A convergent adaptive method for elliptic eigenvalue problems – 2009, 14th Leslie Fox Prize, Warwick
In the last decades, mesh adaptivity has been widely used to improve the accuracy of numerical solutions to many scientific problems. Recently the question of convergence of adaptive methods has received intensive interest and a number of convergence results for the adaptive solution of boundary value problems have appeared.
We proved the convergence of an adaptive linear finite element algorithm for computing eigenvalues and eigenvectors of scalar symmetric elliptic partial differential operators in bounded polygonal or polyhedral domains, subject to Dirichlet boundary data. Our refinement procedure is based on two locally defined quantities, firstly a standard a posteriori error estimator and secondly a measure of the variability (or “oscillation”) of the computed eigenfunction. Our algorithm performs local refinement on all elements on which the minimum of these two local quantities is sufficiently large. We prove that the adaptive method converges provided the initial mesh is sufficiently fine. The latter condition, while absent for adaptive methods for linear symmetric elliptic boundary value problems, commonly appears for nonlinear problems and can be thought of as a manifestation of the nonlinearity of the eigenvalue problem.