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Convergence and optimality of the adaptive Morley element method

This paper is devoted to the convergence and optimality analysis of the adaptive Morley element method for the fourth order elliptic problem. A new technique is developed to establish a quasi-orthogonality which is crucial for the convergence analysis of the adaptive nonconforming method. By introducing a new parameter-dependent error estimator and further establishing a discrete reliability property, sharp convergence and optimality estimates are then fully proved for the fourth order elliptic problem.


Introduction
This paper is devoted to the study of adaptive nonconforming finite element methods for high order elliptic boundary value problems. The adaptive conforming finite element method for the second order elliptic problems has been a subject of extensive studies for many years since the pioneering work of Babuska and Rheinboldt [2], and its theory has become rather mature [32,20,27,28,6,31,30,14]. For the nonconforming method, the a posteriori error theory of the second order elliptic problems has been studied only very recently [18,19,10,9,12,11]; for the fourth order elliptic problem, only the a posteriori error estimate of the Morley element method can be found in the literature [22,3,33] and there have been no works on either convergence or optimality for any finite element methods for fourth order problems.
The main difficulty for the analysis of nonconforming finite element methods arises from the nonconformity of the discrete space and consequently the lack of the Galerkin-orthogonality which is a key ingredient for the convergence analysis of the adaptive conforming method of the second order elliptic problem [20,27,28,30,14]. For the nonconforming linear element of the Poisson equation, a quasi-orthogonality is established instead in [13] by using some special equivalency between the nonconforming linear element and the lowest order Raviart-Thomas element [25]. For the Morley element of the fourth order elliptic problem, however, it is unclear whether such type of equivalency still holds. We also note that the convergence (not to mention optimality) analysis of the adaptive conforming method is still missing for the fourth order elliptic problem in the literature.
This paper is devoted to the convergence and optimality analysis of the adaptive version of the Morley element [26,29,34]. Our analysis is based on an observation that a quasi-orthogonality can be obtained from a crucial local conservative property (that plays a critical role in a general study in [34]), of the Morley element method. Another ingredient is a new parameter dependent estimator which is introduced to analyze optimality of the adaptive nonconforming method. With the help of the discrete reliability which is established by introducing two interpolation operators between two nonconforming spaces, we show convergence and optimality of the adaptive algorithm.
The rest of the paper is organized as follows. In Section 2, we present the Kirchhoff plate problem and the Morley finite element method, and recall a posteriori error analysis due to [22]. In Section 3, we prove the quasi-orthogonality and then show reduction of some total error in Section 4 by introducing a new parameterdependent estimator. To obtain optimality of the adaptive algorithm, we establish the discrete reliability in Section 5. Consequently, we show optimality of the adaptive Morley element method in Section 6. We give a brief comment on the extension of the theory to the Morley element method in three dimensions in Section 7. Also, we discuss the generalization to the nonconforming linear elements in both two and three dimensions therein. This extension gives an alternative analysis of the convergence result from [13]. The paper ends with Section 8 where we give the conclusion and some comments. Let Ω ⊂ R 2 be a bounded domain, E the Young modulus, and ν the Poisson ratio. For all 2 × 2 symmetric matrices, the linear operator C is defined by The bilinear form a(u, v) is defined by , for any u , v ∈ W := H 2 0 (Ω) , where ∇ 2 u is the Hessian matrix of u. The corresponding energy norm is given by (2.2) u 2 C := a(u, u) for any u ∈ W , which is equivalent to the usual norm | · | H 2 (Ω) for any u ∈ W .
We consider the Kirchhoff plate bending problem as follows: Given f ∈ L 2 (Ω), find u ∈ W such that We now present the Morley element. Suppose that Ω is covered exactly by a shape-regular triangulation T h consisting of triangles in 2D, see [16]. E h is the set of all edges in T h , E h (Ω) is the set of interior edges, and E(K) is the set of edges of any given element K in T h ; h K = |K| 1/2 , the size of the element K ∈ T h . ω K is the union of elements K ′ ∈ T h that share an edge with K, and ω e is the union of elements that share a common edge e. Given any edge e ∈ E h (Ω) with the length h e we assign one fixed unit normal ν e := (ν 1 , ν 2 ) and tangential vector τ e := (−ν 2 , ν 1 ). For e on the boundary we choose ν e = ν the unit outward normal to Ω. Once ν e and τ e have been fixed on e, in relation to ν e one defines the elements K − ∈ T h and K + ∈ T h , with e = K + ∩ K − . Given e ∈ E h (Ω) and some R d -valued function v defined in Ω, The discrete space of the Morley finite element method is defined as follows [26,29,34] and e ∇v · ν e ds = 0 on e ∈ E h ∩ ∂Ω}, where M 2,h is the space of piecewise polynomials of degree ≤ 2 over T h which are continuous at all the internal nodes and vanish at all the nodes on the boundary ∂Ω, and ∇ h the discrete gradient operator which is defined elementwise. We define where the discrete Hessian operator ∇ 2 h is defined elementwise with respect to the triangulation T h .
We now consider the finite element discretization of (2.3) as follows: Find u h ∈ W h such that To recall the a posteriori error estimate for the Morley element, we first define an estimator on each element K ∈ T h as For any S h ⊂ T h , we define the estimator over S h by In particular, for S h = T h , we have We further define the oscillation osc(f, T h ) by where f K is the constant projection of f over K. For the estimator (2.9), we have the following reliability and efficiency whose proof can be found in [22].
Here and throughout the paper, we shall follow [35] to use the notation and ≅. When we write then there exist possible constants C 1 , c 2 and C 2 such that Given v ∈ H 2 (T h ) := {v ∈ L 2 (Ω), v| K ∈ H 2 (K), for any K ∈ T h }, we define the following residual (2.12) Res with u H being the solution of the discrete problem (2.6) on T H , which is a nested and coarser mesh to T h ; namely, T h is some refinement of T H . It follows from the discrete problem (2.6) that Lemma 2.2. For any v ∈ W , it holds that The proof of the above lemma can be found in [33,3,22].

Quasi-orthogonality
In this section, we address one difficulty, namely the quasi-orthogonality, in the convergence analysis of the adaptive Morley element method. Our analysis is based on two interpolation operators: the canonical interpolation operator Π h of the nonconforming space W h , and the restriction operator I H from the discrete space W h on the mesh T h to the discrete space W H on the nested and coarser mesh T H of T h .
Here and in what follows, N h denotes the set of nodes of the partition T h . We first define the canonical interpolation operator Π h : W → W h by, Lemma 3.1. Let the interpolation operator Π h be defined as in (3.1). Then, The above properties are immediate from the definition of Π h . Now we define the restriction interpolation operator I H : ∂v h ∂ν e ds , e ∈ E H with e = e 1 ∪ e 2 · · · ∪ e ℓ and e i ∈ E h .
Before analyzing the properties of this interpolation, we state the following simple result.
∂νe | K 2 are two constant functions that must be equal since by the definition of W h e [ ∂v h ∂νe ]ds = 0. Thus v must belong to P 1 (K 1 ∪ K 2 ). The properties of the interpolation operator I H are summarized in the following lemma. (3.5). Then,

Lemma 3.3. Let the interpolation operator I H be defined as in
Proof. The properties of (3.6), and (3.8) directly follow from the definition of the interpolation. We only need to prove (3.7) and the estimate (3.9). We first define σ H = C∇ 2 H v H to assert that In fact, for e ∈ E H \E h , this assertion follows from the fact that σ H is a piecewise constant matrix with respect to T H and the definition of I H in (3.5). For e ∈ E h ∩E H , the assertion follows from ( For the edge e ∈ E h which lies in the interior of some K ∈ T H , we can use the continuity of e ∇ h v h ds over e and the fact σ H is constant over K to show that Whence, we integrate by parts and use (3.10) and (3.11) to conclude (3.7). Now we turn to (3.9). In fact, both sides of (3.9) are semi-norms of the restriction W h (K) of W h on K. If the right hand side vanishes for some v h ∈ W h (K), then v h is a piecewise polynomial of degree≤ 1 on K with respect to T h . It follows from Lemma 3.2 that v h is a polynomial of degree≤ 1 on K. Therefore the left hand side also vanishes for the same v h . The desired result then follows from a scaling argument.

Lemma 3.4. (Quasi-orthogonality) Let
T h be a refinement of T H , and u h and u H be the solutions of (2.6) on T h and T H , respectively. Then, Proof. Let the interpolation operator Π h be defined as in (3.1) Let v h = Π h (u − u h ) and the interpolation I H v h be defined as in (3.5). The combination of (2.6) and (2.12) leads to By (3.8) and (3.9), we have Then, the desired result then follows from the triangle inequality and the approximation property of Π h .
Remark 3.5. For the nonconforming P 1 element of the Poisson equation, the quasiorthogonality was established in [13]. The analysis therein is based on some special equivalency between the nonconforming P 1 and Raviart-Thomas elements. For the Stokes-like problem, the quasi-orthogonality of the nonconforming P 1 element has been first proved in [24] based on some special relation of the nonconforming P 1 and Raviart-Thomas elements. For the Morley element, it is unclear whether there exists similar equivalency or relation so far.
Remark 3.6. This paper is a refined version of a technical report in 2009 [23], where it was the first time in the literature to make use of the conservative property of the nonconforming finite element space to analyze the quasi-orthogonality.

Reduction of a properly defined total error
In the rest of the paper, we shall establish convergence and optimality of our Adaptive Nonconforming Finite Element Method(ANFEM). Our analysis is based on two main ingredients: the strict reduction of some total error between two levels and the discrete reliability of the estimator. To this end, we shall first introduce a modified estimatorη with a undetermined positive constant; we shall then borrow the concept of the total error of [14,13] which contains the energy norm of the error and the scaled estimatorη; we shall finally show reduction of this total error. We shall establish the discrete reliability of the estimator in the next section.
Let us first define our adaptive algorithm. Given an initial shape regular triangulation T 0 , a right-hand side function f ∈ L 2 (Ω), a tolerance ε > 0, and a parameter θ ∈ (0, 1). Hereafter, we shall replace the subscript h by an iteration counter called k.
Solve (2.6) on T k , to get the solution u k .
(3) Mark the minimal element set M k such that (4) Refine each triangle K ∈ M k by the newest vertex bisection and possible further refining to conformity to get T k+1 . k = k + 1.

END ANFEM
In order to prove a strict reduction of some total error, we define the following modified estimator for some positive constant β 1 to be determined later.
Proof. The result can be proved by following the idea in [14]. We give the details only for the readers' convenience. In fact, we have For any K ∈ T H \T h , we only need to consider the case where K is subdivided into and with ρ = 1 − 1 2 1/2 . Taking the positive parameter β with 1 − ρθ < β < 1, the desired result follows by combining the above inequality and the bulk criterion (4.1).

Lemma 4.3.
Let T h be some refinement of T H , then there exists ρ > 0 such that Proof. The proof immediately follows from the definition of the meshsize h K .
Proof. Given any K ∈ T h , it follows from the definitions of η K (u h ) and η K (u H ) in (4.5) that (4.10) With e = K 1 ∩ K 2 ∈ E h , we use the trace theorem and the fact that which gives Applying the Young inequality with any positive constant ǫ and summarizing over all elements in T h completes the proof of the lemma.
Proof. Let δ, γ 1 , and γ 2 , be three positive constants to be chosen later. Applying the Young inequality to Lemma 3.4 and adding the resulting estimate to the inequality (4.9) leads to with the positive constant ρ from (4.8). We note that the bound for η 2 (u H , T h ) is given in Lemma 4.2. Hence with ρ and β from Lemma 4.2. In what follows we shall choose the parameters α, β, γ 1 , γ 2 , and δ to achieve the reduction of the total error. We first set (4.16) (4.17) We choose ǫ to be small enough such that 0 < β < 1. Let the positive constant α with β < α < 1 be determined later, this gives Whence we derive as provided that 0 < δ < 1. Then, the choice of α = γ 1 β+γ 2 +C Rel γ 1 +γ 2 +C Rel (1−δ) > β gives

Discrete reliability
This section is devoted to the discrete reliability of the estimator η(u H , T H ). The analysis needs the prolongation operator I ′ h : W H → W h defined as follows. Given P ∈ N h and e ∈ E h , the nodal patch ω P,H of P and the edge patch ω e,H of e with respect to the mesh T H are defined by, respectively, Define ξ P = card(ω P,H ) and ξ e = card(ω e,H ). We define the prolongation interpolation I ′ h v H ∈ W h by, for any v H ∈ W H , Proof. We can assume that the common edge e shared by K 1 and K 2 lies along the x-axis. Then, v can be expressed as v H | K 1 = a 0 +a 1 x+a 2 y+a 3 xy+a 4 x 2 +a 5 y 2 , and v H | K 2 = b 0 +b 1 x+b 2 y+b 3 xy+b 4 x 2 +b 5 y 2 .
Since ∂v H ∂x is continuous over e, we have a 1 = b 1 and a 4 = b 4 . The continuity of ∂v H ∂y over e gives a 2 = b 2 and Lemma 5.2. Let the interpolation operator I ′ h be defined as in (5.2). Then, Proof. It follows from the definition of H v H τ e ] L 2 (e) = 0 for any e ∈ ∂K ⊂ E H . Finally, the local quasi-uniformity of the mesh together with a scaling argument leads to the estimate (5.4).

Remark 5.3. An easy observation finds that the positive constant in (5.4) depends on the following ratio
In the analysis of optimality of the adaptive finite element method, this dependence is not allowed since we only know that T h is some refinement of T H by the newest vertex bisection and the boundness of µ is not guaranteed.
To overcome the above difficulty, we introduce a modified prolongation operator J h which preserves the local projection property. We need the prolongation operator Π : W H → W C H , where W C H ⊂ W is some conforming finite element space over the mesh T H . Here we take W C H as the Hsieh-Clough-Tocher finite element space over the mesh T H [16,7].
Let F be any (global) degree of freedom of W C H , i.e., F is either the evaluation of a shape function or its first order derivatives at an interior node of T H , or the evaluation of the normal derivative of a shape function at a node on an interior edge.
For v H ∈ W H , we define [8] (5.6) where ω F is the set of triangles in T H that share the degree of freedom F , and |ω F | is the number of elements of ω F . Then a similar argument of [8] proves Define and The main idea herein is to take the mixture of the prolongation operators I ′ h and Π. More precisely, we use Π in the region Ω R where the elements of T H are refined and take I ′ h on Ω C , and we define some mixture in the layer between them. This leads to the prolongation operator J h : Lemma 5.5. It holds true that Proof. We only need to use the scaling argument like that in Lemma 5.2 in the layer Ω\(Ω R ∪ Ω C ). The desired result follows from the estimate (5.7) and the local projection property Lemma 5.6. (Discrete reliability) It holds that Proof. For any v h ∈ W h , we deduce from the discrete problem (2.6) that Thanks to (2.13), (3.8) and (3.9), the residual J 1 can be bounded by a similar argument for the term on the right hand-side of (3.14), which implies Since v h ∈ W h is arbitrary, we apply the Young and Cauchy-Schwarz inequalities in (5.11) to obtain that Taking v h = J h u H and applying (5.9) complete the proof of the Lemma.
We end this section by applying the previous discrete reliability to show a result indicating that the bulk criterion is in some sense a necessary condition for reduction of the energy norm between two levels.
Lemma 5.7. If T h is a refinement of T H such that the following reduction holds for some 0 < α ′ < 1, then there exists 0 < θ * < 1 such that Proof. We start with the following decomposition By the discrete reliability of Lemma 5.6 with the coefficient C Drel , we have It follows from (5.14) that Therefore, we apply the Cauchy-Schwarz inequality to obtain (5.20) Since it is obvious that we combine (5.16)-(5.21) to prove the desired result by the parameter with the efficiency constant C Ef f of the estimator η(u H , T H ) from Lemma 2.1.

Optimality
To analyze the optimality, we follow an idea commonly used in the adaptive finite element literature to introduce a nonlinear approximation class. First, we have the following quasi-optimality.
where the consistency error term is given by It follows from [21, Section 4.1] that Therefore, we define Finally, we choose the nonlinear approximation class as follows: Compared to the adaptive conforming method for the second order elliptic problem [14,30], we have not the following monotone convergence: where T h is some refinement of T H . However, it follows from the quasi-orthogonality in Lemma 3.4, the efficiency of the estimator in Lemma 2.1, and the Young inequality that T H )). Theorem 6.1. Let M k be a set of marked elements with minimal cardinality from Algorithm 4.1, u the solution of Problem (2.3), and (T k , W k , u k ) the sequence of meshes, finite element spaces, and discrete solutions produced by the adaptive finiteelement methods with 0 < θ < C Ef f 2(2(C QO ) 2 +C Drel +1) . Then, the following estimate holds: where the parameter 0 < α ′ < 1 is from Lemma 5.7.
The overlay T * of T ǫ and T k is the smallest refinement of both T ǫ and T k . Let u * be the finite element solution of (2.6) on the mesh T * . Since T * is a refinement of T ǫ , we use, (6.8), and (6.6) to obtain that We deduce from Lemma 5.7 that (6.10) θ * η 2 (u k , T k ) ≤ η 2 (u k , M k, * ), for some θ * ∈ (0, 1).
We note that the step (3) in Algorithm 4.1 with θ ≤ θ * chooses a subset of M k ⊂ T k with minimal cardinality with the same property. Therefore This together with the definition of ǫ leads to which completes the proof. Theorem 6.2. Let the marking step in Algorithm 4.1 select a set M k of marked elements with minimal cardinality, u the solution to Problem (2.6), and (T k , W k , u k ) the sequence of meshes, finite element spaces, and discrete solutions produced by the adaptive finite-element methods with 0 < θ < C Ef f 2(2(C QO ) 2 +C Drel +1) . Then, it holds that [31,30] to obtain that It follows from the efficiency of the estimator that (6.15) u − u j 2 C j + osc 2 (f, T j ) ≅η 2 (u j , T j ), which gives For any 0 ≤ j ≤ N − 1, we use the convergence result from Theorem 4.5 to derive that A combination of (6.14)-(6.17) yields it is easy to prove that Inserting this bound into (6.18) leads to

The extensions of the theory
This section extends the theory to the Morley element in three dimensions and the nonconforming linear elements in both two and three dimensions. 7.1. The Morley element in three dimensions. Let T h be a decomposition of the domain Ω ⊂ R 3 into simplicies. Given any face F , we let ν F denote its unit normal vector. The Morley element in three dimensions is defined and analyzed in [34], where the space reads  Define the estimator on each element K ∈ T h as where × denotes the usual tensor product. The estimator is defined by The following reliability and efficiency of the estimator were proved in [22]. Lemma 7.2. Let K 1 , K 2 ∈ T h be two elements sharing a common face F with three edges e ℓ and midpoints m ℓ , ℓ = 1 , 2 , 3 , and v be a piecewise polynomial of degree ≤ 1 over K 1 ∪ K 2 such that (7.5) v| K 1 (m ℓ ) = v| K 2 (m ℓ ), ℓ = 1 , 2, 3, and Then, v is a polynomial of degree ≤ 1 over K 1 ∪ K 2 .
With these preparations, one can generalize the theories of the quasi-orthogonality of Lemma 3.4, error reduction of Theorem 4.5, the discrete reliability of Lemma 5.6, and the optimality of Theorem 6.2 to the Morley element method in three dimensions.
7.2. The nonconforming linear elements for second order elliptic problems. In this subsection, we let T h be a decomposition of the domain Ω ⊂ R 2 or Ω ⊂ R 3 into simplicies in both two and three dimensions. The nonconforming linear element spaces in both two and three dimensions is defined by, respectively,  The continuous problems read: Given f ∈ L 2 (Ω), find u ∈ H 1 0 (Ω) such that (7.8) (∇u, ∇v) L 2 (Ω) = (f, v) L 2 (Ω) for any v ∈ H 1 0 (Ω). The discrete problems read: Given f ∈ L 2 (Ω), find u h ∈ W h such that The convergence of the adaptive nonconforming linear element methods was first analyzed in [13]. The theory in Sections 3-7 can be extended to this case. This extension gives another analysis of the convergence result from [13].

Conclusion and comments
In this paper, we carry out the convergence and optimality analysis of the Morley element for the fourth order elliptic equation. Moreover, we generalize the theory to the nonconforming linear elements. However, the analysis herein heavily depends on the conservative properties of these two classes of nonconforming elements and the fact that the discrete stress is a piecewise constant tensor. At the present time, it is unclear how to generalize these techniques to other nonconforming schemes of the fourth order elliptic problems.