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.

翻译：本研究提出了一类基于弱Galerkin有限元方法求解特征值问题的增广子空间格式。该增广子空间由粗网格上的协调线性有限元空间与细网格上弱Galerkin有限元空间中的特征函数逼近共同构造而成。基于此增广子空间，在细网格弱Galerkin有限元空间中求解特征值问题可简化为在同一弱Galerkin有限元空间中求解线性边值问题与增广子空间中的低维特征值问题。伴随的误差估计表明，所提出的增广子空间技术关于粗网格尺寸具有二阶收敛率。最后通过若干数值算例验证了所提数值技术的有效性。