The Block Preconditioned Steepest Descent Iteration for Elliptic Operator Eigenvalue Problems

The block preconditioned steepest descent iteration is an iterative eigensolver for subspace eigenvalue and eigenvector computations. An important area of application of the method is the approximate solution of mesh eigenproblems for self-adjoint and elliptic partial differential operators. The subspace iteration allows to compute some of the smallest eigenvalues together with the associated invariant subspaces simultaneously. The building blocks of the iteration are the computation of the preconditioned residual subspace for the current iteration subspace and the application of the Rayleigh-Ritz method in order to extract an improved subspace iterate. The convergence analysis of this iteration provides new sharp estimates for the Ritz values. It is based on the analysis of the vectorial preconditioned steepest descent iteration which appeared in SIAM J. Numer. Anal., 50(6):3188-3207, 2012. Numerical experiments using a finite element discretization of the Laplacian with up to 5 · 107 degrees of freedom and with multigrid preconditioning demonstrate the near-optimal complexity of the method.