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

Johnson, C. G. L.

doi:10.1090/S0025-5718-1974-0362943-X

Estimates near plane portions of the boundary for discrete elliptic boundary problems
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Math. Comp. 28 (1974), 909-935 Request permission

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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