Can the a.c.s. notion and the GLT theory handle approximated PDEs/FDEs with either moving or unbounded domains?

TL;DR

This paper investigates the use of the a.c.s. notion and GLT theory to analyze approximated PDEs and FDEs on moving or unbounded domains, providing theoretical results, examples, and numerical tests.

Contribution

It extends the a.c.s. concept to handle approximations of PDEs/FDEs on unbounded or moving domains, linking spectral distribution and domain exhaustion.

Findings

Results on spectral distribution of matrix sequences with symbols on unbounded domains.

Extension of a.c.s. concept to sequences with different dimensions.

Numerical tests illustrating the theoretical developments.

Abstract

In the current note we consider matrix-sequences $\{B_{n,t}\}_n$ 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 $D_t\subset \R^{d}$ of finite measure, $d\ge 1$. Furthermore, we assume that $ \{ \{ B_{n,t}\} : \, t > 0 \} $ is an approximating class of sequences (a.c.s.) for $ \{ A_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 \} $, 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\} $. 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/FDEs 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.