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

Abstract

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.

On real interpolation, finite differences, and estimatesdepending on a parameter for discretizations of elliptic boundary value problems

“…Concerning the results of this type, we mention again the book [3,Chapter 4] where analogs of estimate depending on a parameter in R n are given. Estimates in a half space for boundary value problems (not depending on a parameter) were given also by Johnson [13], while Thomée had proved the analogs of interior estimates in [21].…”