Advances in Computational Mathematics 15: 107–138, 2001.  2002 Kluwer Academic Publishers. Printed in the Netherlands. A posteriori error control for finite element approximations of elliptic eigenvalue problems Vincent Heuveline and Rolf Rannacher Institute of Applied Mathematics, University of Heidelberg, INF 294, D-69120 Heidelberg, Germany E-mail: {vincent.heuveline;rolf.rannacher}@iwr.uni-heidelberg.de Received 2 February 2001; accepted 3 July 2001 Communicated by Y. Xu We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for non- linear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error rep- resentations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process. Keywords: eigenvalue problem, finite element method, a posteriori error estimate, mesh adap- tation 1. Introduction We consider the Galerkin finite element approximations of symmetric or non- symmetric elliptic eigenvalue problems. As a prototypical problem, we consider the convection–diffusion problem Av :=-∇· (a∇ v) + b ·∇ v + cv = λv in , u = 0 on ∂. (1) For b ≡ 0, the governing operator is nonsymmetric and may possess nonreal eigen- values. In this case complex analysis is required. The Galerkin finite element ap- proximation of (1) is based on its variational formulation. We seek nontrivial pairs {v h ,λ h }∈ H h × C satisfying a(v h ,ϕ h ) = λ h (v h ,ϕ h ) ∀ϕ h ∈ H h , (2) where a(·, ·) is the sesqui-linear form associated to the differential operator A, and (·, ·) is the usual L 2 scalar product. The finite element subspace H h ⊂ H := H 1 0 () consists of piecewise linear or bilinear functions on decompositions T h , of the domain  into cells T (triangles, quadrilaterals, etc.) with diameter h T := diam{T }.