An Adaptive Nonconforming Crouzeix-Raviart Finite Element Algorithm Based On Shifted-Inverse Iteration For Laplace Eigenvalue Problem

Adaptive method of finite element is one of effective numerical methods solving the elliptic eigenvalue problem, and a posteriori error estimates is the theoretical basis of adaptive finite element method. American mathematician Babuska and Rheinboldt first proposed the ideas of a posteriori error estimates and adaptive method of finite element in 1978. These methods have been extensively studied theoretically since Babuska and Rheinboldt and have also been successful in practice. Combining conforming finite element and adaptive method to solve elliptic eigenvalue problem, predecessors have done a lot of research, and it is concluded that the convergence and optimality of this method.As we know, using conforming and nonconforming adaptive method to solve the elliptic eigenvalue problem, and we can get the upper and lower bounds of the accuracy eigenvalues, respectively. Thus the research of nonconforming finite element adaptive method to elliptic eigenvalue problem is meaningful.With this background, for the Laplace eigenvalue problem, this paper combines nonconforming Crouzeix-Raviart element, Shifted-inverse iteration, and the first proposes an nonconforming Crouzeix-Raviart element adaptive finite element method based on residual type a posteriori error estimates. We analyse its convergence and a priori error estimates, and prove the efficiency and reliability of the posteriori error estimator. Finally,we use MATLAB to program and compute in the platform of the finite element package of chen long, and get satisfactory numerical results.