Abstract
This paper is devoted to a Bitsadze-Samarskii type overdetermined multipoint nonlocal boundary value problem (NBVP). This inverse problem is reduced to an auxiliary multipoint NBVP. Stability estimates for the solution of the auxiliary NBVP are established. The finite difference method is applied to get the first and second order of accuracy of approximate solutions of the abstract overdetermined problem. Stability estimates for the solution of difference problems are proved. Then the established abstract results are applied to get stability estimates for the solution of the Bitsadze-Samarskii type overdetermined elliptic multidimensional differential and difference problems with multipoint nonlocal boundary conditions (NBVC). Finally, numerical results with explanation on the realization for two dimensional and three dimensional elliptic overdetermined multipoint NBVPs in test examples are presented.
Highlights
The theory and methods of the solutions of inverse problems of determining the parameter of partial differential equations have been extensively studied by several researchers.The well-posedness of source identification problems for elliptic type differential and difference equations was studied in [ – ]
It is well known that Axh is a self-adjoint positive definite operator
5 Numerical results we present numerical results with explanation on the realization for two dimensional and three dimensional examples of the Bitsadze-Samarskii type overdetermined elliptic multipoint nonlocal boundary value problem (NBVP)
Summary
The theory and methods of the solutions of inverse problems of determining the parameter of partial differential equations have been extensively studied by several researchers (see [ – ] and the references therein). Theorem Under the assumptions that φ, ψ, ζ ∈ D(A), fτ ∈ Cτα,α(H) the solution (p, {uk}Nk=– ) of the difference problems ( ) and ( ) in Cτ (H) × H obeys the following stability estimates:. Theorem Under the assumptions φ, ψ, ζ ∈ D(C), fτ ∈ Cτα,α(H) (α ∈ ( , )) the solution (p, {uk}Nk=– ) of the difference problems ( ) and ( ) obeys the coercive stability estimate uk+ – uk + uk– τ k= Cτ (H). Acting in the space of grid functions uh(x), satisfying the condition uh(x) = for all x ∈ Sh. It is well known that Axh is a self-adjoint positive definite operator. The proof of Theorem is based on the symmetry property of the operator Axh in L h and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L h.
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