This work was partially supported by the AFOSR (through AFRL Cooperative Agreement #18RDCOR018 and grant FA9451-18-2-0031), the Croatian Science Foundation grant HRZZ9345, bilateral Croatian-USA grant (administered jointly by Croatian-MZO and NSF) and NSF grant DMS-1414365. The numerical studies were facilitated by the equipment acquired using NSF’s Major Research Instrumentation grant DMS-1624776.
Finite element method, Eigenvalues, Selfadjoint operators, Hilbert space, Discretization (Mathematics), Elliptic operators
We consider filtered subspace iteration for approximating a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated by a quadrature approximation of an operator-valued contour integral of the resolvent. Resolvents on infinite dimensional spaces are discretized in computable finite-dimensional spaces before the algorithm is applied. This study focuses on how such discretizations result in errors in the eigenspace approximations computed by the algorithm. The computed eigenspace is then used to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff distance between the computed and exact eigenvalue clusters are obtained in terms of the discretization parameters within an abstract framework. A realization of the proposed approach for a model second-order elliptic operator using a standard finite element discretization of the resolvent is described. Some numerical experiments are conducted to gauge the sharpness of the theoretical estimates.
Gopalakrishnan, Jay; Grubišić, Luka; and Ovall, Jeffrey S., "Spectral Discretization Errors in Filtered Subspace Iteration" (2019). Mathematics and Statistics Faculty Publications and Presentations. 231.