Error estimates for discontinuous Galerkin time-stepping schemes for the parabolic p-Laplacian: A quasi-norm approach

Error estimates for arbitrary order fully-discrete schemes for the parabolic $p$-Laplacian are considered. The schemes combine the discontinuous Galerkin time-stepping approach for the temporal discretization with classical conforming finite elements in space. In particular, a symmetric -C\'ea Lemma type- error estimate is established for a suitable quasi-norm, under minimal regularity assumptions on the data. The above estimate leads to error bounds of arbitrary order in space and time provided that the necessary regularity is present, without imposing any restrictions between the temporal and spatial discretization parameters. The symmetric structure of the estimate also leads to various error estimates at partition points as well as for the natural energy $L^p(I;W^{1,p}(\Omega))$ norm. Furthermore, an unconditional $L^{\infty}(I;L^2(\Omega))$ stability and error estimate is proved under minimal regularity assumptions, as well as an optimal $L^{\infty}(I;L^2(\Omega))$ error estimate under a suitable restriction between the temporal and spatial discretization parameters and additional regularity of the solution.