A framework for robust eigenvalue and eigenvector error estimation and Ritz value convergence enhancement

We present a general framework for the a posteriori estimation and enhancement of error ineigenvalue/eigenvector computations for symmetric and elliptic eigenvalue problems, and provide detailedanalysis of a specific and important example within this framework—finite element methods with continuous,affine elements. A distinguishing feature of the proposed approach is that it provides provably efficient andreliable error estimation under very realistic assumptions, not only for single, simple eigenvalues, but also forclusters which may contain degenerate eigenvalues. We reduce the study of the eigenvalue/eigenvector errorestimators to the study of associated boundary value problems, and make use of the wealth of knowledgeavailable for such problems. Our choice of a posteriori error estimator, computed using hierarchical bases,very naturally offers a means not only for estimating error in eigenvalue/eigenvector computations, but alsocheaply accelerating the convergence of these computations—sometimes with convergence rates which arenearly twice that of the unaccelerated approximations.