The spectral Galerkin method for the differential operator eigenvalue problems based on a least-squares form and its Schur complement type implementation methods

The differential operator eigenvalue problems often arise in the field of physics and engineering, such as solid band structure, electron orbitals of atoms or molecules, and quantum bound states. In this paper, the spectral Galerkin method based on a least squares setting is developed for solving the differential operator eigenvalue problems. The proposed scheme leads to a global symmetric positive definite algebraic eigenvalue problem. Two kinds of Schur complement methods are given to deal with the corresponding algebraic equation. Namely, the global block matrix can be decomposed into a local matrix eigenvalue problem. Numerical results are given to verify the effectiveness and high-order accuracy of the proposed scheme. The proposed methods are also effective for solving the three-dimensional problem. We also consider the applications of the proposed methods to solve the eigenvalue problems with a parameter and the $grad\left(div\right)$-differential operator eigenvalue problems