We prove the convergence of an adaptive linear finite element method for computing eigenvalues and eigenfunctions of second-order symmetric elliptic partial differential operators. The weak form is assumed to yield a bilinear form which is bounded and coercive in $H^1$. Each step of the adaptive procedure refines elements in which a standard a posteriori error estimator is large and also refines elements in which the computed eigenfunction has high oscillation. The error analysis extends the theory of convergence of adaptive methods for linear elliptic source problems to elliptic eigenvalue problems, and in particular deals with various complications which arise essentially from the nonlinearity of the eigenvalue problem. Because of this nonlinearity, the convergence result holds under the assumption that the initial finite element mesh is sufficiently fine.
Giani, S., & Graham, I. (2009). A convergent adaptive method for elliptic eigenvalue problems. SIAM Journal on Numerical Analysis, 47(2), 1067-1091. https://doi.org/10.1137/070697264