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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 Lp 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 uh of a discrete elliptic problem to the derivatives of the solution u of the corresponding continuous problem, when it is known that uh converges to u in the maximum norm or in a discrete L norm as h tends to zero. 0. Introduction. Denote the coordinates in Ed+l by x = (x0, x')= (x0, X1, , ), let a = (a0, ) = -(a1 * *, ad) be multi-indices (a1 nonnegative integers), lal 2;a, Ia'I= Ed a1, and set D' = (a/axo)a0 ld and D' = (a/ax)1 . .*(a/axd)?d. Let Q be a bounded domain in the half space Had+ = {x: X0 > 0} such that ai2_ n {xoo} contains an open set F in the plane x0 = 0. Consider the boundary problem Au=f in Q2, (0.1) Bku=g9 on r for k= 1, ,m, where A = Zloel2m acuD? and Bk = z loll m bk f, D' are differential operators with constant coefficients and no lower order terms. If (0.1) is elliptic, i.e., if A is properly elliptic and the Bk satisfy a certain complementary condition, then one can prove the so-called "Schauder estimates up to the boundary" (see Agmon-DouglisNirenberg [ 1] ): PROPOSITION 0.1. For any domain Q' CC Q and any noninteger s > max(2m, ml, . .. , mm), one has Received September 12, 1973. AMS (MOS) subject classifications (1970). Primary 65M10; Secondary 65M10.

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