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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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

Author: C. G. L. Johnson
Journal: Math. Comp. 28 (1974), 909-935
MSC: Primary 65N15
MathSciNet review: 0362943
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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_p}$ and Schauder norms for 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_h}$ of a discrete elliptic problem to the derivatives of the solution u of the corresponding continuous problem, when it is known that $ {u_h}$ converges to u in the maximum norm or in a discrete $ {L_p}$ norm as h tends to zero.

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Keywords: Elliptic difference operator, discrete elliptic boundary problem, discrete $ {L_p}$ and Schauder estimates, convergence of difference quotients
Article copyright: © Copyright 1974 American Mathematical Society