Research Article | Open Access
Local Projection-Based Stabilized Mixed Finite Element Methods for Kirchhoff Plate Bending Problems
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 H1-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 -continuity finite element space which is arduous to construct (cf. ). Alternatively, mixed finite element methods are preferred because -continuity finite element space is sufficient for deflection. Another advantage of mixed finite element methods is that the stress or can be approximated simultaneously.
One kind of mixed finite element methods is based on Ciarlet-Raviart method whose unknowns are and deflection (cf. ). Optimal convergence rate of the approximation to and suboptimal convergence rate of the approximation to of Ciarlet-Raviart method were obtained in [3–5], and a posteriori error analysis was given by . It is worth to mention that mixed discontinuous Galerkin method for biharmonic equation advanced in  is on the basis of Ciarlet-Raviart method. Based on a first-order system and using single face-hybridizable technique in , Cockburn et al. derived a hybridizable and superconvergent DG method in  which improved the convergence rate of the approximation to .
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 , which is substantially difficult to construct since the tensor-valued function must be symmetric and belong to simultaneously. As far as we know, the only standard mixed finite element method of this kind mentioned in  adopts composite element. Fortunately, several -conforming elements have been developed in the last decade. Arnold and Winther designed the first pure polynomial -conforming elements in two dimensions in , which were extended to three dimensions in [12, 13]. The vertex degrees of freedom are unavoidable when using pure polynomial shape function spaces, which is demonstrated in . With regard to this, Guzmán and Neilan constructed -conforming elements by enriching the polynomial shape function spaces with rational bubble functions which can avoid vertex degrees of freedom in . On the other hand, some efforts have been made to lower the requirement of -conforming finite element space for stress. Along this way, Hellan-Herrmann-Johnson method raised in [15–17] is a wonderful mixed method for plate bending problems whose convergence rates for both variables are optimal. Behrens and Guzmán introduced a new mixed method which is based on a system of first-order equations and uses nonsymmetric finite element tensor space to approximate stress in . And a hybrid technique is used for this mixed method to reduce the globally coupled degrees of freedom to only those associated with Lagrange multipliers, which is very efficient in implementation. Moreover, a local postprocessing technique is used to produce new approximation of with superconvergence rate. In the context of DG methods, LCDG method using fully discontinuous finite element space for stress devised in  is also based on stress-deflection formulation.
In this paper, we propose a family of local projection-based stabilized mixed finite element methods for problem (1) based on the stress-deflection variational formulation. The stress tensor will be approximated in Arnold-Winther element spaces  which uses polynomial shape functions, and deflection will be approximated in Lagrangian element spaces [1, 20] in our mixed methods. To ensure the well-posedness of our mixed methods, we use local projection method which has been widely used in Stokes equation (cf. [21, 22]), second-order elliptic problems (cf. [23–25]), and fourth-order obstacle problem (cf. ). According to the error equations, we obtain that the convergence rate of the approximation to stress tensor in energy norm is . And by duality argument, the convergence rate of the approximation to deflection in -norm is shown to be . An error estimator is proposed which is closely related to the second-order system (equilibrium equation and constitutive equation) and nonconformity of the finite element spaces. Using the similar argument as in  by replacing Hsieh-Clough-Tocher element space by Zienkiewicz-Guzmán-Neilan element space proposed in , we prove the reliability of the a posteriori error estimator with all orders. However, by using Hsieh-Clough-Tocher element space which includes the third-order polynomials, the a posteriori error estimator is only proved to be reliable for in . Furthermore, efficiency of the a posteriori error estimator is achieved by bubble function argument.
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 mathematical model of this plate clamped on the boundary under a vertical load is governed by (cf. [28, 29]) where is the unit outward normal to , is the usual gradient operator, stands for the divergence operator acting on tensor-valued or vector-valued functions (cf. ), and with a second-order identity tensor, the trace operator acting on second order tensors, and the Poisson ratio satisfying and . For simplicity hereinafter, we introduce a symmetric and positive definite operator defined as follows: for any second-order tensor , Then it is easy to see that and for any second-order tensor . The stress-deflection mixed variational formulation of problem (1) given in  will be obtained as follows: find such that where , , and Here, the symbol : denotes the double dot product operation of tensors.
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 2. An a priori error analysis and a posteriori error analysis for the stabilized mixed finite element methods are given in Sections 3 and 4, respectively.
2. Stabilized Mixed Finite Element Methods
We will define a family of local projection based stabilized mixed finite element methods for solving problem (1) based on the stress-deflection variational formulation (4)-(5) in this section. For this, we first introduce some notations frequently used later on. Denote the space of all symmetric tensor by . Given a bounded domain and a nonnegative integer , let be the usual Sobolev space of functions on , and let be the usual Sobolev space of functions taking values in . The corresponding norm and seminorm are denoted, respectively, by and . If is , we abbreviate them by and , respectively. Let be the closure of with respect to the norm . We also denote by the Sobolev space consisting of all functions whose divergence is square-integrable. For an integer , stands for the set of all polynomials in with the total degree no more than , and denotes the tensor or vector version of for being or , respectively. For any vector field , the symmetric part of the gradient of be given by with meaning the transpose of the second-order tensor.
Let be a regular family of triangulations of (cf. [1, 20]); and . Let be the union of all edges of the triangulation and the union of all interior edges of the triangulation . For any , denote by its length. Based on the triangulation , let the finite element spaces be given by with . The so-called Arnold-Winther element space was designed by Arnold and Winther , which is the first -conforming finite element space with pure polynomials in two dimensions in history. For any , the degrees of freedom for Arnold-Winther element space are given as follows : (i)the values of three components of at each vertex of , (ii)the values of the moments of degree at most of on each edge of , (iii)the values of the moments for all , where with Given an integer , define the elementwise finite element spaces Denote by the elementwise orthogonal projection operator. And let be the vector version of . It follows from  that Arnold-Winther element space has the following property:
Proof. Since and are both finite dimensional, it is enough to show that if , then , . Taking in (12) and in (13), and adding two equalities, it follows that . Thus and in from the definition of and . Then (12) is reduced to According to (11), we get , which means that is piecewise constant on . Therefore, for .
3. A Priori Error Analysis
For a function with for all , let and be the usual broken -type norm and seminorm of : If is a vector-valued or tensor-valued function, the previous symbols are defined in the similar manners. For a vector or tensor , its length is or . Moreover, define an energy norm for and a seminorm for as Throughout this paper, we also use “” to mean that “,” where is a generic positive constant independent of , which may take different values at different appearances. And means and .
Let be the Clément interpolation operator introduced in . Then we define an interpolation operator based on the degrees of freedom of Arnold-Winther element space as follows (cf. ): for any , , For any , by the definition of , it holds that (cf. ) Then we define the second interpolation operator in the following way (cf. [4, 5]): given , for any element , any vertex of , and any edge of , satisfies According to Proposition 5.1 in  and (11), we have for all ,
Lemma 2. For all , with a nonnegative integer, and all , one has the estimates
For any with a nonnegative integer, we obtain from Lemma 2 and triangle inequality
3.2. A Priori Error Analysis
Using (21), the last error equations can be rewritten as
Proof. Choosing in (26) and in (27), and subtracting (27) from (26), we get Then we have from (19) Using Cauchy-Schwarz inequality and triangle inequality, Thus which together with triangle inequality, Lemma 2, and (24) ends the proof.
Using the usual duality argument, we can additionally derive error estimate of in the -norm. To this end, we assume that is a convex bounded polygonal domain hereafter in the section. Let be the solution of the auxiliary problem: Since is convex, we know with the bound (cf. [32, 33]) Denote by the corresponding stabilized mixed finite element solution: As (26)-(27), we have the following error equations for auxiliary problem: By Theorem 3 and regularity (35), we also have
Proof. Taking in (37), it holds that On the other side, choosing in (26), we get Then it follows from the last two equalities that According to Cauchy-Swarchz inequality, triangle inequality, Lemma 2, (38), and regularity (35), Therefore, the proof is finished from last inequality, Theorem 3, and (24).
Theorem 5. Let be a convex bounded polygonal domain. Assume that the solution for mixed formulation (4)-(5) satisfies and for some positive integer , and let be the solution of stabilized mixed finite element methods (12)-(13). Then
Proof. Taking in (34), we have from Lemma 4 Choosing in error equation (26), it holds that Picking in (33) and using (21) and error equation (25), From the last two equalities and Cauchy-Swarchz inequality, By (24), Lemma 2, Theorem 3, and regularity (35), which together with (45) gives Combined with triangle inequality and Lemma 2, we finish the proof.
4. A Posteriori Error Analysis
For any interior edge , let and be the two adjacent triangles sharing edge . Denote by and the unit outward normals to the common edge of the triangles and , respectively. For any vector-valued function , write and . Then define jump on as follows: If an edge lies on the boundary , jump is defined by
Based on the triangulation , let And equip with a broken energy norm as follows: Define a seminorm for as For any and , define error estimator as And the oscillation is defined as
For any vertex and edge of triangulation , denote and by the set of triangles in sharing common vertex and edge , respectively. For any subset of , let be the cardinalities of . For any edge of triangulation , means the union of elements in .
To show the reliability of the error estimator introduced previously, we need a -conforming finite element space and corresponding connection operator. Here we intend to use the Zienkiewicz-Guzmán-Neilan finite element space associated with (cf. [1, 27]), by reason that the degrees of freedom only involve the values and integrations of function and first-order derivatives, no any higher-order derivatives. For any , denote by the three vertices and edges of and the three corresponding barycentric coordinates such that . Define the triangle-bubble function and edge-bubble functions as for , where and are taken as values modulo 3, respectively. Moreover, define rational edge bubble functions as () Then the local and global Zienkiewicz-Guzmán-Neilan finite element spaces given in  are where In the case , is understood as . The local degrees of freedom are taken as
Now we can construct a connection operator by averaging (for details see [6, 34]) in the following way: Given , for every interior vertex , interior edge , and triangle of triangulation , and for every vertex , edge on , According to the similar arguments in [6, 34, 35], we can get the following estimate for connection operator :
4.2. A Posteriori Error Analysis
First, let us consider the reliability of the a posteriori error estimator. We will follow the similar argument as in  by replacing Hsieh-Clough-Tocher element space by Zienkiewicz-Guzmán-Neilan element space proposed in .
Proof. Letting , then . Using integration by parts, (5) and (13), we have Then by Cauchy-Swarchz inequality, Lemma 2, and (23), we get It follows from Cauchy-Swarchz inequality, triangular inequality, and (67) that where means the discrete version of associated with triangulation . Combining the last two inequalities, we obtain by the fact that Thus, we have Then triangular inequality and (67) imply Together with triangular inequality, it holds that Finally, noting the fact on , it follows from the last two inequality that which together with the definition of ends the proof.
Then, we study the efficiency of the a posteriori error estimator by bubble function argument.
Proof. Let . It is obvious that and Using integration by parts and (5) with , it follows that Together with standard scaling argument, Cauchy-Swarchz inequality, and inverse inequality, we have which together with (78) shows Therefore, we can obtain (77) by the last inequality and triangular inequality.
Proof. Let such that is common shared edge of and , and define edge bubble function as (cf. [19, 36]) where and for are barycentric coordinates of and associated with two end points of , respectively. Set . is defined by extending the jump to constantly along the normal to . Thus is a piecewise polynomial of degree on and . It is easy to check that By standard scaling argument, integration by parts, and (5) with , it follows that Then from Cauchy-Schwarz inequality, (77), inverse inequality, and (84), Therefore, we can conclude (82) by canceling on both sides and the fact that .
This work was partly supported by the NNSFC (Grants nos. 11126226 and 11171257) and Zhejiang Provincial Natural Science Foundation of China (Y6110240, LY12A01015).
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