Jikun Zhao

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.

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.

A posteriori error estimation based on conservative flux reconstruction for nonconforming finite element approximations to a singularly perturbed reaction-diffusion problem on anisotropic meshes

Abstract Based on conservative flux reconstruction, we derive a posteriori error estimates for the nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem on anisotropic meshes, since the solution exhibits boundary or interior layers and this anisotropy is reflected in a discretization by using anisotropic meshes. Without the assumption that the meshes are shape-regular, our estimates give the upper bounds on the error only containing the alignment measure factor, and therefore, could provide actual numerical bounds if the alignment measure was approximated very well. Two resulting error estimators are presented to be equivalent to each other up to a data oscillation, one of which can be directly constructed without solving local Neumann problems and provide computable error bound. Numerical experiments confirm that our estimates are reliable and efficient as long as the singularly perturbed problem is discretized by a suitable mesh which leads to a small alignment measure.

The Morley-Type Virtual Element for Plate Bending Problems

We propose a simple nonconforming virtual element for plate bending problems, which has few local degrees of freedom and provides the optimal convergence in

A posteriori error analysis of nonconforming finite element methods for convection–diffusion problems☆

H^2

A posteriori error estimates for nonconforming streamline-diffusion finite element methods for convection-diffusion problems

H2-norm. Moreover, we prove the optimal error estimates in