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On a nonlocal boundary value problem for two dimensional elliptic equation. (English) Zbl 1046.65084

The paper treats a scalar second order elliptic boundary value problem in the unit square. The differential operator has constant coefficients and is assumed to satisfy the standard ellipticity estimate. On three of the sides, the Dirichlet boundary condition is applied and on the fourth side, the solution is constrained to annihilate a weighted mean value along lines orthogonal to that boundary. The author then proves that all conditions of the Lax-Milgram lemma are fulfilled which implies that the problem possess a unique weak solution in a weighted Sobolev space. Here the weight is the same as in the weighted mean value condition.

In the second part of the paper the author constructs a finite difference scheme for this problem by using a type of average operators. It is shown that under certain conditions, the difference scheme converges in a discrete Sobolev norm to the continuous solution at a prescribed rate. The continuous and discrete case are treated in a similar fashion. This is beneficial for the reader since the discrete case tends to be quite technical.

No numerical tests are supplied.


MSC:
65N06Finite difference methods (BVP of PDE)
35J25Second order elliptic equations, boundary value problems
65N12Stability and convergence of numerical methods (BVP of PDE)