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.

1. Introduction

To design conforming finite element method for fourth-order elliptic partial differential equation, it requires C¹-continuity finite element space which is arduous to construct (cf. [1]). Alternatively, mixed finite element methods are preferred because C⁰-continuity finite element space is sufficient for deflection. Another advantage of mixed finite element methods is that the stress or Δu can be approximated simultaneously.

One kind of mixed finite element methods is based on Ciarlet-Raviart method whose unknowns are Δu and deflection u (cf. [2]). Optimal convergence rate of the approximation to u and suboptimal convergence rate of the approximation to Δu of Ciarlet-Raviart method were obtained in [3–5], and a posteriori error analysis was given by [6]. It is worth to mention that mixed discontinuous Galerkin method for biharmonic equation advanced in [7] is on the basis of Ciarlet-Raviart method. Based on a first-order system and using single face-hybridizable technique in [8], Cockburn et al. derived a hybridizable and superconvergent DG method in [9] which improved the convergence rate of the approximation to Δu.

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 H(div, Ω, 𝕊), which is substantially difficult to construct since the tensor-valued function must be symmetric and belong to H(div, Ω) simultaneously. As far as we know, the only standard mixed finite element method of this kind mentioned in [10] adopts composite element. Fortunately, several H(div, Ω, 𝕊)-conforming elements have been developed in the last decade. Arnold and Winther designed the first pure polynomial H(div, Ω, 𝕊)-conforming elements in two dimensions in [11], 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 [11]. With regard to this, Guzmán and Neilan constructed H(div, Ω, 𝕊)-conforming elements by enriching the polynomial shape function spaces with rational bubble functions which can avoid vertex degrees of freedom in [14]. On the other hand, some efforts have been made to lower the requirement of H(div, Ω, 𝕊)-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 [18]. 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 u with superconvergence rate. In the context of DG methods, LCDG method using fully discontinuous finite element space for stress devised in [19] 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 [11] 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. [26]). According to the error equations, we obtain that the convergence rate of the approximation to stress tensor in energy norm is O(h^{min {m−1+α,k−1−α}}). And by duality argument, the convergence rate of the approximation to deflection in H¹-norm is shown to be O(h^{min {m−1+2α,k−1}}). 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 [6] by replacing Hsieh-Clough-Tocher element space by Zienkiewicz-Guzmán-Neilan element space proposed in [27], 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 k = 2,3 in [6]. 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 Ω ⊂ ℝ². The mathematical model of this plate clamped on the boundary under a vertical load f ∈ L²(Ω) is governed by (cf. [28, 29])

()

where n is the unit outward normal to ∂Ω, ∇ is the usual gradient operator, ∇· stands for the divergence operator acting on tensor-valued or vector-valued functions (cf. [29]), and

()

with ℐ a second-order identity tensor, tr the trace operator acting on second order tensors, and ν ∈ L^∞(Ω) the Poisson ratio satisfying inf _x∈Ων > 0 and sup _x∈Ων < 0.5. For simplicity hereinafter, we introduce a symmetric and positive definite operator 𝒞 defined as follows: for any second-order tensor τ,

()

Then it is easy to see that σ = 𝒞𝒦(u) and 𝒞⁻¹τ = (1/(1 − ν))τ − (ν/(1 − ν²))(tr τ)ℐ for any second-order tensor τ. The stress-deflection mixed variational formulation of problem (1) given in [10] will be obtained as follows: find (σ, u) ∈ Σ × V such that

()

where Σ : = H(div, Ω, 𝕊),

, 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 2 × 2 tensor by 𝕊. Given a bounded domain G ⊂ ℝ² and a nonnegative integer m, let H^m(G) be the usual Sobolev space of functions on G, and let H^m(G, 𝕊) be the usual Sobolev space of functions taking values in 𝕊. The corresponding norm and seminorm are denoted, respectively, by ∥·∥_m,G and | · |_m,G. If G is Ω, we abbreviate them by ∥·∥_m and | · |_m, respectively. Let

be the closure of

with respect to the norm ∥·∥_m,G. We also denote by H(div, G, 𝕊) the Sobolev space consisting of all L²(G, 𝕊) functions whose divergence is square-integrable. For an integer l ≥ 0, P_l(G) stands for the set of all polynomials in G with the total degree no more than l, and P_l(G, 𝕏) denotes the tensor or vector version of P_l(G) for 𝕏 being 𝕊 or ℝ², respectively. For any vector field v, the symmetric part of the gradient of v be given by

()

with (·) ^T meaning the transpose of the second-order tensor.

Let {𝒯_h} _h>0 be a regular family of triangulations of Ω (cf. [1, 20]);

and h_K : = diam (K). Let ℰ_h be the union of all edges of the triangulation 𝒯_h and

the union of all interior edges of the triangulation 𝒯_h. For any e ∈ ℰ_h, denote by h_e its length. Based on the triangulation 𝒯_h, let the finite element spaces be given by

()

with k ≥ 3. The so-called Arnold-Winther element space Σ_h was designed by Arnold and Winther [11], which is the first H(div, Ω, 𝕊)-conforming finite element space with pure polynomials in two dimensions in history. For any K ∈ 𝒯_h, the degrees of freedom for Arnold-Winther element space Σ_h are given as follows [11]:

(i)
the values of three components of τ at each vertex of K,
(ii)
the values of the moments of degree at most k − 2 of τn on each edge e of K,
(iii)
the values of the moments ∫_K τ : ς dx for all ς ∈ N_k−2(K),

where N_k−2(K): = ε(P_k−2(K, ℝ²)) + M_k−2(K) with

()

Given an integer l ≥ 0, define the elementwise finite element spaces

()

Denote by Q_l : L²(Ω) → X_l the elementwise L² orthogonal projection operator. And let Q_l be the vector version of Q_l. It follows from [11] that Arnold-Winther element space Σ_h has the following property:

()

Now we can define a family of local projection based stabilized mixed finite element methods corresponding to the mixed formulation (4)-(5) for problem (1) as follows: find (σ_h, u_h) ∈ Σ_h × V_h such that

()

where

()

with 0 ≤ α ≤ 1.

Next, let us illustrate the well-posedness of stabilized mixed finite element method (12)-(13).

Theorem 1. Stabilized mixed finite element methods (12)-(13) are uniquely solvable.

Proof. Since Σ_h and V_h are both finite dimensional, it is enough to show that if f = 0, then σ_h = 0, u_h = 0. Taking τ = σ_h in (12) and v = u_h in (13), and adding two equalities, it follows that a(σ_h, σ_h) + c(u_h, u_h) = 0. Thus σ_h = 0 and ∇u_h = Q_k−2(∇u_h) ∈ X_k−2 in Ω from the definition of a(·, ·) and c(·, ·). Then (12) is reduced to

()

According to (11), we get ∇u_h = 0, which means that u_h is piecewise constant on 𝒯_h. Therefore, u_h = 0 for

3. A Priori Error Analysis

In this section, we provide an a priori error analysis for stabilized mixed finite element methods (12)-(13).

3.1. Preliminaries

For a function v ∈ L²(Ω) with v|_K ∈ H^m(K) for all K ∈ 𝒯_h, let ∥v∥_m,h and |v|_m,h be the usual broken H^m-type norm and seminorm of v:

()

If v is a vector-valued or tensor-valued function, the previous symbols are defined in the similar manners. For a vector or tensor v, its length |v| is (v · v) ^1/2 or (v : v) ^1/2. Moreover, define an energy norm for L²(Ω, 𝕊) and a seminorm for

()

Throughout this paper, we also use “≲⋯” to mean that “≤C⋯,” where C is a generic positive constant independent of h, which may take different values at different appearances. And A ≈ B means A≲B and B≲A.

Let R_h : L²(Ω, 𝕊) → Σ_h∩H¹(Ω, 𝕊) be the Clément interpolation operator introduced in [30]. Then we define an interpolation operator Π_h : Σ → Σ_h based on the degrees of freedom of Arnold-Winther element space as follows (cf. [11]): for any τ ∈ Σ, K ∈ 𝒯_h,

()

For any τ ∈ Σ, by the definition of Π_h, it holds that (cf. [11])

()

Then we define the second interpolation operator I_h : V∩H²(Ω) → V_h in the following way (cf. [4, 5]): given w ∈ V∩H²(Ω), for any element K ∈ 𝒯_h, any vertex a of K, and any edge e of K, I_hw ∈ V_h satisfies

()

According to Proposition 5.1 in [31] and (11), we have for all w ∈ V∩H²(Ω),

()

For simplicity, we still write I_h, Q_k−2 and Π_h for I_h|_K, Q_k−2|_K and Π_h|_K. The error estimates for interpolation operators I_h, Q_k−2, and Π_h are summarized in the following lemma (cf. [1, 4, 5, 11, 20, 31]).

Lemma 2. For all v ∈ H^m+2(K), τ ∈ H^m(Ω, 𝕊) with m a nonnegative integer, and all K ∈ 𝒯_h, one has the estimates

()

For any v ∈ H^m+2(Ω) with m a nonnegative integer, we obtain from Lemma 2 and triangle inequality

()

3.2. A Priori Error Analysis

Subtracting (12)-(13) from (4)-(5), we obtain the following error equations:

()

Using (21), the last error equations can be rewritten as

()

Theorem 3. Assume that the solution (σ, u) for mixed formulation (4)-(5) satisfies σ ∈ H^m(Ω, 𝕊) and u ∈ H^m+2(Ω) for some positive integer m, and let (σ_h, u_h) be the solution of stabilized mixed finite element methods (12)-(13). Then

()

Proof. Choosing τ = Π_hσ − σ_h in (26) and v = I_hu − u_h 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 u_h in the H¹(Ω)-norm. To this end, we assume that Ω is a convex bounded polygonal domain hereafter in the section. Let

be the solution of the auxiliary problem:

()

Since Ω 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

()

Lemma 4. Let Ω be a convex bounded polygonal domain. Assume that the solution (σ, u) for mixed formulation (4)-(5) satisfies σ ∈ H^m(Ω, 𝕊) and u ∈ H^m+2(Ω) for some positive integer m; then

()

Proof. Taking v = I_hu − u_h 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 Ω be a convex bounded polygonal domain. Assume that the solution (σ, u) for mixed formulation (4)-(5) satisfies σ ∈ H^m(Ω, 𝕊) and u ∈ H^m+2(Ω) for some positive integer m, and let (σ_h, u_h) be the solution of stabilized mixed finite element methods (12)-(13). Then

()

Proof. Taking v = I_hu − u_h in (34), we have from Lemma 4

()

Choosing

in error equation (26), it holds that

()

Picking τ = σ − σ_h 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

In this section, we intend to investigate the a posteriori error estimates of stabilized mixed finite element methods (12)-(13).

4.1. Preliminaries

For any interior edge

, let K⁺ and K⁻ be the two adjacent triangles sharing edge e. Denote by n⁺ and n⁻ the unit outward normals to the common edge e of the triangles K⁺ and K⁻, respectively. For any vector-valued function ϕ, write

and

. Then define jump on e as follows:

()

If an edge e lies on the boundary ∂Ω, jump is defined by

()

Based on the triangulation 𝒯_h, let

()

And equip W with a broken energy norm as follows:

()

Define a seminorm for

()

For any τ ∈ Σ_h and v ∈ V_h, define error estimator as

()

And the oscillation is defined as

()

For any vertex p and edge e of triangulation 𝒯_h, denote 𝒯_p and 𝒯_e by the set of triangles in 𝒯_h sharing common vertex p and edge e, respectively. For any subset 𝒮 of 𝒯_h, let |𝒮| be the cardinalities of 𝒮. For any edge e of triangulation 𝒯_h, ω_e means the union of elements in 𝒯_e.

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 𝒯_h (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 K ∈ 𝒯_h, denote by

the three vertices and edges of K and

the three corresponding barycentric coordinates such that λ_i(p_j) = δ_ij. Define the triangle-bubble function and edge-bubble functions as

()

for i = 1,2, 3, where i + 1 and i + 2 are taken as values modulo 3, respectively. Moreover, define rational edge bubble functions as (i = 1,2, 3)

()

Then the local and global Zienkiewicz-Guzmán-Neilan finite element spaces given in [27] are

()

where

()

In the case k = 2,

is understood as P₀(K). The local degrees of freedom are taken as

()

Now we can construct a connection operator L_h : V_h → U_h by averaging (for details see [6, 34]) in the following way: Given w ∈ V_h, for every interior vertex p, interior edge e, and triangle K of triangulation 𝒯_h,

()

and for every vertex p, edge e on ∂Ω,

()

According to the similar arguments in [6, 34, 35], we can get the following estimate for connection operator L_h:

()

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 [6] by replacing Hsieh-Clough-Tocher element space by Zienkiewicz-Guzmán-Neilan element space proposed in [27].

Theorem 6. Let (σ, u) and (σ_h, u_h) be solutions of mixed formulation (4)-(5) and stabilized mixed finite element methods (12)-(13), respectively. Then

()

Proof. Letting w = u − L_hu_h, 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 𝒦_h means the discrete version of 𝒦 associated with triangulation 𝒯_h. Combining the last two inequalities, we obtain by the fact that σ = 𝒞(𝒦(u))

()

Thus, we have

()

Then triangular inequality and (67) imply

()

Together with triangular inequality, it holds that

()

Finally, noting the fact [∇u] = 0 on ℰ_h, it follows from the last two inequality that

()

which together with the definition of η_h ends the proof.

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

Lemma 7. Let (σ, u) and (σ_h, u_h) be solutions to mixed formulation (4)-(5) and stabilized mixed finite element methods (12)-(13), respectively. Then for all K ∈ 𝒯_h,

()

Proof. Let . It is obvious that and

()

Using integration by parts and (5) with v = ψ_K, 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.

Lemma 8. Let (σ, u) and (σ_h, u_h) be solutions to mixed formulation (4)-(5) and stabilized mixed finite element methods (12)-(13), respectively. Then for all ,

()

Proof. Let K₁, K₂ ∈ 𝒯_h such that e is common shared edge of K₁ and K₂, and define edge bubble function as (cf. [19, 36])

()

where

and

for i = 1,2 are barycentric coordinates of K₁ and K₂ associated with two end points of e, respectively. Set J_e : = [∇·σ_h]|_e. E_h(J_e) is defined by extending the jump J_e to ω_e constantly along the normal to e. Thus E_h(J_e) is a piecewise polynomial of degree k − 2 on ω_e and

. It is easy to check that

()

By standard scaling argument, integration by parts, and (5) with v = ψ_e, it follows that

()

Then from Cauchy-Schwarz inequality, (77), inverse inequality, and (84),

()

Therefore, we can conclude (82) by canceling ∥J_e∥_0,e on both sides and the fact that h_e ≈ h_K.

Theorem 9. Let (σ, u) and (σ_h, u_h) be solutions for mixed formulation (4)-(5) and stabilized mixed finite element methods (12)-(13), respectively. Then

()

Proof. Noting that σ = 𝒞(𝒦(u)), it follows from triangular inequality that

()

Hence, the proof is finished by combining the last inequality, (77), (82), and the definition of η_h(σ_h, u_h, f) and ∥|u − u_h|∥.

Acknowledgments

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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Local Projection-Based Stabilized Mixed Finite Element Methods for Kirchhoff Plate Bending Problems

Abstract

1. Introduction

2. Stabilized Mixed Finite Element Methods

3. A Priori Error Analysis

3.1. Preliminaries

3.2. A Priori Error Analysis

4. A Posteriori Error Analysis

4.1. Preliminaries

4.2. A Posteriori Error Analysis

Acknowledgments

References

References

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Local Projection-Based Stabilized Mixed Finite Element Methods for Kirchhoff Plate Bending Problems

Abstract

1. Introduction

2. Stabilized Mixed Finite Element Methods

3. A Priori Error Analysis

3.1. Preliminaries

3.2. A Priori Error Analysis

4. A Posteriori Error Analysis

4.1. Preliminaries

4.2. A Posteriori Error Analysis

Acknowledgments

References

References

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