Superconvergent Two-Grid Methods for Elliptic Eigenvalue Problems

Abstract

Abstract. Some numerical algorithms for elliptic eigenvalue problems are proposed, analyzed, and numerically tested. The methods combine advantages of the two-grid algorithm [J. Xu and A. Zhou, Math. Comp, 70(2001) To reduce the computational cost of eigenvalue problems, Xu and Zhou introduced a two-grid discretization scheme [42]. Later on, similar ideas were applied to non self-adjoint eigenvalue problems [22] and semilinear elliptic eigenvalue problems [11]. Furthermore, it also has been generalized to three-scale discretization [16] and multilevel discretization [24]. Recently, a new shifted-inverse power method based two-grid scheme was proposed in [21,43].To improve accuracy of eigenvalue approximation, many methods have been proposed. In [37], Shen and Zhou introduced a defect correction scheme based on averaging recovery, like a global L 2 projection and a Clément-type operator. In [34], Naga, Zhang, and Zhou used Polynomial Preserving Recovery to enhance eigenvalue approximation. In [40], Wu and Zhang further showed polynomial preserving recovery can even enhance eigenvalue approximation on adaptive meshes. The idea was further studied in [31,15]. Alternatively, Racheva and Andreev proposed a two-space method to achieve better eigenvalue approximation [36] and it was also applied to biharmonic eigenvalue problem [1].In this paper, we propose some fast and efficient solvers for elliptic eigenvalue problems. We combine ideas of the two-grid method, two-space method, shiftedinverse power method, and PPR recovery enhancement to design our new algorithms. The first purpose is to introduce two superconvergent two-grid methods for eigenvalue problems. Our first algorithm is a combination of the shifted-inverse power based two-

Gradient recovery for elliptic interface problem: II. Immersed finite element methods

“…Remark 3.9. One of the most practical applications of gradient recovery techniques is to construct asymptotically exact a posteriori error estimators [1,3,19,31,41,42] for adaptive computational methods. Based on the recovery operator R h , one can define a local a posteriori error estimator on element T ∈ T h as…”

Polynomial preserving recovery on boundary

“…It can be used for mesh smoothing, a posteriori error estimate [12,20,22,23,25], and adaptive finite element method even with anisotropic meshes [2,9,11,16]. More recently, the gradient recovery technique was applied to improve eigenvalue approximation as well [8,14,15,18].…”

A $C^0$ linear finite element method for two fourth-order eigenvalue problems

“…We want to remark that lower bound of eigenvalue is very important in practice, and many efforts have been made to obtain eigenvalue approximation from below. The readers are referred to Armentano & Durán (2004), Guo et al (2016), Yang et al (2010) and Zhang et al (2007) for other ways to approximate eigenvalue from below. Our numerical experiments 2132 H. CHEN ET AL.…”