Guaranteed lower Dirichlet eigenvalue bounds (GLB) can be computed for the m-th Laplace operator with a recently introduced extra-stabilized nonconforming Crouzeix–Raviart (\(m=1\)) or Morley (\(m=2\)) finite element eigensolver. Striking numerical evidence for the superiority of a new adaptive eigensolver motivates the convergence analysis in this paper with a proof of optimal convergence rates of the GLB towards a simple eigenvalue. The proof is based on (a generalization of) known abstract arguments entitled as the axioms of adaptivity. Beyond the known a priori convergence rates, a medius analysis is enfolded in this paper for the proof of best-approximation results. This and subordinated \(L^2\) error estimates for locally refined triangulations appear of independent interest. The analysis of optimal convergence rates of an adaptive mesh-refining algorithm is performed in 3D and highlights a new version of discrete reliability.

可以使用最近引入的超稳定非相容 Crouzeix–Raviart ( \(m=1\) ) 或 Morley ( \(m=2\) ) 有限元来计算第 m个拉普拉斯算子的保证狄利克雷特征值下界 (GLB )元素本征解算器。新型自适应特征值求解器优越性的有力数值证据推动了本文的收敛分析，并证明了 GLB 向简单特征值的最佳收敛率。该证明基于称为适应性公理的已知抽象论证（的概括）。除了已知的先验收敛率之外，本文还进行了中值分析，以证明最佳近似结果。这个和局部细化三角测量的从属\(L^2\)误差估计似乎具有独立的意义。自适应网格细化算法的最佳收敛速度的分析是在 3D 中进行的，并强调了新版本的离散可靠性。