Abstract
An a posteriori error bound for the maximum (pointwise) error for the interior penalty discontinuous Galerkin method for a standard elliptic model problem on polyhedral domains is presented. The computational domain is not required to be Lipschitz, thus allowing for domains with cracks and other irregular polyhedral domains. The proof is based on the direct use of Green's functions and varies substantially from the approach used in previous proofs of similar $L_\infty$ estimates for (continuous) finite element methods in the literature. Numerical experiments indicating the good behavior of the resulting a posteriori bounds within an adaptive algorithm are also presented.
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