Pointwise error estimates of the local discontinuous Galerkin method for a second order elliptic problem

In this paper we derive some pointwise error estimates for the local discontinuous Galerkin (LDG) method for solving second-order elliptic problems in $R^N$ ($N\geq 2$). Our results show that the pointwise errors of both the vector and scalar approximations of the LDG method are of the same order as those obtained in the $L^2$ norm except for a logarithmic factor when the piecewise linear functions are used in the finite element spaces. Moreover, due to the weighted norms in the bounds, these pointwise error estimates indicate that when at least piecewise quadratic polynomials are used in the finite element spaces, the errors at any point $z$ depend very weakly on the true solution and its derivatives in the regions far away from $z$. These localized error estimates are similar to those obtained for the standard conforming finite element method.