Error Estimates for the Discontinuous Galerkin Methods for Parabolic Equations

We analyze the classical discontinuous Galerkin method for a general parabolic equation. Symmetric error estimates for schemes of arbitrary order are presented. The ideas we develop allow us to relax many assumptions freqently required in previous work. For example, we allow different discrete spaces to be used at each time step and do not require the spatial operator to be self adjoint or independent of time. Our error estimates are posed in terms of projections of the exact solution onto the discrete spaces and are valid under the minimal regularity guaranteed by the natural energy estimate. These projections are local and enjoy optimal approximation properties when the solution is sufficiently regular.