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

Abstract

In this paper, we consider pointwise error estimation for the linear finite element approximation to −Δu + u = f in Ω, u = g on Γ, where Ω is a bounded domain in $\mathbb {R}^{N}$ (N = 2,3) with smooth boundary Γ. The domain Ω is approximated by a polyhedron Ω_h with boundary Γ_h, while the Dirichlet data g is approximated by the Lagrange interpolation or L²-projection of its transformation to Γ_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^1,1-norm, while the latter errors are further converted to the local weighted H¹- and L²-norms estimates by using the dyadic annuli decomposition. Finally, we obtain the convergence rates $O(h^{2}|\log h|)$ for the $L^{\infty }$-norm error and O(h) for the $W^{1,\infty }$-norm error. Numerical examples are provided to confirm our theoretical findings.