Augmented Subspace Scheme for Eigenvalue Problem by Weak Galerkin Finite Element Method
Yue Feng, Zhijin Guan, Hehu Xie, Chenguang Zhou
TL;DR
This paper integrates augmented subspace techniques with the weak Galerkin finite element method to solve elliptic eigenvalue problems. By coupling a coarse conforming space $W_H$ with fine WG approximations in an augmented subspace, the fine-level eigenproblem reduces to a linear boundary-value solve plus a low-dimensional eigenproblem, achieving a second-order convergence rate in the coarse mesh size $H$. The authors provide comprehensive error analyses for energy and $b_h$-norms, including explicit dependence on eigenvalue gaps, and validate the theory with Laplace eigenproblem experiments on $P_0/P_0$ and $P_1/P_1$ WG spaces. The approach supports parallelization and suggests efficient, scalable eigensolvers for WG discretizations, extending multilevel correction ideas to nonstandard WG settings. Overall, the work advances robust and efficient eigensolvers for WG discretizations with practical impact on large-scale eigenvalue computations in engineering and physics.
Abstract
This study proposes a class of augmented subspace schemes for the weak Galerkin (WG) finite element method used to solve eigenvalue problems. The augmented subspace is built with the conforming linear finite element space defined on the coarse mesh and the eigenfunction approximations in the WG finite element space defined on the fine mesh. Based on this augmented subspace, solving the eigenvalue problem in the fine WG finite element space can be reduced to the solution of the linear boundary value problem in the same WG finite element space and a low dimensional eigenvalue problem in the augmented subspace. The proposed augmented subspace techniques have the second order convergence rate with respect to the coarse mesh size, as demonstrated by the accompanying error estimates. Finally, a few numerical examples are provided to validate the proposed numerical techniques.