A note on the compactness properties of discontinuous Galerkin time discretizations

Sergio Gómez

Paper

A note on the compactness properties of discontinuous Galerkin time discretizations

Abstract

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

, and

satisfying

compactly and

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.