The weak Galerkin method for eigenvalue problems

Abstract: This article is devoted to computing the eigenvalue of the Laplace eigenvalue problem by the weak Galerkin (WG) finite element method with emphasis on obtaining lower bounds. The WG method is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. We establish the optimal-order error estimates for the WG finite element approximation for the eigenvalue problem. Comparing with the classical nonconforming finite element method which can just provide lower bound approximation by linear elements with only the second order convergence, the WG methods can naturally provide lower bound approximation with a high order convergence (larger than $2$). Some numerical results are also presented to demonstrate the efficiency of our theoretical results.