Convergence estimates for essentially positive type discrete Dirichlet problems

Bramble, James H.; Hubbard, B. E.; Thomée, Vidar

doi:10.1090/s0025-5718-1969-0266444-7

“…There are many results in the literature (cf. [5]) for difference approximations of the Dirichlet problem in a plane domain £2, (7 5~\ Au = f in £2, u = g on 9 £2, which near F reduce to the following difference approximation:…”

Section: Estimates For a Boundarymentioning

confidence: 99%

Estimates near plane portions of the boundary for discrete elliptic boundary problems

Johnson¹

1974

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Abstract. We consider an elliptic difference operator together with certain boundary difference operators near a plane portion of the boundary parallel to some coordinate direction.We prove discrete analogues of known estimates in L and Schauder normsfor elliptic boundary problems. The discrete estimates are then used to prove results about convergence near plane portions of the boundary of difference quotients of solutions u" of a discrete elliptic problem to the derivatives of the solution u of the corresponding continuous problem, when it is known that un converges to u in the maximum norm or in a discrete L" norm as h tends to zero. Key words and phrases. Elliptic difference operator, discrete elliptic boundary problem, discrete L and Schauder estimates, convergence of difference quotients.

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“…The methods in [5,10] are necessarily totally different from ours. Earlier work in the 1960's dealt with extensions of L and Schauder estimates to elliptic difference equations; see for example [3,4,13], also [6] for a more modern treatment. Unlike the hypotheses of the latter theories, our coefficients are random in the sense that their values at neighboring mesh points are independent of the mesh length.…”

Section: Introductionmentioning

confidence: 99%

Linear Elliptic Difference Inequalities with Random Coefficients

Kuo¹,

Trudinger²

1990

Mathematics of Computation

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“…The methods in [5,10] are necessarily totally different from ours. Earlier work in the 1960's dealt with extensions of L and Schauder estimates to elliptic difference equations; see for example [3,4,13], also [6] for a more modern treatment. Unlike the hypotheses of the latter theories, our coefficients are random in the sense that their values at neighboring mesh points are independent of the mesh length.…”

Section: Introductionmentioning

confidence: 99%