A DiscontinuoushpFinite 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 orderspin 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 developpriorierror estimates, convergence proofs, and stability estimates. The results of numerical experiments onh- andp-convergence rates for representative two-dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence.

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