Local pointwise convergence of the 3D finite element

Abstract

For an elliptic problem with variable coefficients in three dimensions, this article discusses local pointwise convergence of the three-dimensional (3D) finite element. First, the Green’s function and the derivative Green’s function are introduced. Secondly, some relationship of norms such as L²-norms, W^1,∞-norms, and negative-norms in locally smooth subsets of the domain Ω is derived. Finally, local pointwise convergence properties of the finite element approximation are obtained.