Volume 63, pp. 424-451, 2025.
Revisiting the notion of approximating class of sequences for handling approximated PDEs on moving or unbounded domains
Andrea Adriani, Alec Jacopo Almo Schiavoni-Piazza, Stefano Serra-Capizzano, and Cristina Tablino-Possio
Abstract
In the current work we consider matrix sequences $\{B_{n,t}\}_n$, with matrices of increasing sizes, depending on $n$, and equipped with a parameter $t>0$. For every fixed $t>0$, we assume that each $\{B_{n,t}\}_n$ possesses a canonical spectral/singular values symbol $f_t$, defined on $D_t\subset \mathbb{R}^{d}$, which are sets of finite measure, for $d\ge 1$. Furthermore, we assume that $ \{ \{ B_{n,t}\}_n : \, t > 0 \} $ is an approximating class of sequences (a.c.s.) for $ \{ A_n \}_n $ and that $ \bigcup_{t > 0} D_t = D $ with $ D_{t + 1} \supset D_t $. Under such assumptions and via the notion of a.c.s, we prove results on the canonical distributions of $ \{ A_n \}_n $, whose symbol, when it exists, can be defined on the, possibly unbounded, domain $D$ of finite or even infinite measure. We then extend the concept of a.c.s.~to the case where the approximating sequence $ \{ B_{n,t}\}_n $ has possibly a different dimension than the one of $ \{ A_n\}_n $. This concept seems to be particularly natural when dealing, e.g., with the approximation both of a partial differential equation (PDE) and of its (possibly unbounded or moving) domain $D$, using an exhausting sequence of domains $\{ D_t \}$. Examples coming from approximated PDEs with either moving or unbounded domains are presented in connection with the classical and the new notion of a.c.s., while numerical tests and a list of open questions conclude the present work.
Full Text (PDF) [2.2 MB], BibTeX , DOI: 10.1553/etna_vol63s424
Key words
discretization of PDEs, moving/unbounded domains, spectral distribution of matrix sequences, (generalized) approximating class of sequences, generalized locally Toeplitz (GLT) matrix sequences, GLT theory
AMS subject classifications
15A18, 65N, 47B
Links to the cited ETNA articles
| [5] | Vol. 53 (2020), pp. 28-112 Giovanni Barbarino, Carlo Garoni, and Stefano Serra-Capizzano: Block generalized locally Toeplitz sequences: theory and applications in the unidimensional case |
| [6] | Vol. 53 (2020), pp. 113-216 Giovanni Barbarino, Carlo Garoni, and Stefano Serra-Capizzano: Block generalized locally Toeplitz sequences: theory and applications in the multidimensional case |
| [45] | Vol. 59 (2023), pp. 1-8 Alec Jacopo Almo Schiavoni-Piazza and Stefano Serra-Capizzano: Distribution results for a special class of matrix sequences: joining approximation theory and asymptotic linear algebra |
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