Abstract
In this study, the Bitsadze-Samarskii type nonlocal boundary-value problem with integral condition for an elliptic differential equation in a Hilbert space H with self-adjoint positive definite operator A is considered. The second order of the accuracy difference scheme for the approximate solutions of this nonlocal boundary-value problem is presented. The well-posedness 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.
Highlights
1 Introduction In Bitsadze and Samarskii [ ] 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
We introduce the Hilbert space L h = L ([, ]h) of the grid functions φh(x) = {φn}Mn=– defined on [, ]h, equipped with the norms φh L h =
We introduce the Hilbert space L h = L ( h) of the grid functions φh(x) = {φ(h m, . . . , hmmm)} defined on h, equipped with the norms φh L h =
Summary
In Bitsadze and Samarskii [ ] 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. Under the assumption ( ) the solution of the difference scheme ( ) satisfies the following stability estimates: The proof of Theorem is based on Theorem and the symmetry properties of the difference operator Axh defined by ( ) in L h. Under the assumption ( ) the solution of the difference scheme ( ) satisfies the following almost coercivity estimates: The proof of Theorem is based on Theorem , on the estimate ( ), on the symmetry properties of the difference operator Axh defined by ( ) in L h, and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L h. The proof of Theorem is based on Theorem , on the symmetry properties of the difference operator Axh defined by the formula ( ), and on Theorem on the coercivity inequality for the solution of the elliptic difference equation in L h
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