Elliptic difference equations and interior regularity

Thomée, Vidar; Westergren, Bertil

doi:10.1007/bf02161842

Cited by 60 publications

(18 citation statements)

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“…In § 2. we define the discrete Sobolev spaces and show a number properties of these spaces, most of them being équivalents of well-known properties of the continuous Sobolev spaces. The properties proved in this section can also be found, possibly in slightly different forms, in the existing literature ( [5] mainly, [12], [13], [14]), but there the results are either less gênerai or stated without (satisfactory) proof. In particular, the author has never found proofs of the lemmas 2.4 and 2.6 (for s ^ N) in the literature.…”

Section: 22)mentioning

confidence: 67%

Discrete Sobolev spaces and regularity of elliptic difference schemes

Stevenson¹

1991

ESAIM: M2AN

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Section: 22)mentioning

confidence: 67%

Discrete Sobolev spaces and regularity of elliptic difference schemes

Stevenson¹

1991

ESAIM: M2AN

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“…Then by using an interpolation inequality (e.g Thom_e and Westergren [11]) the first inequality of the lemma is easily proven. The second estimate is proved in a similar manner; the proof will be omitted.…”

Section: Estimates On the Normal Differencesmentioning

confidence: 99%

Regularity Estimates up to the Boundary for Elliptic Systems of Difference Equations

Strikwerda

Wade

Bube

1990

SIAM J. Numer. Anal.

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We prove regularity estimates up to the boundary for solutions of elliptic systems of finite difference equations. The regularity estimates, obtained for boundary-fitted coordinate systems on domains with smooth boundary, involve discrete Sobolev norms and are proved using pseudo-difference operators to treat systems with variable coefficients. The elliptic systems of difference equations and the boundary conditions which are considered are very general in form. We prove that regularity of a regular elliptic system of difference equations is equivalent to the nonexistence of "eigensolutions". The regularity estimates obtained are analogous to those in the theory of elliptic systems of partial differential equations, and to the results of Gustafsson, Kreiss, and SundstrSm [1972] and others for hyperbolic difference equations.

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“…It is well-known (cf. [6]) that such a finite différence operator L h is consistent with For mesh-functions we define, with k a non-negative integer, the following norms IYK* ha / |»|*.«.fc = S \^v\h,cï with |ü| ftjQ |rl<fc and for 0 < e < 1, For mesh-functions with finite support we also use the inner product…”

Section: Ellipttc Unite Difference Operatorsmentioning

confidence: 99%