A locally optimal preconditioned Newton-Schur method is proposed for solving symmetric elliptic eigenvalue problems. Firstly, the Steklov-Poincar operator is used to project the eigenvalue problem on the domain \Omega onto the nonlinear eigenvalue subproblem on \Gamma, which is the union of subdomain boundaries. Then, the direction of correction is obtained via applying a non-overlapping domain decomposition method on \Gamma. Four different strategies are proposed to build the hierarchical subspace U_{k+1} over the boundaries, which are based on the combination of the coarse-subspace with the directions of correction. Finally, the approximation of eigenpair is updated by solving a local optimization problem on the subspace U_{k+1}. The convergence rate of the locally optimal preconditioned Newton-Schur method is proved to be \Gamma =1-c_{0}T_{h,H}^{-1}, where c_{0} is a constant independent of the fine mesh size h, the coarse mesh size H and jumps of the coefficients; whereas T_{h,H} is the constant depending on stability of the decomposition. Numerical results confirm our theoretical analysis.  

　　Publication:  

　　Mathematics of Computation, 92 (2023), 2655-2684  

　　http://dx.doi.org/10.1090/mcom/3860  

　　Author:  

　　Wenbin Chen  

　　School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, People’s Republic of China  

　　Nian Shao  

　　School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China  

　　Xuejun Xu  

　　LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China  

　　Email: xxj@lsec.cc.ac.cn