Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds

Abstract

AbstractThe 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 $$C_{\text {st},1}$$ C st , 1 that is found not robust as the polynomial degree p increases. This is related to the $$H^1$$ H 1 stability bound of the $$L^{2}$$ L 2 projection onto polynomials of degree at most p and its growth $$C_{\textrm{st, 1}}\propto (p+1)^{1/2}$$ C st, 1 ∝ ( p + 1 ) 1 / 2 as $$p \rightarrow \infty $$ p → ∞ . A similar estimate for the Galerkin projection holds with a p-robust constant $$C_{\text {st},2}$$ C st , 2 and $$C_{\text {st},2} \le 2$$ C st , 2 ≤ 2 for right-isosceles triangles. This paper utilizes the new inequality with the constant $$C_{\text {st},2}$$ C 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 $$L^{2}$$ 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.