Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds

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.