Error estimates for discontinuous galerkin approximations of implicit parabolic equations

We analyze the classical discontinuous Galerkin method for "implicit" parabolic equations. Symmetric error estimates for schemes of arbitrary order are presented. The ideas developed allow certain assumptions frequently required in previous work to be relaxed. For example, different discrete spaces are allowed at each time step, and the spatial operator is not required to be self-adjoint or independent of time. 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.