Finite Element Pointwise Results on Convex Polyhedral Domains

Abstract

The main goal of the paper is to establish that the $L^1$ norm of jumps of the normal derivative across element boundaries and the $L^1$ norm of the Laplacian of a piecewise polynomial finite element function can be controlled by corresponding weighted discrete $H^2$ norm on convex polyhedral domains. In the finite element literature such results are only available for piecewise linear elements in two dimensions and the extension to convex polyhedral domains is rather technical. As a consequence of this result, we establish almost pointwise stability of the Ritz projection and the discrete resolvent estimate in $L^\infty$ norm.