Revisiting the notion of approximating class of sequences for handling approximated PDEs on moving or unbounded domains

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.