An augmented subspace method for computing interior multiple and clustered eigenvalues of symmetric elliptic operators

Source abstract

Abstract In this paper, with the help of finite element space on the coarse mesh, we design a type of efficient augmented subspace method to solve the large-scale algebraic eigenvalue problem resulting from finite element discretization of symmetric elliptic eigenvalue problems. The augmented subspace method may efficiently compute interior multiple and clustered eigenvalues of symmetric elliptic operators. It can transform solution of eigenvalue problems defined on fine mesh into several linear source problems on the same mesh, solved by some steps of multigrid or domain decomposition iterations, and a small-dimensional eigenvalue problem in the augmented subspace on every outer iteration. Using some new estimates we give a rigorous theoretical analysis for the proposed method, which shows that the convergence factor of the augmented subspace method is independent of fine mesh size and the gaps among the target eigenvalues (including multiple and clustered eigenvalues), depending on the distance from both ends of the target eigenvalues to others. Numerical examples are finally provided to validate the proposed method.