In this report we introduce an extended discontinuous Galerkin (XDG) method. Our XDG scheme is based on the Babuska-Zlamal approach and we apply it to a class of prototype elliptic boundary value problems that have solutions consisting of smooth functions perturbed by a set of high frequency modes which occupy a narrow band. The XDG scheme we study is enriched by trigonometric functions that cover the range of these perturbations. A theoretical error analysis is provided that shows the method converges and gives specifics on its accuracy. Computations with the XDG scheme further demonstrate the efficacy of this approach.