A Scalable Two-Level Domain Decomposition Eigensolver for Periodic Schrödinger Eigenstates in Anisotropically Expanding Domains

TL;DR

This paper introduces a two-level domain decomposition preconditioner that addresses this issue for the linear Schr\"odinger eigenvalue problem, even in the presence of a vanishing eigen value gap in non-uniform, expanding domains.

Abstract

Accelerating iterative eigenvalue algorithms is often achieved by employing a spectral shifting strategy. Unfortunately, improved shifting typically leads to a smaller eigenvalue for the resulting shifted operator, which in turn results in a high condition number of the underlying solution matrix, posing a major challenge for iterative linear solvers. This paper introduces a two-level domain decomposition preconditioner that addresses this issue for the linear Schrödinger eigenvalue problem, even in the presence of a vanishing eigenvalue gap in non-uniform, expanding domains. Since the quasi-optimal shift, which is already available as the solution to a spectral cell problem, is required for the eigenvalue solver, it is logical to also use its associated eigenfunction as a generator to construct a coarse space. We analyze the resulting two-level additive Schwarz preconditioner and obtain a condition number bound that is independent of the domain's anisotropy, despite the need for only one basis function per subdomain for the coarse solver. Several numerical examples are presented to illustrate its flexibility and efficiency.

Figures (7)

• Figure 1: (a) Geometric setup of $\Omega_L$ with $p=2$ expanding and $q=1$ fixed directions with dimensions $L=5.5,\ell=2$. (b) Iteration number estimates for an inner-outer eigenvalue algorithm using IPM/CG for the Laplacian EVP on $(0,L)\times(0,1)$ using finite differences ($h=1/10$) and different shifts $\sigma$. Note that the arbitrary scaling of $n_{\textnormal{tot}}$ is only applied for better visualization.
• Figure 1: Finite element representation of the (a) factorization principle from \ref{['eq:factorization-short']} and the (b) coarse space basis functions from \ref{['eq:algebraic-coarse-space']} for the equal-weights partition of unity and an overlap region of two elements between subdomains.
• Figure 1: Sketch of the non-overlapping $\{\Omega_i'\}_{i=1}^8$ (black border), overlapping $\{\Omega_i\}_{i=1}^8$ (white border), and periodic neighborhood decomposition $\{\tilde{\Omega}_i\}_{i=1}^8$ (cross-hatch) of $\Omega_4 := (0,4) \times (0,2)$ for an overlap (dark shades) of (a)$\delta=1$ and (b)$\delta=2$ layers of elements. An increase of the periodic neighborhood from $\tilde{\Omega}_1 = (0,2) \times (0,2)$ for $\delta=1$ to $\tilde{\Omega}_1 = (0,3) \times (0,2)$ for $\delta=2$ can be observed.
• Figure 1: CG residual norms for varying domain length $L$ using the AS preconditioner with no (left) and the PerFact coarse space (right).
• Figure 2: A union of disks domain $\Omega_N$ for $N=4$ with (a) the applied symmetric potential $V$ (truncated to $[-10,-1]$ for visualization) and an exemplary $\mathbb{P}_1$ mesh and (b) the resulting first eigenfunction $\phi$. Both color scales divided the listed interval into 14 colors.

Theorems & Definitions (33)

• Definition 2.1
• Remark 2.2
• Definition 3.1: finite coloring bastianMultilevelSpectralDomain2022
• Definition 3.2: stable decomposition spillaneAbstractRobustCoarse2014 bastianMultilevelSpectralDomain2022
• Lemma 3.3: partition of unity
• Proof 1
• Remark 3.4
• Definition 3.5: periodic neighborhood
• Definition 3.6: periodic neighborhood intersection multiplicity
• Definition 3.7: PerFact coarse space