hp-Finite Elements for Elliptic Eigenvalue Problems: Error Estimates Which Are Explicit with Respect to Lambda, h, and p

Convergence rates for finite element discretizations of elliptic eigenvalue problems in the literature usually are of the following form: If the mesh width $h$ is fine enough, then the eigenvalues, resp., eigenfunctions, converge at some well-defined rate. In this paper, we will determine the maximal mesh width $h_{0}$—more precisely the minimal dimension of a finite element space—so that the asymptotic convergence estimates hold for $h\leq h_{0}$. This mesh width will depend on the size and spacing of the exact eigenvalues, the spatial dimension, and the local polynomial degree of the finite element space. For example, in the one-dimensional case, the condition $\lambda^{3/4}h_{0}\lesssim1$ is sufficient for piecewise linear finite elements to compute an eigenvalue $\lambda$ with optimal convergence rates as $h_{0}\geq h\rightarrow0$. It will turn out that the condition for eigenfunctions is slightly more restrictive. Furthermore, we will analyze the dependence of the ratio of the errors of the Galerkin approximation and of the best approximation of an eigenfunction on $\lambda$ and $h$. In this paper, the error estimates for the eigenvalue/-function are limited to the selfadjoint case. However, the regularity theory and approximation property cover also the nonselfadjoint case and, hence, pave the way towards the error analysis of nonselfadjoint eigenvalue/-function problems.