Enhancing eigenvalue approximation by gradient recovery on adaptive meshes

Gradient recovery has been widely used for a posteriori error estimates (see Ainsworth & Oden, 2000; Babuška & Strouboulis, 2001; Chen & Xu, 2007; Fierro & Veeser, 2006; Zhang, 2007; Zienkiewicz et al., 2005; Zienkiewicz & Zhu, 1987, 1992a,b). Recently, it has been employed to enhance the eigenvalue approximations by the finite-element method under certain mesh conditions (see Naga et al., 2006; Shen & Zhou, 2006). However, it was unclear whether the gradient recovery is effective for adaptive meshes. In this paper we prove that under adaptive meshes the recovered gradient still helps to improve the convergence rate of the eigenvalue approximation. Our proof relies on the superconvergence result for adaptive meshes established in Wu & Zhang (2007). Furthermore, we demonstrate the effectiveness of the gradient recovery in enhancing eigenvalue approximation by two typical numerical examples: the Laplace operator on the L-shaped domain and the cracked domain. In both cases, we observe superconvergence rates for eigenvalue approximation. We would like to indicate that the enhancement can be applied to more general problems, see, e.g., Shen & Zhou (2006). Likewise, our results on adaptive meshes in this paper can be generalized. Due to the nonquasi-uniform nature of adaptive meshes, error estimates depend on the total degrees of freedom N , rather than the mesh size h (see Bangerth & Rannacher, 2003; Binev et al., 2004; Dörfler, 1996; Morin et al., 2002; Verfürth, 1995). Therefore, the standard analysis for h-version finite-element eigenvalue approximation cannot be used directly and some modifications are needed.