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arXiv · 2509.20039

A note on the compactness properties of discontinuous Galerkin time discretizations

Sergio Gómez
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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 $X$, $B$, 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.

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Sergio Gómez. 2025-09-24. A note on the compactness properties of discontinuous Galerkin time discretizations. https://doi.org/10.1016/j.nonrwa.2026.104639

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