Overlapping Schwarz preconditioned eigensolvers for spectral element discretizations

Model generalized eigenproblems associated with self-adjoint differential operators in nonstandard homogeneous or heterogeneous domains are considered. Their numerical approximation is based on Gauss-Lobatto-Legendre conforming spectral elements defined by Gordon-Hall transfinite mappings. The resulting discrete eigenproblems are solved iteratively with a Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method, accelerated by an overlapping Schwarz preconditioner. Several numerical tests show the good convergence properties of the proposed preconditioned eigensolver, such as its scalability and quasi-optimality in the discretization parameters, which are analogous to those obtained for overlapping Schwarz preconditioners for linear systems.