BEST APPROXIMATION PROPERTY IN THE${\mathrm{W}}_{\mathrm{\infty }}^{1}$NORM FOR FINITE ELEMENT METHODS ON GRADED MESHES

We consider finite element methods for a model second-order elliptic equation on a general bounded convex polygonal or polyhedral domain. Our first main goal is to extend the best approximation property of the error in the ${\mathrm{W}}_{\mathrm{\infty }}^{1}$ norm, which is known to hold on quasi-uniform meshes, to more general graded meshes. We accomplish it by a novel proof technique. This result holds under a condition on the grid which is mildly more restrictive than the shape regularity condition typically enforced in adaptive codes. The second main contribution of this work is a discussion of the properties of and relationships between similar mesh restrictions that have appeared in the literature.