Pointwise error estimates for linear finite element approximation to elliptic Dirichlet problems in smooth domains

In this paper, we consider pointwise error estimation for the linear finite element approximation to -delta u + u = f in omega, u = g on gamma, where omega is a bounded domain in R-N (N = 2,3) with smooth boundary gamma. The domain omega is approximated by a polyhedron omega(h) with boundary gamma(h), while the Dirichlet data g is approximated by the Lagrange interpolation or L-2-projection of its transformation to gamma(h). By using a duality argument, the pointwise errors are converted to the approximation errors plus the finite element errors for two classes of regularized Green's functions under the W-1,W-1-norm, while the latter errors are further converted to the local weighted H-1- and L-2-norms estimates by using the dyadic annuli decomposition. Finally, we obtain the convergence rates O(h(2)|log h|) for the L-infinity-norm error and O(h) for the W-1,W-infinity-norm error. Numerical examples are provided to confirm our theoretical findings.