Can the a.c.s. notion and the GLT theory handle approximated PDEs/FDEs with either moving or unbounded domains? (2405.20150v1)

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.