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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.20039 |
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| _version_ | 1866915879493566464 |
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| author | Gómez, Sergio |
| author_facet | Gómez, Sergio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20039 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the compactness properties of discontinuous Galerkin time discretizations Gómez, Sergio Numerical Analysis 46B50, 35A35, 65M12, 65M60 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. |
| title | A note on the compactness properties of discontinuous Galerkin time discretizations |
| topic | Numerical Analysis 46B50, 35A35, 65M12, 65M60 |
| url | https://arxiv.org/abs/2509.20039 |