This paper is devoted to a new error analysis of nonconforming finite element methods. Compared with the classic error analysis in literature, only weak continuity, the F-E-M-Test for nonconforming finite element spaces, and basic H
 m
 regularity for exact solutions of 2m-th order elliptic problems under consideration are assumed. The analysis is motivated by ideas from a posteriori error estimates and projection average operators. One main ingredient is a novel decomposition for some key average terms on (n − 1)-dimensional faces by introducing a piecewise constant projection, which defines the generalization to more general nonconforming finite elements of the results in literature. The analysis and results herein are conjectured to apply for all nonconforming finite elements in literature.

A new a priori error estimate of nonconforming finite element methods

We propose a class of 12 degrees of freedom triangular plate bending elements with quadratic rate of convergence. They may be viewed as the second order Specht triangle, while the Specht triangle is one of the best first order plate bending element. The convergence result is proved under minimal smoothness assumption on the solution. Numerical results for both the smooth solution and nonsmmoth solution confirm the theoretical prediction. Mathematics subject classification: 65N38, 65N30.

/pdf/the-quadratic-specht-triangle-277df7zy13.pdf

The Quadratic Specht Triangle

/pdf/a-family-of-nonconforming-rectangular-elements-for-strain-1pzwitkq86.pdf

A Family of Nonconforming Rectangular Elements for Strain Gradient Elasticity

This work is devoted to the high accuracy analysis of a discrete Stokes complex over rectangular meshes with a simple structure. The 0-form in the complex is a non $C^0$ nonconforming element space for biharmonic problems. This plate element contains only 12 degrees of freedom (DoFs) over a rectangular cell with a zeroth order weak continuity for the normal derivative, therefore only the lowest convergence order can be obtained by a standard consistency error analysis. Nevertheless, we prove that, if the rectangular mesh is uniform, an $O(h^2)$ convergence rate in discrete $H^2$-norm will be achieved. Moreover, based on a duality argument, it has an $O(h^3)$ convergence order in discrete $H^1$-norm if the solution region is convex. The 1-form and 2-form constitute a divergence-free pair for incompressible flow. We also show its higher accuracy than that derived from a usual error estimate under uniform partitions, which explains the phenomenon observed in our previous work. Numerical tests verify our theoretical results.

High accuracy analysis of a nonconforming discrete Stokes complex over rectangular meshes

A new $$C^0$$
 nonconforming quasi three degree element with 13 freedoms is introduced to solve biharmonic problem. The given finite element space consists of piecewise polynomial space $$P_3$$
 and some bubble functions. Different from non-
 $$C^0$$
 nonconforming scheme, a smoother discrete solution can be obtained by this method. Compared with the existed 16 freedoms finite element method, this scheme uses less freedoms. As the finite elements are not affine equivalent each other, the associated interpolating error estimation is technically proved by introducing another affine finite elements. With this space to solve biharmonic problem, the convergence analysis is demonstrated between true solution and discrete solution. Under a stronger hypothesis that true solution $$u\in H_0^2(\Omega )\cap H^4(\Omega )$$
 , the scheme is of linear order convergence by the measurement of discrete norm $$\Vert \cdot \Vert _h$$
 . Some numerical results are included to further illustrate the convergence analysis.

A nonconforming scheme with piecewise quasi three degree polynomial space to solve biharmonic problem

C0-nonconforming elements for fourth order elliptic problem

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C0-nonconforming elements for fourth order elliptic problem