and of the grid functions defined on , equipped with the norms

and

Theorem 3.1Letτandbe sufficiently small positive numbers. Then, solutions of difference scheme (26) satisfy the following stability and almost coercivity estimates:

whereMis independent not only ofτ, h, but also of, and, .

The proof of Theorem 3.1 is based on Theorem 2.1, Theorem 2.2, the symmetry properties of the difference operator defined by formula (24) in , the estimate

and the following theorem in :

Theorem 3.2For the solution of the elliptic difference problem

the following coercivity inequality holds[23]:

HereMis independent ofhand.

Theorem 3.3Letτandbe sufficiently small positive numbers. Then, the solutions of difference scheme (26) satisfy the following coercivity stability estimates:

Here, Mis independent not only ofτ, h, but also of, and, .

The proof of Theorem 3.3 is based on the abstract Theorem 2.3, Theorem 3.2, and the symmetry properties of the difference operator defined by formula (24).

The theoretical statements for the solution of these difference schemes are supported by the results of numerical experiments of the nonlocal boundary value problem

for the elliptic-parabolic equation. The exact solution of this problem is

For the comparison, the errors computed by the following formula

are recorded for different values of N and M, where represents the exact solution and represents the numerical solution at . The results are shown in Table 1 for respectively.

Therefore, the results indicate that the second order of accuracy difference scheme is more accurate than the first order of accuracy difference scheme.

The authors declare that they have no competing interests.

The manuscript was drafted by OG and it is based on his PhD thesis. AA is the supervisor of the thesis and gave detailed comments on the manuscript. All authors read and approved the final manuscript.

The authors are very grateful to Prof. P. E. Sobolevskii (Jerusalem, Israel) for valuable comments to the improvement of this article.

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