OF COMPUTATION Volume 81, Number 279, July 2012, Pages 1247–1262 S 0025-5718(2012)02602-0 Article electronically published on February 29, 2012

FINITE ELEMENT METHODS FOR THE DISPLACEMENT OBSTACLE PROBLEM OF CLAMPED PLATES

SUSANNE C. BRENNER, LI-YENG SUNG, AND YI ZHANG

Abstract. We study finite element methods for the displacement obstacle problem of clamped Kirchhoff plates. A unified convergence analysis is pro- vided for C1 finite element methods, classical nonconforming finite element methods and C0 interior penalty methods. Under the condition that the ob- stacles are sufficiently smooth and that they are separated from each other and the zero displacement boundary constraint, we prove that the convergence in the energy norm is O(h) for convex domains.

1. Introduction 2 Let Ω ⊂ R be a bounded convex polygonal domain, f ∈ L2(Ω), ψ1,ψ2 ∈ 2 C (Ω) ∩ C(Ω),¯ ψ1 <ψ2 on Ω,¯ and ψ1 < 0 <ψ2 on ∂Ω. In this paper we consider the following displacement obstacle problem for a clamped Kirchhoff plate: Find u ∈ K such that (1.1) u =argminG(v), v∈K where { ∈ 2 ≤ ≤ } (1.2) K = v H0 (Ω) : ψ1 v ψ2 on Ω , 1 (1.3) G(v)= a(v, v) − (f,v), 2  (1.4) a(w, v)= D2w : D2vdx and (f,v)= fvdx. Ω Ω  2 2 2 Here D w : D v = i,j=1 wxixj vxixj is the Frobenius inner product between the Hessian matrices of w and v. · · 2 Since a( , ) is symmetric, bounded and coercive on H0 (Ω) and K is a nonempty 2 closed convex subset of H0 (Ω), by the standard theory (cf. [41, 38, 45, 32]) the obstacle problem (1.1) has a unique solution that is also uniquely determined by the (1.5) a(u, v − u) ≥ (f,v − u) ∀ v ∈ K.

Received by the editor November 13, 2010 and, in revised form, November 14, 2010 and May 4, 2011. 2010 Mathematics Subject Classification. Primary 65K15, 65N30. Key words and phrases. Displacement obstacle, clamped Kirchhoff plate, fourth order, varia- tional inequality, finite element, discontinuous Galerkin. This work was supported in part by the National Science Foundation under Grant No. DMS- 10-16332 and by the Institute for Mathematics and its applications with funds provided by the National Science Foundation.

c 2012 American Mathematical Society Reverts to public domain 28 years from publication 1247

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Remark 1.1. One can also use an equivalent formulation of (1.1) where the bilinear · · · · form a( , )isgivenbya(w, v)= Ω(Δw)(Δv) dx. However, the choice of a( , )in (1.4) is more appropriate for nonconforming finite element methods because the corresponding norms provide more local information.

Finite element methods for second order displacement obstacle problems have been investigated in [29, 20, 24, 34, 49, 36, 51]. An important ingredient in their a priori error analysis is the H2 (and higher) regularity of the solution of the obstacle problem which holds if the domain and the data are sufficiently smooth. The main difference/difficulty in the apriorierror analysis of finite element methods for the fourth order obstacle problem (1.1) is the lack of H4 regularity for the solution u. ∈ 3 ∩ 2 It is known (cf. [30, 31, 21, 46, 32]) that u Hloc(Ω) C (Ω) under the assump- tions on ψi.Sinceψi ∈ C(Ω)¯ and ψ1 < 0 <ψ2 on ∂Ω, we have ψ1

Remark 1.2. The C2 regularity result in [31, 21] is obtained for the one-obstacle problem and f = 0. But it can be extended in a standard fashion to the two- obstacle problem (1.1) under our assumptions on ψ1, ψ2 and f (cf. Appendix A). 2 Note also that u does not belong to C (Ω) in general if we only assume ψ1 ≤ ψ2 in Ω (cf. [22]).

Remark 1.3. Mixed finite element methods for fourth order obstacle problems were discussed in [34] without convergence rates. Nonconforming finite element methods were studied in [50], where an O(h) error estimate was obtained for convex do- 2 mains. However, in that paper Δ u is erroneously treated as a function in L2(Ω). Discontinuous Galerkin methods were also investigated in [4] under a mistaken H4 regularity of u.

In this paper we take advantage of the fact that H2(Ω) is compactly embedded in C(Ω)¯ to give a convergence analysis that uses only the minimization/variational inequality formulations of the continuous and discrete problems. We prove that the error in the energy norm is of magnitude O(h). The rest of the paper is organized as follows. We introduce three types of finite element methods in Section 2 and present a general framework for the analysis of these methods. In Section 3 we study an auxiliary obstacle problem that plays a key role in the convergence analysis of the finite element methods, which is carried out in Section 4. We end the paper with some concluding remarks in Section 5 and three appendices. The extension of the C2 regularity result to the two-obstacle problem is discussed in Appendix A. Appendix B contains an estimate for the Morley element and Appendix C provides a concise unified analysis of finite element methods for the biharmonic problem using the tools introduced in Section 2.

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2. A general framework for finite element methods for the obstacle problem

Let Th be a regular triangulation of Ω with mesh size h. The piecewise Sobolev 3 space H (Ω, Th)isdefinedby  3 T { ∈  ∈ 3 ∀ ∈T } H (Ω, h)= v L2(Ω) : vT = v T H (T ) T h .

3 0 Remark 2.1. The energy space H (Ω, Th) is needed for C interior penalty methods 2 T { ∈ (cf. Example 2.4). We can use the larger space H (Ω, h)= v L2(Ω) : vT =  ∈ 2 ∀ ∈T} v T H (T ) T h for the classical nonconforming finite element methods (cf. Example 2.3).

Let Vh be a finite element space associated with Th and ah(·, ·) be a symmetric 3 on H (Ω, Th) such that ∀ ∈ 2 ∩ 3 T (2.1) ah(w, v)=a(w, v) v, w H0 (Ω) H (Ω, h).   · 2 ∩ 3 T We assume there exists a norm h on Vh + H0 (Ω) H (Ω, h) such that   | |≤     ∀ ∈ 2 ∩ 3 T (2.2) ah(v, w) C1 v h w h v, w Vh + H0 (Ω) H (Ω, h) , ≥  2 ∀ ∈ (2.3) ah(v, v) C2 v h v Vh. From here on we use C (with or without subscripts) to denote a generic positive constant independent of h that can take different values at different appearances. Let Vh be the set of the vertices of Th. WeassumethefunctionsinVh are ∈V 2 −→ continuous at the vertices p h and there exists an operator Πh : H0 (Ω) Vh such that ∀ ∈ 2 ∈V (2.4) (Πhζ)(p)=ζ(p) ζ H0 (Ω),p h,  −  ≤ | | ∀ ∈ 3 ∩ 2 (2.5) ζ Πhζ h Ch ζ H3(Ω) ζ H (Ω) H0 (Ω).

−→ 2 ∩ 3 T Furthermore, we assume there exists an operator Eh : Vh H0 (Ω) H (Ω, h) ∈ ∈ 3 ∩ 2 such that, for any v Vh and ζ H (Ω) H0 (Ω),

(2.6) (Ehv)(p)=v(p) ∀ p ∈Vh,  1 2 2 2 2  −  | − | | | 2 ≤   (2.7) v Ehv L2(Ω) + h v Ehv H1(T ) + h Ehv H (Ω) Ch v h, T ∈Th

(2.8) |ah(ζ,v − Ehv)|≤Ch|ζ|H3(Ω)vh, 2  −  | − | 1 | − | 2 (2.9) ζ EhΠhζ L2(Ω) + h ζ EhΠhζ H (Ω) + h ζ EhΠhζ H (Ω) 3 ≤ Ch |ζ|H3(Ω).

1 ⊂ 2 Example 2.2 (C Finite Elements [9, 5, 52, 23]). We take Vh H0 (Ω) to be the Hsieh-Clough-Tocher macro element space or the quintic Argyris finite element space, ah(·, ·)=a(·, ·), ·h = a(·, ·)=|·|H2(Ω),Πh to be a quasi-local inter- polation operator [25, 47, 33], and Eh to be the natural injection. The properties (2.6)–(2.9) are trivial in this case. In the case where Ω can be triangulated by rectangles, we can also use the Bogner-Fox-Schmit element.

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Example 2.3 (Classical Nonconforming Finite Elements [2, 7, 43, 27, 48]). Let Vh ⊂ L2(Ω) be the Morley finite element. We take  2 2 ah(w, v)= D w : D vdx, T ∈T T h ·h = ah(·, ·), Πh to be the interpolation operator determined by the function values at the vertices and the of the normal on the edges, and Eh to be an enriching operator defined by averaging. Details about the construction of Eh can be found in [13, 14], where the properties (2.6), (2.7) and (2.9) are established. The derivation of estimate (2.8) is given in Appendix B. Similar constructions (cf. [12]) can also be carried out for the Zienkiewicz element and the de Veubeke element, and in the case where Ω can be triangulated by rectangles, also for the Adini element and the incomplete biquadratic element. Example 2.4 (C0 Interior Penalty Methods [28, 17]). These are discontinuous ⊂ 1 Galerkin methods. We take Vh H0 (Ω) to be a Pk Lagrange finite element space (k ≥ 2) and   2 2 2 2 ah(w, v)= D w : D vdx+ {{∂ w/∂n }}[[∂v/∂n]] ds T e T ∈Th e∈Eh   + {{∂2v/∂n2}}[[∂w/∂n]] ds + σ |e|−1 [[∂w/∂n]][[∂v/∂n]] ds, e e e∈Eh e∈Eh 2 2 where Eh is the set of the edges in Th, {{∂ v/∂n }} (resp. [[∂v/∂n]]) is the average (resp. jump) of the second (resp. first) order normal of v across an edge, |e| is the length of the edge e and σ>0 is a penalty parameter. The norm ·h is defined by

2 2 2 2 2 −1 2 v = |v| 2 + |e|{{∂ v/∂n }} + |e| [[∂v/∂n]] . h H (T ) L2(e) L2(e) T ∈Th e∈Eh e∈Eh The penalty parameter σ is assumed to be large enough so that (2.3) holds. We take Πh to be the nodal interpolation operator and Eh to be an enriching operator defined by averaging. Details of the construction of Eh can be found in [17], where the properties (2.6), (2.7) and (2.9) are established. Property (2.8) is a simple consequence of integration by parts and (2.7). Remark 2.5. The estimates (2.7) and (2.8) are useful tools for analyzing finite element methods for the biharmonic problem (cf. Appendix C). The estimates (2.7) and (2.9) are also useful for the analysis of fast solvers for the resulting discrete problems [11, 13, 14, 15].

The finite element method for (1.1) (equivalently (1.5)) is: Find uh ∈ Kh such that

(2.10) uh =argminGh(v), v∈Kh where

(2.11) Kh = {v ∈ Vh : ψ1(p) ≤ v(p) ≤ ψ2(p) ∀ p ∈Vh}, 1 and G (v)= a (v, v) − (f,v). h 2 h

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Since ah(·, ·) is symmetric positive definite on Vh, it follows from the standard theory that the discrete obstacle problem (2.10) has a unique solution uh,whichis also characterized by the following variational inequality:

(2.12) ah(uh,v− uh) ≥ (f,v − uh) ∀ v ∈ Kh.

The finite element function uh provides an approximation to u. A preliminary error estimate is given in the following lemma.

Lemma 2.6. There exist positive constants C† and C‡ independent of h such that

 − 2 ≤  − 2 − − − (2.13) u uh h C† Πhu u h + C‡ ah(u, Πhu uh) (f,Πhu uh) . Proof. From (2.2), (2.3), (2.12) and the arithmetic-geometric mean inequality, we have Π u − u 2 ≤ C a (Π u − u , Π u − u ) h h h 1 h h h h h = C a (Π u − u, Π u − u )+a (u, Π u − u ) − a (u , Π u − u ) 1 h h h h h h h h h h h ≤ C2Πhu − uhΠhu − uhh + C1 ah(u, Πhu − uh) − (f,Πhu − uh) 1 ≤ Π u − u 2 + C Π u − u2 + C a (u, Π u − u ) − (f,Π u − u ) , 2 h h h 3 h h 1 h h h h h and hence

 − 2 ≤  − 2 − − − Πhu uh h 2C3 Πhu u h +2C1 ah(u, Πhu uh) (f,Πhu uh) . The estimate (2.13) then follows from the triangle inequality.  In view of (2.5), it only remains to find an estimate for the term

ah(u, Πhu − uh) − (f,Πhu − uh) on the right-hand side of (2.13).

3. An auxiliary obstacle problem

The analysis of the term ah(u, Πhu−uh)−(f,Πhu−uh) should of course involve the variational inequality (1.5). But it is not easy to connect Kh directly to K. However, by (1.2), (2.6) and (2.11), we have

Ehv ∈ K˜h ∀ v ∈ Kh, ˜ ⊂ 2 where Kh H0 (Ω)isdefinedby ˜ { ∈ 2 ≤ ≤ ∀ ∈V } (3.1) Kh = v H0 (Ω) : ψ1(p) v(p) ψ2(p) p h . Thus we are led to consider the following intermediate obstacle problem: Find u˜h ∈ K˜h such that

(3.2)u ˜h =argminG(v), v∈K˜ h where the functional G is given by (1.3)–(1.4). The unique solution of (3.2) satisfies the following variational inequality:

(3.3) a(˜uh,v− u˜h) ≥ (f,v − u˜h) ∀ v ∈ K˜h.

Our strategy is to first estimate the difference between u andu ˜h in this section and then combine it with the variational inequalities (1.5) and (3.3) to complete the convergence analysis in Section 4.

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Note that K is a subset of K˜h and the problems (1.1) and (3.2) involve the same functional G. We can therefore consider u as an internal approximation ofu ˜h and it is known (cf. [6] and Remark 3.5) that the distance betweenu ˜h and u can be bounded by the square root of the distance betweenu ˜h and K. Below we will estimate the distance betweenu ˜h and K through several lemmas. Lemma 3.1. There exists a positive constant C independent of h such that

u˜hH2(Ω) ≤ C.

Proof. Since K ⊂ K˜h, we deduce from (3.2) that

(3.4) G(˜uh) ≤ G(u) and hence, by the Cauchy-Schwarz inequality, a Poincar´e-Friedrichs inequality [44] and the arithmetic-geometric mean inequality,  1| |2 ≤ u˜h H2(Ω) G(u)+ fu˜h dx 2 Ω ≤     ≤  2 1| |2 G(u)+ f L (Ω) u˜h L (Ω) G(u)+C f + u˜h 2 . 2 2 L2(Ω) 4 H (Ω) 

Lemma 3.2. The solution u˜h of (3.2) converges uniformly on Ω to the solution u of (1.1) as h ↓ 0.

Proof. It suffices to show that given any sequence hn ↓ 0 there exists a subsequence h such thatu ˜ converges uniformly on Ω to u as k →∞. nk hnk By Lemma 3.1 and the weak sequential compactness of bounded subsets of a

Hilbert space [40], there exists a subsequence hnk such that 2 (3.5)u ˜ converges weakly to some u∗ ∈ H (Ω) as k →∞. hnk 0 Since H2(Ω) is compactly embedded in C(Ω)¯ by the Rellich-Kondrachov Theorem [1], the weak convergence ofu ˜ implies that hnk

(3.6)u ˜h converges uniformly on Ω to u∗ as k →∞. nk  Inviewof(3.1)andthefactthattheset V is dense in Ω for any k,we j≥k hnj obtain by (3.6) the relation ψ1 ≤ u∗ ≤ ψ2 on Ω, i.e., u∗ ∈ K. Since the functional G is convex and continuous, it is also weakly lower semi- continuous (cf. [19]). Therefore we have, by (3.4) and (3.5),

G(u∗) ≤ lim inf G(˜uh ) ≤ G(u). k→∞ nk

We conclude from (1.1) that u∗ = u and henceu ˜ converges uniformly on Ω to u hnk by (3.6). 

Let Ii (i =1, 2) be the coincidence set of the obstacle problem (1.1) defined by

(3.7) Ii = {x ∈ Ω: u(x)=ψi(x)}.

The boundary condition of u and the assumptions on ψi imply that the compact sets ∂Ω, I1 and I2 are mutually disjoint. For any positive τ, let the compact set Ii,τ (i =1, 2) be defined by

(3.8) Ii,τ = {x ∈ Ω:¯ dist(x, Ii) ≤ τ}.

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We can choose τi > 0(i =1, 2) small enough so that the compact sets I1,2τ1 , I2,2τ2 and ∂Ω remain mutually disjoint.

Lemma 3.3. There exist positive numbers h0, γ1 and γ2 such that

(3.9) u˜h(x) − ψ1(x) ≥ γ1 if x ∈ Ω¯ and dist (x, I1) ≥ τ1,

(3.10) ψ2(x) − u˜h(x) ≥ γ2 if x ∈ Ω¯ and dist (x, I2) ≥ τ2,

provided h ≤ h0.

Proof. Since u − ψ1 (resp. ψ2 − u) is strictly positive on the compact set {x ∈ Ω:dist(¯ x, I1) ≥ τ1} (resp. {x ∈ Ω:dist(¯ x, I2) ≥ τ2}), this is an immediate consequence of Lemma 3.2. 

Let Ih be the nodal interpolation operator for the conforming P1 finite element space associated with Th. We can rewrite the displacement constraints onu ˜h as

(3.11) Ihψ1 ≤Ihu˜h ≤Ihψ2 on Ω¯.

Let the number δh,i be defined by  −I I −  (3.12) δh,i = (˜uh hu˜h)+( hψi ψi) ∞ for i =1, 2. L (Ii,τi ) ∈ 2 Since ψi C (Ω) and the compact set Ii,τi is disjoint from ∂Ω, Taylor’s Theorem implies that  −I  ≤ 2 (3.13) ψi hψi ∞ Ch for i =1, 2. L (Ii,τi ) Moreover, from the standard interpolation error estimate (cf. [24, 16])  −I  ≤ | | ∀ ∈ 2 (3.14) ζ hζ L∞(Ω) Ch ζ H2(Ω) ζ H (Ω) and Lemma 3.1, we have  −I  ≤ | | ≤ u˜h hu˜h L∞(Ω) Ch u˜h H2(Ω) Ch, and hence

(3.15) δh,i ≤ Ch for i =1, 2. Lemma 3.4. There exists a positive constant C independent of h such that | − |2 ≤ (3.16) u u˜h H2(Ω) C(δh,1 + δh,2) provided h is sufficiently small.

Proof. Let h0, γ1 and γ2 be as in Lemma 3.3. Without loss of generality, we may assume h ≤ h0. We have, by (3.11) and (3.12),

(3.17) u˜h(x) − ψ1(x)

=˜uh(x) −Ihu˜h(x)+Ihu˜h(x) −Ihψ1(x)+Ihψ1(x) − ψ1(x) ≥− ∀ ∈ δh,1 x I1,τ1 ,

(3.18) ψ2(x) − u˜h(x)

= ψ2(x) −Ihψ2(x)+Ihψ2(x) −Ihu˜h(x)+Ihu˜h(x) − u˜h(x) ≥− ∀ ∈ δh,2 x I2,τ2 .

In view of (3.15) we may also assume that, for h ≤ h0,

(3.19) max δh,i < min γi. i=1,2 i=1,2

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∞ Let φi (i =1, 2) be a function in C (Ω)¯ such that

(3.20) 0 ≤ φi ≤ 1onΩ¯,

(3.21) φi =1 onIi,τi , ¯ \ (3.22) φi =0 onΩ Ii,2τi . We claim that

(3.23)u ˆh =˜uh + δh,1φ1 − δh,2φ2 ∈ K.

Since the compact sets I1,2τ1 , I2,2τ2 and ∂Ω are mutually disjoint, it is clear from ∈ 2 (3.22) thatu ˆh H0 (Ω). It only remains to show that

(3.24) ψ1(x) ≤ uˆh(x) ≤ ψ2(x) ∀ x ∈ Ω, which we will establish by considering the following three possibilities: (i) x ∈ \ ∪ ∈ ∈ Ω I1,2τ1 I2,2τ2 ,(ii) x I1,2τ1 and (iii) x I2,2τ2 .   ∈ \ ∪ In view of (3.9) and (3.10), it is trivial that (3.24) holds for x Ω I1,2τ1 I2,2τ2 . ∈ For the case where x I1,2τ1 , we note that (3.10), (3.19), (3.20) and (3.22) imply

uˆh(x)=˜uh(x)+δh,1φ1(x) ≤ ψ2(x) − γ2 + δh,1 <ψ2(x). ∈ \ Furthermore, if x I1,2τ1 I1,τ1 ,then

uˆh(x)=˜uh(x)+δh,1φ1(x) ≥ ψ1(x)+γ1 >ψ1(x) ∈ by (3.9) and (3.20). Finally, if x I1,τ1 ,then

uˆh(x)=˜uh(x)+δh,1 ≥ ψ1(x) ∈ by (3.17) and (3.21). Therefore (3.24) holds for x I1,2τ1 . ∈ The proof that (3.24) holds for x I2,2τ2 is completely analogous. From (1.3), (1.4), and Lemma 3.1, we have 1 G(ˆu )= a(˜u + δ φ − δ φ , u˜ + δ φ − δ φ ) h 2 h h,1 1 h,2 2 h h,1 1 h,2 2 − (f,u˜h + δh,1φ1 − δh,2φ2) 1 (3.25) = a(˜u , u˜ ) − (f,u˜ )+a(˜u ,δ φ − δ φ ) 2 h h h h h,1 1 h,2 2 1 + a(δh,1φ1 − δh,2φ2,δh,1φ1 − δh,2φ2) − (f,δh,1φ1 − δh,2φ2) 2   ≤ G(˜uh)+C δh,1 + δh,2 . From (1.1), (1.3), (1.4), (3.3), (3.23) and (3.25) we then obtain

1 2 1 1 |u − u˜ | 2 = a(u, u) − a(˜u , u˜ ) − a(˜u ,u− u˜ ) 2 h H (Ω) 2 2 h h h h 1 1 ≤ a(u, u) − a(˜uh, u˜h) − (f,u − u˜h) 2 2   = G(u) − G(˜uh) ≤ G(ˆuh) − G(˜uh) ≤ C δh,1 + δh,2 . 

Remark 3.5. According to (3.23) the distance betweenu ˜h and K is bounded by

|δh,1φ1 − δh,2φ2|H2(Ω) ≤ C(δh,1 + δh,2),

and hence the estimate (3.16) shows precisely that the distance betweenu ˜h and u is bounded by the square root of the distance betweenu ˜h and K.

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2 We now take a closer look at the numbers δh,1 and δh,2.Sinceu ∈ C (Ω) and

the compact set Ii,τi is disjoint from ∂Ω, we have, by Taylor’s Theorem,  −I  ≤ 2 (3.26) u hu ∞ Ch for i =1, 2. L (Ii,τi ) Combining (3.14) and (3.26) we find  −I  ≤ − −I −   −I  u˜h hu˜h L∞(I ) (u u˜h) h(u u˜h) L∞(Ω) + u hu L∞(I ) i,τi   i,τi 2 ≤ C h|u − u˜h|H2(Ω) + h , which together with (3.12) and (3.13) implies   2 (3.27) δh,i ≤ C h|u − u˜h|H2(Ω) + h for i =1, 2. It now follows from (3.16), (3.27) and the arithmetic-geometric mean inequality that

2 2 1 2 |u − u˜ | 2 ≤ Ch + |u − u˜ | 2 , h H (Ω) 2 h H (Ω) and hence we have

(3.28) |u − u˜h|H2(Ω) ≤ Ch. Finally, we note that (3.27) and (3.28) imply 2 (3.29) δh,i ≤ Ch for i =1, 2, which improves the estimate (3.15).

4. Convergence analysis With Lemma 3.4, the estimate (3.29), and the two variational inequalities (1.5) and (3.3) at our disposal, we can now complete the convergence analysis of the finite element methods. In our analysis we will also need the estimate | |≤ | | | | ∀ ∈ 3 ∈ 2 (4.1) a(ζ,v) C ζ H3(Ω) v H1(Ω) ζ H (Ω),v H0 (Ω), which follows immediately from (1.4) and integration by parts. Lemma 4.1. There exists a positive constant C independent of h such that (4.2) a (u, Π u − u ) − (f,Π u − u ) h h  h h h   1 1    ≤ 2 2  −  2 C h + δh,1 + δh,2 Πhu uh h + h + δh,1 + δh,2 ,

where δh,i is defined in (3.12). Proof. We have   (4.3) ah(u, Πhu − uh)=a(u, EhΠhu − Ehuh)+ah u, Πhu − uh − Eh(Πhu − uh) because of (2.1), and it follows from (2.8) that      (4.4) ah u, Πhu − uh − Eh(Πhu − uh) ≤ ChΠhu − uhh. Moreover, we have

(4.5) a(u, EhΠhu − Ehuh)=a(˜uh,EhΠhu − Ehuh)+a(u − u˜h,EhΠhu − Ehuh),

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and it follows from (2.7) and (3.16) that     (4.6) a(u − u˜h,EhΠhu − Ehuh) ≤|u − u˜h|H2(Ω)|Eh(Πhu − uh)|H2(Ω)  1 1  ≤ 2 2  −  C δh,1 + δh,2 Πhu uh h. Now we use (3.3) to derive

(4.7) a(˜uh,EhΠhu − Ehuh)=a(˜uh, u˜h − Ehuh)+a(˜uh,EhΠhu − u˜h)

≤ (f,u˜h − Ehuh)+a(˜uh,EhΠhu − u˜h). Note that

(4.8) a(˜uh,EhΠhu − u˜h)=a(˜uh − u, EhΠhu − u˜h)+a(u, EhΠhu − u)

+ a(u, u − u˜h) and     (4.9) a(˜uh − u, EhΠhu − u˜h)+a(u, EhΠhu − u)

≤|u − u˜h|H2(Ω)|(˜uh − u)+(u − EhΠhu)|H2(Ω)

+ C|u| 3 |u − E Π u| 1  H (Ω) h h H (Ω)  2 2 ≤ C |u − u˜ | 2 + |u − E Π u| 2 + |u| 3 |u − E Π u| 1  h H (Ω)  h h H (Ω) H (Ω) h h H (Ω) 2 ≤ C h + δh,1 + δh,2 by (2.9), (3.16) and (4.1). Without loss of generality, we may assume that h is sufficiently small so that (3.23) holds. Therefore we can use (1.5) to obtain

a(u, u − u˜h)=a(u, u − uˆh)+δh,1a(u, φ1) − δh,2a(u, φ2)

(4.10) ≤ (f,u − uˆh)+δh,1a(u, φ1) − δh,2a(u, φ2) =(f,E Π u − u˜ )+(f,u − E Π u) h h h  h h 

− δh,1 (f,φ1) − a(u, φ1) + δh,2 (f,φ2) − a(u, φ2) , and, we have, because of (2.9),    −  ≤   −  ≤ 3 (4.11) (f,u EhΠhu) f L2(Ω) u EhΠhu L2(Ω) Ch . Combining (4.3)–(4.11), we find   (4.12) ah(u, Πhu − uh) − f,Eh(Πhu − uh) 1 1 ≤ 2 2  −  2 C (h + δh,1 + δh,2) uh Πhu h +(h + δh,1 + δh,2) . The estimate (4.2) follows from (2.7) and (4.12).  Theorem 4.2. There exists a positive constant C independent of h such that

(4.13) u − uhh ≤ Ch. Proof. It follows from (2.5), (2.13), Lemma 4.1, and the arithmetic-geometric mean inequality that  1 1     − 2 ≤  − 2 2 2  −  2 u uh h C u Πhu h + h + δh,1 + δh,2 Πhu uh h + h + δh,1 + δh,2  1 1     ≤ 2 2  −   −  2 C h + δh,1 + δh,2 Πhu u h + u uh h + h + δh,1 + δh,2   1 ≤ C h2 + δ + δ + u − u 2 h,1 h,2 2 h h

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and hence    − 2 ≤ 2 u uh h C h + δh,1 + δh,2 , which together with (3.29) implies the estimate (4.13). 

5. Concluding remarks The results in this paper are also valid for the one-obstacle problem. In fact, the analysis of the one-obstacle problem is slightly simpler since we only have to deal with one coincidence set. For simplicity we have assumed Ω to be convex. But the results can be extended ∈ 2+α ∈ 1 to a nonconvex polygonal domain Ω, in which case u H (Ω) for some α ( 2 , 1) α (cf. [8, 35, 26, 39]), and the error estimate (4.13) becomes u − uhh ≤ Ch . It is also possible to extend the results in this paper to obstacle problems with 4 nonhomogeneous boundary conditions. Let g ∈ H (Ω) such that ψ1

Appendix A. C2 regularity for the two-obstacle problem First we will extend the C2 regularity result in [31, 21] to the one-obstacle problem with nonhomogeneous data. Let D ⊂ R2 be a smooth bounded domain 2 (not necessarily connected), f ∈ L2(D), ψ ∈ C (D¯), ψ<0on∂D,and { ∈ 2 ≥ } K0,ψ = v H0 (D):v ψ on D . Consider the following obstacle problem: Find z0,ψ,f ∈ K0,ψ such that

(A.1) z0,ψ,f =argminGD(v), v∈K0,ψ

where the functional GD is defined as in (1.3)–(1.4), but with Ω replaced by D. 2 Lemma A.1. The solution z0,ψ,f of (A.1) belongs to C (D¯). · · ∈ 2 Proof. Let aD ( , ) be defined as in (1.4) but with Ω replaced by D,andφ H0 (D) be defined by ∀ ∈ 2 (A.2) aD (φ, v)=(f,v) v H0 (D). It follows from elliptic regularity (cf. [3, 42, 37]) and the Sobolev embedding theo- rem [1] that φ ∈ H4(D) ⊂ C2(D¯). Let ψ∗ = ψ − φ and { ∈ 2 ≥ } K0,ψ∗ = w H0 (D): w ψ∗ on D .

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∈ 2 ¯ ∈ ∈ Then we have ψ∗ C (D), ψ∗ < 0on∂D,andw K0,ψ∗ if and only if v = w +φ K0,ψ. By a simple calculation using (A.2) we find 1 1 a (w, w)=G (v)+ a (φ, φ), 2 D D 2 D − ∈ and hence z∗ = z0,ψ,f φ K0,ψ∗ satisfies 1 z∗ =argmin aD (w, w). w∈K0,ψ∗ 2 2 2 It follows from the C regularity result in [31, 21] that z∗ ∈ C (D¯) and hence 2 z0,ψ,f = z∗ + φ ∈ C (D¯). 

Next we consider the obstacle problem of finding zg,ψ,f ∈ Kg,ψ such that

(A.3) zg,ψ,f =argminGD(v), v∈Kg,ψ where g ∈ H4(D) ⊂ C2(D¯) satisfies g>ψon ∂D and { ∈ 2 − ∈ 2 ≥ } Kg,ψ = v H (D): v g H0 (D)andv ψ on D . 2 Lemma A.2. The solution zg,ψ,f of (A.3) belongs to C (D¯).

Proof. Let ψ† = ψ − g and { ∈ 2 ≥ } K0,ψ† = w H0 (D): w ψ† . ∈ 2 ¯ ∈ ∈ Then we have ψ† C (D), ψ† < 0on∂D,andw K0,ψ† if and only if v = w +g Kg,ψ. By another simple calculation we find 1 1 a (w, w) − (f − Δ2g, w)=G (v) − a (g, g)+(f,g) 2 D D 2 D − ∈ and hence z† = zg,ψ,f g K0,ψ† satisfies   1 − − 2 z† =argmin aD (w, w) (f Δ g, w) , ∈ w K0,ψ† 2

i.e., z† = z0,ψ†,(f−Δ2g) in the notation of (A.1). Therefore Lemma A.1 implies that 2 2 z† ∈ C (D¯) and hence zg,ψ,f = z† + g ∈ C (D¯).  We are now ready to tackle the two-obstacle problem defined by (1.1)–(1.3). 2 Since (1.5) implies Δ u = f on Ω \(I1 ∪ I2), from interior elliptic regularity (cf. ∈ 4 \ ∪ 2 [3, 42, 37]) we know that u Hloc Ω (I1 I2) and hence u is C outside the coincidence sets I1 and I2. Let D ⊂ Ω be a smooth domain such that I1 ⊂ D and D¯ is disjoint from ∂Ω 4 and I2.Sinceu belongs to the H (N )inanopenneighborhoodN of ∂D,wehaveu = g on ∂D,whereg ∈ H4(D) is the product of u and a smooth cut-off function. Because ∂D is outside the coincidence set I1, we also have g>ψ1 on ∂D.   We claim that uD = u D satisfies

(A.4) uD =argminGD(v), v∈K∗ where { ∈ 2 − ∈ 2 ≤ ≤ } K∗ = v H (D): v g H0 (D)andψ1 v ψ2 on D .

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∈ Indeed, if there exists a function v∗ K∗ such that GD(v∗)

belongs to K and G(u∗)

Next we claim that uD also satisfies

(A.5) uD =argminGD(v), ∈ v Kg,ψ1 where { ∈ 2 − ∈ 2 ≥ } Kg,ψ1 = v H (D): v g H0 (D)andv ψ1 on D .

We will establish (A.5) by showing that uD satisfies the equivalent variational in- equality − ≥ − ∀ ∈ (A.6) aD (uD ,v uD ) (f,v uD ) v Kg,ψ1 . ∈ Let v Kg,ψ1 be arbitrary. Since uD is strictly less than ψ2 on the compact set ¯ − D, the function wt =(1 t)uD + tv belongs to K∗ for any sufficiently small non- ≤ negative number t and GD(uD ) GD(wt)forsucht because of (A.4). It follows that (d/dt)GD(wt) is non-negative at t = 0, which is equivalent to (A.6). ∈ 2 ¯ In view of (A.5) and Lemma A.2, we have uD = zg,ψ1,f C (D). Similarly, we 2 can prove that u is C in a neighborhood of the coincidence set I2. We conclude that u ∈ C2(Ω).

Appendix B. Estimate (2.8) for the Morley finite element space

 A function v belongs to the Morley finite element space Vh if and only if (i)  ∈ ∈T T v T P2(T ) for all T h, (ii) v is continuous at the vertices of h and vanishes at the vertices along ∂Ω, and (iii) the normal derivative of v is continuous at the midpoints of the edges in Th and vanishes at the midpoints of the edges along ∂Ω. Note that ∇v (and hence ∇(v − Ehv)) is continuous at the midpoints of the edges in Eh and vanishes at the midpoints of the edges along ∂Ω. ∈ 3 ∩ 2 ∈ Let ζ H (Ω) H0 (Ω), v Vh and vT be the restriction of v to a triangle T ∈Th. Using integration by parts and the midpoint rule, we find  2 2 ah(ζ,v − Ehv)= D ζ : D (v − Ehv) dx ∈T T  T h 

− ∇ ·∇ − 2 ∇ − ⊗ = Δ( ζ) (v Ehv) dx + D ζ : (vT Ehv) nT ds T ∂T T ∈Th T ∈Th  = − Δ(∇ζ) ·∇(v − Ehv) dx ∈T T T h 

2 2 + D ζ−(D ζ)e :[[∇(v − Ehv) ⊗ n]] e ds, e e∈Eh

2 2 where nT is the unit outward normal along ∂T, (D ζ)e is the mean of D ζ over the ∇ − × ∇ − ⊗ edge e and [[ (v Ehv) n]] e is the sum of (vT Ehv) nT over the triangles that share e as a common edge.

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It follows that  1/2 | − |≤| | | − |2 ah(ζ,v Ehv) Δζ H1(Ω) v Ehv H1(T ) T ∈Th  1/2 1/2 + |e|−1D2ζ − (D2ζ) 2 |e|[[∇(v − E v) ⊗ n]]2 e L2(e) h L2(e) e∈Eh e∈Eh  1/2 ≤ | | | − |2 ≤ | |   C ζ H3(Ω) v Ehv H1(T ) Ch ζ H3(Ω) v h, T ∈Th where we have used the Cauchy-Schwarz inequality, the trace theorem (with scal- ing), the Bramble-Hilbert lemma [10] and (2.7).

Appendix C. Finite element methods for the biharmonic problem 2 When the obstacles are absent we have K = H0 (Ω), Kh = Vh, and the finite element methods in Section 2 become finite element methods for the biharmonic problem. A concise unified analysis of these methods can be performed using the tools introduced in Section 2. ∈ 2 ∈ Let z H0 (Ω) and zh Vh satisfy ∀ ∈ 2 (C.1) a(z,v)=(f,v) v H0 (Ω),

(C.2) ah(zh,v)=(f,v) ∀ v ∈ Vh, 3 where f ∈ L2(Ω). The convexity of Ω implies that z ∈ H (Ω) and

  3 ≤   (C.3) z H (Ω) CΩ f L2(Ω). We have a standard abstract error estimate [16, Lemma 10.1.1]  (C.4) z − zhh ≤ C inf z − vh +sup ah(z,w) − (f,w) /wh v∈V h w∈Vh\{0}

that follows from (2.2), (2.3) and (C.2), and for any w ∈ Vh,wehave (C.5) a (z,w) − (f,w)=a (z,w − E w) − (f,w − E w) h h h h 2 ≤ | | 3     C h z H (Ω) + h f L2(Ω) w h by (2.1), (2.7), (2.8) and (C.1). Combining (2.5) and (C.3)–(C.5), we find  −  ≤   z zh h Ch f L2(Ω). References

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Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803 E-mail address: [email protected] Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803 E-mail address: [email protected] Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803 E-mail address: [email protected]

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