Discrete Sobolev spaces and regularity of elliptic difference schemes. (English) Zbl 0733.65058

Author’s summary: This paper is concerned with the regularity of elliptic finite difference schemes with respect to discrete (fractional order) Sobolev spaces. For schemes arising from discretizations that are from the same “type” at the boundary as in the interior, it proves the discrete equivalent of J. Nečas’ regularity theorem for differential operators on Lipschitz regions [Rev. Roum. Math. Pures Appl. 9, 47-69 (1964; Zbl 0196.407)]. A different proof was given by W. Hackbusch [Ark. Mat. 19, 71-95 (1981; Zbl 0462.65058)]. However, the proof here is shorter and more transparent.
In case of a curved boundary usually different discretizations are applied in points near the boundary. For schemes of this kind, it is shown by using Nečas’ theorem for the corresponding “unperturbed” scheme, that “minimal” regularity implies the stronger regularity from Nečas’ theorem. Finally, conditions sufficient for minimal regularity are given.


65N06 Finite difference methods for boundary value problems involving PDEs
35J40 Boundary value problems for higher-order elliptic equations
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