Well-Posed and Ill-Posed Boundary Value Problems for PDEView this Special Issue
FDM for Elliptic Equations with Bitsadze-Samarskii-Dirichlet Conditions
A numerical method is proposed for solving nonlocal boundary value problem for the multidimensional elliptic partial differential equation with the Bitsadze-Samarskii-Dirichlet condition. The first and second-orders of accuracy stable difference schemes for the approximate solution of this nonlocal boundary value problem are presented. The stability estimates, coercivity, and almost coercivity inequalities for solution of these schemes are established. The theoretical statements for the solutions of these nonlocal elliptic problems are supported by results of numerical examples.
Many problems in fluid mechanics, dynamics, elasticity, and other areas of engineering, physics, and biological systems lead to partial differential equations of elliptic type. The role played by coercive inequalities in the study of local boundary-value problems for elliptic and parabolic differential equations is well known (see, e.g., [1, 2]).
In the present paper, we consider the Bitsadze-Samarskii type nonlocal boundary value problem for elliptic differential equations in a Hilbert space with self-adjoint positive definite operator . It is known (see, e.g., [3–11]) that various nonlocal boundary value problems for elliptic equations can be reduced to the boundary value problem (1.1). The simply nonlocal boundary value problem was presented and investigated for the first time by Bitsadze and Samarskii . Further, methods of solutions of Bitsadze-Samarskii nonlocal boundary value problems for elliptic differential equations have been studied extensively by many researchers (see [13–21] and the references given therein).
A function is called a solution of problem (1.1) if the following conditions are satisfied.(i) is twice continuously differentiable on the segment . Derivatives at the endpoints of the segmentare understood asthe appropriate unilaterial derivatives.(ii)The element belongs to for all , and the function is continuous on .(iii) satisfies the equation and nonlocal boundary condition in (1.1).
Let be the open unit cube in with boundary . In present paper, we are interested in studying the stable difference schemes for the numerical solution of the following nonlocal boundary value problem for the multidimensional elliptic equation Here , and are given smooth functions, is a large positive constant and .
In the present paper, the first and second-orders of accuracy difference schemes are presented for the approximate solution of problem (1.2). The stability and coercive stability estimates for the solution of these difference schemes are obtained. A numerical method is proposed for solving nonlocal boundary value problem for the multidimensional elliptic partial differential equation with the Bitsadze-Samarskii-Dirichlet condition. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of two-dimensional elliptic partial differential equations.
2. Difference Schemes: Well-Posedness
The discretization of problem (1.2) is carried out in two steps. In the first step let us define the grid sets as follows: We introduce the Hilbert space of the grid functions defined on , equipped with the norm and the Hilbert space defined on , equipped with the norm Finally, we introduce the Banach spaces and of grid abstract function defined on with values , equipped with the following norms:
To the differential operator generated by the problem (1.2) we assign the difference operator by the formula acting in the space of grid functions , satisfying the condition for all . It is known that is a self-adjoint positive definite operator in (). With the help of , we arrive at the nonlocal boundary value problem for an infinite system of the following ordinary differential equations: In the second step, we replaced problem (2.6) by the first-order of accuracy difference scheme as follows: and the second order of accuracy difference scheme as follows: Now, we will study the well-posedness of (2.7) and (2.8). We have the following theorem on stability of (2.7) and (2.8).
Theorem 2.2. Let and be sufficiently small positive numbers. Then the solutions of difference schemes (2.7) and (2.8) satisfy the following almost coercive stability estimate: where does not depend on , and .
Proof. The proof of (2.14) is based on the formulas (2.11), (2.12), (2.13), the symmetry properties of the difference operator defined by the formula (2.5) and on the following theorem on well-posedness of the elliptic difference problem.
Theorem 2.3. For the solutions of the elliptic difference problem the following coercivity inequality holds (see ): where M does not depend on and .
Proof. The proof of (2.17) is based on the formula for (2.7) and for (2.8), the symmetry properties of the difference operator defined by the formula (2.5) and on Theorem 2.3 on well-posedness of the elliptic difference problem.
3. Numerical Analysis
We have not been able to obtain a sharp estimate for the constants figuring in the stability inequality. Therefore, we will give the following results of numerical experiments of the Bitsadze-Samarskii-Dirichlet problem: for the two-dimensional elliptic equation.
The exact solution of this problem is For the approximate solution of problem (3.1), we consider the set of a family of grid points depending on small parameters and as follows:
We have system of linear equations in (3.4) and we will write them in the following matrix form: Here, where and We seek a solution of the matrix equation in following form: where are square matrices and are column matrices defined by (see, ) as follows: where
So, again we have system of linear equations in (3.11) and we will write them in the following matrix form: where