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 Cₒₓ, ₁ that is found not robust as the polynomial degree p increases. This is related to the H¹ stability bound of the L² projection onto polynomials of degree at most p and its growth C ₒₓ, ₁ (p+1) ^1/2 as p. A similar estimate for the Galerkin projection holds with a p-robust constant Cₒₓ, ₂ and Cₒₓ, ₂ 2 for right-isosceles triangles. This paper utilizes the new inequality with the constant Cₒₓ, 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 L² 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.