Open Access

Carsten Carstensen, Benedikt Gräßle, Ngoc Tien Tran

• 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 Cst,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 Cst,1 ∝ (p + 1)1/2 as p → ∞. A similar estimate for the Galerkin projection holds with a p-robust constant Cst,2 and Cst,2 ≤ 2 for right-isosceles triangles. This paper utilizes the new inequality with the constant Cst,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 prioriThe 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 Cst,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 Cst,1 ∝ (p + 1)1/2 as p → ∞. A similar estimate for the Galerkin projection holds with a p-robust constant Cst,2 and Cst,2 ≤ 2 for right-isosceles triangles. This paper utilizes the new inequality with the constant Cst,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 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.…