hp - Finite Elements for Elliptic Eigenvalue Problems

Convergence rates for finite element discretisations of elliptic eigenvalue problems in the literature usually are of the 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 h0 — more precisely the minimal dimension of a finite element space — so that the asymptotic convergence estimates hold for h ≤ h0. 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 λ3/4h0 1 is sufficient for piecewise linear finite elements to compute an eigenvalue λ with optimal convergence rates as h0 ≥ h → 0. It will turn out that the condition for eigenfunctions is slightly more restrictive. Furthermore, we will analyse the dependence of the ratio of the errors of the Galerkin approximation and of the best approximation of an eigenfunction on λ 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 non-selfadjoint case and, hence, pave the way towards the error analysis of nonselfadjoint eigenvalue/-function problems. 1 Eigenvalue problems for second order elliptic problems In this paper, we shall deal with the numerical approximation of eigenvalue problems for linear second order partial differential equations. Let Ω ⊂ Rd be a bounded Lipschitz domain with boundary Γ and let Hk (Ω) denote the usual Sobolev space equipped with the scalar product (·, ·)Hk(Ω) and norm ‖·‖Hk(Ω). For simplicity we restrict to the pure Dirichlet problem and denote by H1 0 (Ω) the subspace of H1 (Ω) consisting of all functions with vanishing boundary traces. We introduce the usual ∗([email protected]), Institut für Mathematik, Universität Zürich, Winterthurerstr 190, CH-8057 Zürich, Switzerland