A note on the compactness properties of discontinuous Galerkin time discretizations of nonlinear evolution problems

This work extends the discrete compactness results of Walkington (SIAM J. Numer. Anal., 47(6):4680–4710, 2010) for high-order discontinuous Galerkin time discretizations of parabolic problems to more general function space settings. In particular, we show a discrete version of the Aubin–Lions–Simon lemma that holds for general Banach spaces X, B , and Y satisfying X ↪ B compactly and B ↪ Y continuously. Our proofs rely on the properties of a time reconstruction operator and remove the need for quasi-uniform time partitions assumed in previous works. Thus, we provide a useful and flexible tool for the analysis of high-order discontinuous Galerkin time discretizations of complex nonlinear partial differential equations.