Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds

TL;DR

The paper develops a modified hybrid high-order (HHO) eigensolver to compute guaranteed lower eigenvalue bounds (GLB) for the Laplacian. A pivotal advance is a p-robust stabilization strategy that leverages a bound $C_{ m st,2}$, enabling a simple, robust parameter choice and a GLB criterion $oldsymbol{\sigma_2^2 ext{max}igraceeta, h_{ ext{max}}^2 ext{min}igraceoldsymbol{\lambda_h,oldsymbol{\lambda}}igraceig floor oldsymbol{ extless}oldsymbol{oldsymbol{\alpha}}$, with $eta=oldsymbol{oldsymbol{\alpha}}/oldsymbol{oldsymbol{\sigma_2^2}}$. The authors provide a rigorous a priori error analysis and a stabilization-free a posteriori error estimator, together with an adaptive AFEM that attains optimal convergence rates for high-order discretizations on challenging domains. Numerical experiments on L-shaped, isospectral, and dumbbell-slit domains corroborate the theoretical findings, demonstrating that higher polynomial degrees yield tighter GLBs and that the estimator drives effective refinement. Overall, the work offers a robust framework for high-order GLB computation with reliable a posteriori control and practical adaptivity for complex geometries.

Abstract

The higher-order guaranteed lower eigenvalue bounds of the Laplacian in the recent work by Carstensen, Ern, and Puttkammer [Numer. Math. 149, 2021] require a parameter $C_{\mathrm{st},1}$ that is found $\textit{not}$ robust as the polynomial degree $p$ increases. This is related to the $H^1$ stability bound of the $L^2$ projection onto polynomials of degree at most $p$ and its growth $C_{\rm st, 1}\propto (p+1)^{1/2}$ as $p \to \infty$. A similar estimate for the Galerkin projection holds with a $p$-robust constant $C_{\mathrm{st},2}$ and $C_{\mathrm{st},2} \le 2$ for right-isosceles triangles. This paper utilizes the new inequality with the constant $C_{\mathrm{st},2}$ to design a modified hybrid high-order (HHO) eigensolver that directly computes guaranteed lower eigenvalue bounds under the idealized hypothesis of exact solve of the generalized algebraic eigenvalue problem and a mild explicit condition on the maximal mesh-size in the simplicial mesh. A key advance is a $p$-robust parameter selection. The analysis of the new method with a different fine-tuned volume stabilization allows for a priori quasi-best approximation and improved $L^2$ error estimates as well as a stabilization-free reliable and efficient a posteriori error control. The associated adaptive mesh-refining algorithm performs superior in computer benchmarks with striking numerical evidence for optimal higher empirical convergence rates.

Figures (11)

• Figure 1: Approximations of $C_{\rm st,1}^2$ and $C_{\rm st,2}^2$ as a function of the polynomial degree $p$ on the equilateral triangle , the right-isosceles triangle , $\coloneqq \mathrm{conv}\{(0,0), (1.5,0), (0,1)\}$, and $\coloneqq\mathrm\{(0,0), (1,0), (-1/2, 1)\}$.
• Figure 2: Dependence of $C_{\rm st,2}^2$ on the interior angle $\omega$ of the isosceles triangle $T=\mathrm{conv}\{(0,0), (1,0), (\cos(\omega),\sin(\omega))\}$
• Figure 3: Initial triangulations of the L-shaped domain in \ref{['sec:L-shape']}, the isospectral drum in \ref{['sec:isospectral-domain']}, and the dumbbell-slit domain in \ref{['sec:dumbbellslit']}.
• Figure 4: Polynomial degrees $p = 0, \dots, 4$ in the numerical benchmarks of \ref{['sec:numerical-examples']}.
• Figure 5: Convergence history plot of $\lambda(1) - \mathrm{GLB}(1)$ (left) and $\eta^2$ (right) for polynomial degrees $p$ from \ref{['fig:legend']} on uniform (dashed line) and adaptive (solid line) triangulations of the L-shaped domain in \ref{['sec:L-shape']}.

Theorems & Definitions (27)

• Theorem 2.1
• proof
• Lemma 2.2: $p$-robust stability
• Theorem 2.3: stability constant
• proof
• Example 2.4: numerical example
• Conjecture
• Lemma 3.1: commutativity
• Lemma 3.2: discrete norm
• proof