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

TL;DR

The paper develops a unified framework to analyze discretizations of PDEs/FDEs on moving or unbounded domains through approximating class of sequences (a.c.s.) and its generalization (g.a.c.s.), linking the spectral and singular value distributions of discretization matrices to symbol functions via GLT theory. It extends classical results to exhaustions of domains with finite or infinite measure and to cases where discretization matrices have varying dimensions, enabling analysis of FE and FD schemes on complex domains. Theoretical results are supported by numerical experiments on (i) bounded intervals, (ii) two-dimensional unbounded finite-measure domains, and (iii) variable-coefficient problems, showing that the eigenvalue/spectral distributions converge to the predicted symbols. This work provides a practical pathway to predict and accelerate solvers for large-scale discretizations and highlights open questions for infinite-measure domains and non-normal spectra. The approach has potential implications for robust solver design in simulations with evolving geometries and nonlocal operators.

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.

Figures (65)

• Figure 1: Graph over $[0,\pi]$ of the eigenvalue functions $\lambda_i(\mathbf f_2(\theta)),\ i=1,2,$ and of the pairs $\left(\frac{k\pi}{n+1},\lambda_k(K_n^{(2)})\right),\ k=1,\ldots,n,$ and $\left(\frac{(2n-k)\pi}{n+1},\lambda_k(K_n^{(2)}) \right),\ k=n+1,\ldots,2n-1,$ for $n=40$.
• Figure 2: Eigenvalues distribution of $A_n$ for different $h$ values together with the sampling of $f(\theta_1,\theta_2)=(2-2\cos \theta_1)+(2-2\cos \theta_2)$.
• Figure 3: Minimal distance of eigenvalues of $A_n$ from $f(\theta_1,\theta_2)=(2-2\cos \theta_1)+(2-2\cos \theta_2)$ for different $h$ values.
• Figure 4: Eigenvalues distribution of $C_{n,t}$ for different $h$ values together with the sampling of $f(\theta_1,\theta_2)=(2-2\cos \theta_1)+(2-2\cos \theta_2)$ and $t=2$.
• Figure 5: Minimal distance of eigenvalues of $C_{n,t}$ from $f(\theta_1,\theta_2)=(2-2\cos \theta_1)+(2-2\cos \theta_2)$ and $t=2$ for different $h$ values.

Theorems & Definitions (16)

• Definition 2.1
• Definition 2.2
• Theorem 2.1
• Theorem 3.1: TillinotaTillicomplexDNS
• Theorem 4.1
• proof
• Theorem 4.2
• Definition 4.1
• Remark 4.1
• Theorem 4.3