Shaochun Chen

Conforming Rectangular Mixed Finite Elements for Elasticity

We present a new family of rectangular mixed finite elements for the stress-displacement system of the plane elasticity problem. Based on the theory of mixed finite element methods, we prove that they are stable and obtain error estimates for both the stress field and the displacement field. Using the finite element spaces in this family, an exact sequence is established as a discrete version of the elasticity complex in two dimensions. And the relationship between this discrete version and the original one is shown in a commuting diagram.

ACCURACY ANALYSIS FOR QUASI-WILSON ELEMENT

Abstract In this paper, it is shown that Quasi-Wilson element possesses a very special property i.e. the consistency error is of order O(h2), one order higher than that of Wilson element.

The nonconforming virtual element method for plate bending problems

We develop the nonconforming virtual element method for linear plate bending problems. A class of nonconforming virtual elements is constructed, which is C0-continuous. Like the classical nonconforming plate elements, it relaxes the continuity requirement for the function space to some extent. Further, the virtual element is constructed for any order of accuracy and adapts to complicate element geometries. We present a general framework on the error analysis for the nonconforming virtual element method, highlighting the main difference with the conforming one.

A locking-free anisotropic nonconforming finite element for planar linear elasticity problem*

The main aim of this article is to study the approximation of a locking-free anisotropic nonconforming finite element for the pure displacement boundary value problem of planar linear elasticity. The optimal error estimates are obtained by using some novel approaches and techniques. The method proposed in this article is robust in the sense that the convergence estimates in the energy and L2-norms are independent of the Lame parameter λ.

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction-diffusion problems

The numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems. The quasi-optimal order error estimates are proved in the @e-weighted H^1-norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in @e-weighted H^1-norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.

Second-order locking-free nonconforming elements for planar linear elasticity

In this paper, we present two nonconforming finite elements for the pure displacement planar elasticity problem. Both of them are locking-free and have two order of convergence. Some numerical results attest the validity of our theoretical analysis.

Anisotropic error bounds of Lagrange interpolation with any order in two and three dimensions

Abstract In this paper, using the Newton’s formula of Lagrange interpolation, we present a new proof of the anisotropic error bounds for Lagrange interpolation of any order on the triangle, rectangle, tetrahedron and cube in a unified way.

A locking-free nonconforming triangular element for planar elasticity with pure traction boundary condition

A new nonconforming triangular element for the equations of planar linear elasticity with pure traction boundary conditions is considered. By virtue of construction of the element, the discrete version of Korns second inequality is directly proved to be valid. Convergence rate of the finite element methods is uniformly optimal with respect to @l. Error estimates in the energy norm and L^2-norm are O(h^2) and O(h^3), respectively.

The simplest conforming anisotropic rectangular and cubic mixed finite elements for elasticity

In this paper, we construct two simplest conforming rectangular elements for the linear elasticity problem under the Hellinger-Reissner variational principle. One is a rectangular element in 2D with only 8 degrees of freedom for the stress and 2 degrees of freedom for the displacement. Another one is a cubic element in 3D with only 18?+?3 degrees of freedom. We prove that the two elements are stable and anisotropic convergent. Numerical test is presented to illustrate the element is stable and effective.

Robust a posteriori error estimates for conforming and nonconforming finite element methods for convection-diffusion problems

A posteriori error estimation is carried out within a unified framework for various conforming and nonconforming finite element methods for convection-diffusion problems. Our main contribution is finding an appropriate norm to measure the error, which incorporates a discrete energy norm, a discrete dual semi-norm of the convective derivative and jumps of the approximate solution over element faces (edges in two dimensions). The error estimator is shown to be robust with respect to the Peclet number in the sense of the modified norm. Based on a general error decomposition, we show that the key ingredient of error estimation is the estimation on the consistency error related to the particular numerical scheme, and the remaining terms can be bounded in a unified way. The numerical results are presented to illustrate the robustness and practical performance of the estimator in an adaptive refinement strategy.