Hailong Guo

Superconvergent Two-Grid Methods for Elliptic Eigenvalue Problems

Some numerical algorithms for elliptic eigenvalue problems are proposed, analyzed, and numerically tested. The methods combine advantages of the two-grid algorithm (Xu and Zhou in Math Comput 70(233):17–25, 2001), the two-space method (Racheva and Andreev in Comput Methods Appl Math 2:171–185, 2002), the shifted inverse power method (Hu and Cheng in Math Comput 80:1287–1301, 2011; Yang and Bi in SIAM J Numer Anal 49:1602–1624, 2011), and the polynomial preserving recovery enhancing technique (Naga et al. in SIAM J Sci Comput 28:1289–1300, 2006). Our new algorithms compare favorably with some existing methods and enjoy superconvergence property.

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

Abstract This is the second paper on the study of gradient recovery for elliptic interface problem. In our previous work Guo and Yang (2016) [17] , we developed a novel gradient recovery technique for finite element method based on the body-fitted mesh. In this paper, we propose new gradient recovery methods for two immersed interface finite element methods: symmetric and consistent immersed finite method (Ji et al. (2014) [23] ) and Petrov–Galerkin immersed finite element method (Hou et al. (2004) [22] , and Hou and Liu (2005) [20] ). Compared to the body-fitted mesh based gradient recovery method, the new methods provide a uniform way of recovering gradient on regular meshes. Numerical examples are presented to confirm the superconvergence of both gradient recovery methods. Moreover, they provide asymptotically exact a posteriori error estimators for both immersed finite element methods.

Gradient Recovery for the Crouzeix---Raviart Element

A gradient recovery method for the Crouzeix–Raviart element is proposed and analyzed. The proposed method is based on local discrete least square fittings. It is proven to preserve quadratic polynomials and be a bounded linear operator. Numerical examples indicate that it can produce a superconvergent gradient approximation for both elliptic equations and Stokes equations. In addition, it provides an asymptotically exact posteriori error estimators for the Crouzeix–Raviart element.

Hessian recovery for finite element methods

In this article, we propose and analyze an effective Hessian recovery strategy for the Lagrangian finite element of arbitrary order

Polynomial preserving recovery on boundary

k

Polynomial Preserving Recovery for High Frequency Wave Propagation

. We prove that the proposed Hessian recovery preserves polynomials of degree

A spectral collocation method for eigenvalue problems of compact integral operators

k+1

Gradient recovery for elliptic interface problem: III. Nitsche's method

on general unstructured meshes and superconverges at rate

A

O(h^k)

C^0

on mildly structured meshes. In addition, the method preserves polynomials of degree