Based on stress-deflection variational formulation, we propose a family of local projection-based stabilized mixed finite element methods for Kirchhoff plate bending problems. According to the error equations, we obtain the error estimates of the approximation to stress tensor in energy norm. And by duality argument, error estimates of the approximation to deflection in ^{1}-norm are achieved. Then we design an a posteriori error estimator which is closely related to the equilibrium equation, constitutive equation, and nonconformity of the finite element spaces. With the help of Zienkiewicz-Guzmán-Neilan element spaces, we prove the reliability of the a posteriori error estimator. And the efficiency of the a posteriori error estimator is proved by standard bubble function argument.

To design conforming finite element method for fourth-order elliptic partial differential equation, it requires

One kind of mixed finite element methods is based on Ciarlet-Raviart method whose unknowns are

Another kind of mixed finite element methods for Kirchhoff plate bending problems is based on stress-deflection formulation. Standard stress-deflection mixed finite element methods require the finite element space for stress belonging to

In this paper, we propose a family of local projection-based stabilized mixed finite element methods for problem (

In the end of this section, let us describe the Kirchhoff plate bending problem. Assume that a thin plate occupies a bounded polygonal domain

The rest of this paper is organized as follows. A family of local projection based stabilized mixed finite element methods based on stress-deflection variational formulation for Kirchhoff plate bending problems is proposed in Section

We will define a family of local projection based stabilized mixed finite element methods for solving problem (

Let

the values of three components of

the values of the moments of degree at most

the values of the moments

Now we can define a family of local projection based stabilized mixed finite element methods corresponding to the mixed formulation (

Next, let us illustrate the well-posedness of stabilized mixed finite element method (

Stabilized mixed finite element methods (

Since

In this section, we provide an a priori error analysis for stabilized mixed finite element methods (

For a function

Let

For simplicity, we still write

For all

For any

Subtracting (

Using (

Assume that the solution

Choosing

Using the usual duality argument, we can additionally derive error estimate of

Let

Taking

Let

Taking

In this section, we intend to investigate the a posteriori error estimates of stabilized mixed finite element methods (

For any interior edge

Based on the triangulation

For any vertex

To show the reliability of the error estimator introduced previously, we need a

Now we can construct a connection operator

First, let us consider the reliability of the a posteriori error estimator. We will follow the similar argument as in [

Let

Letting

Then, we study the efficiency of the a posteriori error estimator by bubble function argument.

Let

Let

Let

Let

Let

Noting that

This work was partly supported by the NNSFC (Grants nos. 11126226 and 11171257) and Zhejiang Provincial Natural Science Foundation of China (Y6110240, LY12A01015).

^{0}finite element approximations of Darcy equations