The higher-order guaranteed lower eigenvalue bounds of the Laplacian in the recent work by Carstensen et al. (Numer Math 149(2):273–304, 2021) require a parameter \(C_{\text {st},1}\) that is found 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 \(C_{\textrm{st, 1}}\propto (p+1)^{1/2}\) as \(p \rightarrow \infty \) . A similar estimate for the Galerkin projection holds with a p-robust constant \(C_{\text {st},2}\) and \(C_{\text {st},2} \le 2\) for right-isosceles triangles. This paper utilizes the new inequality with the constant \(C_{\text {st},2}\) to design a modified hybrid high-order 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.
