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

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 $D_t\subset {\mathbb{R}}^d$ 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.

57 pages, 64 figures