Lower eigenvalue bounds with hybrid high-order methods

TL;DR

The work presents hybrid high-order (HHO) eigensolvers that yield guaranteed lower eigenvalue bounds (GLB) for SPD PDE eigenproblems, with constants derived from local embeddings that are independent of the polynomial degree. The method introduces a single reconstruction operator, enabling degree-robust GLB and compatibility with adaptive mesh refinement. A general GLB theorem is established, linking GLB(j) to the discrete eigenvalue via GLB(j)=min{1,1/(α+βλ_h(j))}λ_h(j) under explicit conditions, and the framework is extended from the Laplace problem to Steklov and linear elasticity, including local embedding optimizations. Numerical experiments on Laplace, Steklov, and elasticity demonstrate optimal convergence under adaptive refinement and illustrate how improved bounds on local embedding constants reduce preasymptotic effects, enhancing practical applicability of GLB in engineering simulations.

Abstract

This paper proposes hybrid high-order eigensolvers for the computation of guaranteed lower eigenvalue bounds. These bounds display higher order convergence rates and are accessible to adaptive mesh-refining algorithms. The involved constants arise from local embeddings and are available for all polynomial degrees. Applications include the linear elasticity and Steklov eigenvalue problem.

Figures (12)

• Figure 1: (a) Convergence history plot of $\lambda_\mathrm{C}(1) - \mathrm{GLB}(1)$ and (b) adaptive mesh with 736 triangles ($k = 2$) for the Laplace eigenvalue problem in \ref{['sec:num_ex_laplace']}
• Figure 2: Convergence history plot of $\lambda_\mathrm{C}(1) - \mathrm{GLB}(1)$ for the Steklov eigenvalue problem in \ref{['sec:num-ex-steklov']}
• Figure 3: (a) Initial triangulation and (b) convergence history plot of $\lambda_\mathrm{C}(1) - \mathrm{GLB}(1)$ for the linear elasticity eigenvalue problem in \ref{['sec:ex-linear-elast']}
• Figure 5: Improved convergence history plot of $\lambda_\mathrm{C}(1) - \mathrm{GLB}(1)$ for the linear elasticity eigenvalue problem in \ref{['sec:ex-le-impr']}
• Figure : $6.9898 \leq \gamma$

Theorems & Definitions (34)

• Lemma 2.1: coercivity
• proof
• Remark 2.2: spectral correctness
• Lemma 3.1: trace inequality
• proof
• Remark 3.2: simplicial cells
• Theorem 3.3: GLB for Laplace
• Lemma 3.4: linear independency
• proof : Proof of (a)
• proof : Proof of \ref{['thm:dLEB']}