Volume 59, pp. 179-201, 2023.

On the numerical solution of an elliptic problem with nonlocal boundary conditions

Zorica Milovanović Jeknić, Bratislav Sredojević, and Dejan Bojović

Abstract

In this paper we consider a class of non-standard elliptic transmission problems in disjoint domains. As a model example, we consider an area consisting of two non-adjacent rectangles. In each subarea, a boundary-value problem of elliptic type is considered, where the interaction between their solutions is described by nonlocal integral conjugation conditions. An a priori estimate for its weak solution in an appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate, compatible with the smoothness of the input data, up to a slowly increasing logarithmic factor of the mesh size, is obtained.

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Key words

transmission problem, boundary-value problem, nonlocal boundary condition, finite differences, Sobolev spaces, convergence rate estimates

AMS subject classifications

65N12, 65N15

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