Interior maximum-norm estimates for finite element methods. II

Abstract

We consider bilinear forms A ( ∙ , ∙ ) A( \bullet , \bullet ) connected with second-order elliptic problems and assume that for u h {u_h} in a finite element space S h {S_h} , we have A ( u − u h , χ ) = F ( χ ) A(u - {u_h},\chi ) = F(\chi ) for χ \chi in S h {S_h} with local compact support. We give local estimates for u − u h u - {u_h} in L ∞ {L_\infty } and W ∞ 1 W_\infty ^1 of the type "local best approximation plus weak outside influences plus the local size of F ".