Shipeng Mao

A Convergent Nonconforming Adaptive Finite Element Method with Quasi-Optimal Complexity

In this paper, we prove convergence and quasi-optimal complexity of a simple adaptive nonconforming finite element method. In each step of the algorithm, the iterative solution of the discrete system is controlled by an adaptive stopping criterion, and the local refinement is based on either a simple edge residual or a volume term, depending on an adaptive marking strategy. We prove that this marking strategy guarantees a strict reduction of the error, augmented by the volume term and an additional oscillation term, and quasi-optimal complexity of the generated sequence of meshes.

An optimally convergent adaptive mixed finite element method

We prove convergence and optimal complexity of an adaptive mixed finite element algorithm, based on the lowest-order Raviart–Thomas finite element space. In each step of the algorithm, the local refinement is either performed using simple edge residuals or a data oscillation term, depending on an adaptive marking strategy. The inexact solution of the discrete system is controlled by an adaptive stopping criterion related to the estimator.

Quasi-Optimality of Adaptive Nonconforming Finite Element Methods for the Stokes Equations

We prove convergence and quasi-optimal complexity of adaptive nonconforming low-order finite element methods for the Stokes equations, covering the Crouzeix-Raviart discretization on triangular and tetrahedral meshes, as well as the Rannacher-Turek discretization on two- and three-dimensional rectangular meshes. Hanging nodes are allowed in order to ease local mesh refinement. The adaptive algorithm is based on standard a posteriori error estimators consisting of two parts: a volume residual and an edge term measuring the nonconformity of the velocity approximation. We use an adaptive marking strategies, which, in each step of the iteration, takes only the dominant term into account. This paper can be regarded as an extension of [R. Becker, S. Mao, and Z.-C. Shi, SIAM J. Numer. Anal., 47 (2010), pp. 4639-4659] to the Stokes problem, but the analysis here does not make use of any relationship between mixed and nonconforming finite element methods.

High accuracy analysis of two nonconforming plate elements

We prove the superconvergence of Morley element and the incomplete biquadratic nonconforming element for the plate bending problem. Under uniform rectangular meshes, we obtain a superconvergence property at the symmetric points of the elements and a global superconvergent result by a proper postprocessing method.

On the Interpolation Error Estimates for

In this paper, we study the relation between the error estimate of the bilinear interpolation on a general quadrilateral and the geometric characters of the quadrilateral. Some explicit bounds of the interpolation error are obtained based on some sharp estimates of the integral over

Q_1

\frac{1}{|J|^{p-1}}

Quadrilateral Finite Elements

for

Crystal structure of sodium indium (monohydrogenmonophosphate- dihydrogenmonoborate-monophosphate), NaIn(BP2O7(OH)3)

1\leq p\leq\infty

Superconvergence of Finite Element Approximations for the Fractional Diffusion-Wave Equation

on the reference element, where

Crystal structure of ammonium indium (monophosphate-hydrogenmonoborate-monophosphate), (NH4)In[BP2O8(OH)]

J