Optimal 𝐿^{∞} estimates for the finite element method on irregular meshes

Abstract

Uniform estimates for the error in the finite element method are derived for a model problem on a general triangular mesh in two dimensions. These are optimal if the degree of the piecewise polynomials is greater than one. Similar estimates of the error are also derived in L p {L^p} . As an intermediate step, an L 1 {L^1} estimate of the gradient of the error in the finite element approximation of the Green’s function is proved that is optimal for all degrees.