A note on the compactness properties of discontinuous Galerkin time discretizations
Provides a theoretical tool for analysts working on convergence of high-order DG time discretizations for nonlinear PDEs; incremental extension of existing results.
This work extends discrete compactness results for high-order discontinuous Galerkin time discretizations to general Banach spaces, proving a discrete Aubin-Lions-Simon lemma without requiring quasi-uniform time partitions. This provides a flexible tool for analyzing nonlinear PDEs.
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 $X \hookrightarrow B$ compactly and $B \hookrightarrow Y$ 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.