Error estimates for the discontinuous Galerkin methods for parabolic equations

The classical discontinuous Galerkin method for a general parabolic equation is analyzed. Symmetric error estimates for schemes of arbitrary order are presented. The ideas developed below relax many assumptions required in previous work. For example, different discrete spaces may be used at each time step, and the spatial operator need not 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.