A discontinuous hp finite element method for diffusion problems

We present an extension of the discontinuous Galerkin method which is applicable to the numerical solution of diffusion problems. The method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries. Within each element, arbitrary spectral approximations can be constructed with different orders p in each element. We demonstrate that the method is elementwise conservative, a property uncharacteristic of high-order finite elements. For clarity, we focus on a model class of linear second-order boundary value problems, and we develop a priori error estimates, convergence proofs, and stability estimates. The results of numerical experiments on h- and p-convergence rates for representative two-dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence. (C) 1998 Academic Press