A Domain Decomposition Method for Nonconforming Finite Element Approximations of Eigenvalue Problems

Since the nonconforming finite elements (NFEs) play a significant role in approximating PDE eigenvalues from below, this paper develops a new and parallel two-level preconditioned Jacobi-Davidson (PJD) method for solving the large scale discrete eigenvalue problems resulting from NFE discretization of 2mth \((m=1,2)\) $(m=1,2)$ order elliptic eigenvalue problems. Combining a spectral projection on the coarse space and an overlapping domain decomposition (DD), a parallel preconditioned system can be solved in each iteration. A rigorous analysis reveals that the convergence rate of our two-level PJD method is optimal and scalable. Numerical results supporting our theory are given.