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 {Bn,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 {Bn,t}n possesses a canonical spectral/singular values symbol ft, defined on Dt ⊂ Rd, which are sets of finite measure, for d ≥ 1. Furthermore, we assume that {{Bn,t}n : t > 0} is an approximating class of sequences (a.c.s.) for {An}n and that St>0 Dt = D with Dt+1 ⊃ Dt. Under such assumptions and via the notion of a.c.s, we prove results on the canonical distributions of {An}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 {Bn,t}n has possibly a different dimension than the one of {An}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 {Dt}. 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.