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1 Original citation: Burke, Siobhan, Ortner, Christoph and Süli, Endre. (2010) An adaptive finite element approximation of a variational model of brittle fracture. SIAM Journal on Numerical Analysis, Vol.48 (No.3). pp Permanent WRAP url: Copyright and reuse: The Warwick Research Archive Portal (WRAP) makes this work of researchers of the University of Warwick available open access under the following conditions. Copyright and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available. Copies of full items can be used for personal research or study, educational, or not-forprofit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. Publisher s statement: SIAM A note on versions: The version presented in WRAP is the published version or, version of record, and may be cited as it appears here. For more information, please contact the WRAP Team at: publications@warwick.ac.uk

2 SIAM J. NUMER. ANAL. Vol. 48, No. 3, pp c 2010 Society for Industrial and Applied Mathematics AN ADAPTIVE FINITE ELEMENT APPROXIMATION OF A VARIATIONAL MODEL OF BRITTLE FRACTURE SIOBHAN BURKE, CHRISTOPH ORTNER, AND ENDRE SÜLI Abstract. The energy of the Francfort Marigo model of brittle fracture can be approximated, in the sense of Γ-convergence, by the Ambrosio Tortorelli functional. In this work, we formulate and analyze two adaptive finite element algorithms for the computation of its (local) minimizers. For each algorithm, we combine a Newton-type method with residual-driven adaptive mesh refinement. We present two theoretical results which demonstrate convergence of our algorithms to local minimizers of the Ambrosio Tortorelli functional. Key words. adaptive finite element method, brittle fracture, free-discontinuity problem, Ambrosio Tortorelli functional AMS subject classifications. 65N30, 65N50, 74R10 DOI / Introduction. Beginning with the work of Francfort and Marigo 25], a variational approach to the theory of quasi-static brittle fracture mechanics has experienced rapid and successful development. Upon recasting Griffith s idea of balancing energy release rate with a fictitious surface energy 27] as an energy minimization problem, Francfort and Marigo were able to formulate a model that was free of the usual constraints of fracture mechanics such as a predefined and piecewise smooth crack path. With the help of the theory of free-discontinuity problems 3], this model was soon shown to be well-posed in a surprisingly general setting 18, 19, 24]. We briefly review the model in section 1.1. The model of Francfort and Marigo is posed in terms of the minimization of a highly irregular energy functional, which also occurs in image segmentation where it is known as the Mumford Shah functional 30]. Several methods have been proposed in the literature, which regularize this energy in order to render the problem accessible to numerical simulation 10, 16, 31]. Such methods typically use the theory of Γ- convergence 12] to construct approximating functionals whose minimizers converge to those of the original functional. In our experience, the Ambrosio Tortorelli approximation 4, 5] is one of the most promising approaches. A particularly nice feature of the Ambrosio Tortorelli functional is that its minimization can be reduced to the solution of elliptic boundary value problems that are straightforward to discretize, for example, by a finite element method. This approach has been used successfully by Bourdin, Francfort, and Marigo 11] and Bourdin 8, 9] for the simulation of problems that are usually inaccessible to traditional methods. A brief review of the Ambrosio Tortorelli approximation is given in section 1.2. The Ambrosio Tortorelli approximation can be understood as a phase-field model for the crack set. To resolve the phase-field variable, the mesh near the crack has Received by the editors November 17, 2008; accepted for publication (in revised form) April 26, 2010; published electronically July 1, This work was supported by the EPSRC research programme New Frontiers in the Mathematics of Solids (OxMOS). Mathematical Institute, University of Oxford, Oxford OX1 3LB, United Kingdom (burke@maths. ox.ac.uk, ortner@maths.ox.ac.uk, suli@maths.ox.ac.uk). 980

3 AN ADAPTIVE FEM IN BRITTLE FRACTURE 981 to be significantly finer than the mesh that would be required to resolve the elastic deformation. Since we do not know the crack path in advance, it is therefore a natural idea to use an adaptively refined mesh. Using established techniques of adaptive finite element theory 36], we derive a residual estimate for the gradient of the Ambrosio Tortorelli functional (see section 3.1). In section 3.2, we formulate two adaptive algorithms for which we use the residual estimates to steer the mesh refinement. In section 4, we prove that the second of these algorithms converges to a critical point of the Ambrosio Tortorelli functional. We also prove that the first algorithm converges to a critical point under the assumption that the associated residuals converge to zero. We conclude with an implementation of the adaptive algorithms for two computational examples in section 5. In order to lay out the main ideas, our analysis in the present work is restricted to linearized elasticity in antiplane deformation and to linear finite elements. We will extend our results to more general approximations and a wider range of models in future work The Francfort Marigo model of brittle fracture. In order to introduce the Francfort Marigo model of brittle fracture, we briefly define the space of special functions of bounded variation 3, 22]. A knowledge of this space is not necessary for the main ideas contained in the paper. For p 1, ] and an open domain in R N,weuseL p () to denote the standard Lebesgue spaces on and H 1 () to denote the standard Hilbertian Sobolev space on. The N-dimensional Lebesgue and Hausdorff measures are denoted by L N and H N, respectively. We say that a function f L 1 () has bounded variation if { } sup fdivϕ dx : ϕ C 1 0 (; RN ), ϕ 1 <. The space of functions of bounded variation is denoted BV(). A function of bounded variation can exhibit discontinuities that are reflected in its distributional gradient. Given a function f BV(), the distributional derivative of f, denoted Df, is a Radon measure, which can be decomposed as Df = fl N +(f + (x) f (x)) ν f (x)h N 1 J(f)+D c f, where f is called the approximate gradient of f, J(f) isthejump set of f, ν f is the unit normal to J(f), f ± are the inner and outer traces of f on J(u) with respect to ν f,andd c f is called the Cantor part of the derivative. We refer the reader to 3] for the precise definitions of these terms. The space of special functions of bounded variation is then defined as SBV() := {f BV() : D c f =0}. We are now in a position to describe the Francfort Marigo model of brittle fracture 25]. The crack-free reference configuration of a linearly elastic body is denoted by. The set is taken to be an open, bounded, and connected domain with Lipschitz boundary (we shall relax this assumption later). For each u SBV() and for each Hausdorff measurable set Γ, we define the

4 982 SIOBHAN BURKE, CHRISTOPH ORTNER, AND ENDRE SÜLI bulk, surface, and total energy, respectively, by (1.1) E B (u) := u 2 dx, (1.2) (1.3) \J(u) E S (Γ) := κh N 1 (Γ), and { EB (u)+e E(u, Γ) := S (Γ) if H N 1 (J(u) \ Γ) = 0, + otherwise. The energy functional E(u, Γ) reflects Griffith s principle that, to create a crack, one has to spend an amount of elastic energy that is proportional to the area of the crack created 27]. The constant of proportionality κ is called fracture toughness and is often also denoted G c (critical energy release rate). The crack set Γ and the jump set J(u) are decoupled in the definition of the total energy in order to be able to impose irreversibility of the crack evolution. The quasi-static evolution will be obtained upon minimizing E subject to several constraints, which we introduce momentarily. We wish to study how the body evolves in time under the action of a varying load g(t), which is applied on an open subset D of positive N-dimensional Lebesgue measure. We assume that g L (0,T;W 1, ()) W 1,1 (0,T;H 1 ()), and we define A(g(t)) := { u SBV() : u D = g(t) D }. The fact that the Dirichlet condition is imposed on a set of positive N-dimensional Lebesgue measure is mostly technical and ensures that the jump set on the Dirichlet boundary D is well defined. Since we work in antiplane deformation, the boundary displacement is applied in a direction perpendicular to the plane in which the initial configuration of the domain lies. We call N := \ D the Neumann boundary. It is most intuitive to introduce the Francfort Marigo model through a time discretization. Let 0 = t 0 <t 1 < <t F = T be a discretization of the time interval 0,T], with Δt := max {t k t k 1 : k =1,...,F}. Given an initial crack Γ(0) (which should be the jump set of an SBV function u(0)), we seek (u(t k ), Γ(t k )), k =1,...,F, such that the following properties hold: 1. irreversibility: Γ(t k ) Γ(t k 1 ); 2. global stability: E(u(t k ), Γ(t k )) E(v, Γ) for all v A(g(t k )) and Γ Γ(t k 1 ). In practice, this formulation requires the successive solution of the global minimization problems (1.4) u(t k ):= argmin E B (v)+e S (J(v) Γ(t k 1 )), Γ(t k ):=J(u(t k )) Γ(t k 1 ). v A(g(t k )) In this formulation, the problem is accessible to the direct method of the calculus of variations to prove the existence of solutions to the time-discrete Francfort Marigo model 1, 2, 3, 20]. It remains to describe the limit of the time-discrete evolution as Δt 0. In full generality, it was first shown by Francfort and Larsen 24] (an earlier result is due to Dal Maso and Toader 19] and the extension to finite elasticity was developed by Dal Maso, Francfort, and Toader 18]) that it is indeed possible to extract weakly convergent subsequences of the discrete solutions and thus prove the existence of a trajectory u BV(0,T; SBV()) with crack set Γ(t) = s<t J(u(s)), t (0,T], such that the following properties hold:

5 AN ADAPTIVE FEM IN BRITTLE FRACTURE irreversibility: Γ(s) Γ(t) for all s, t 0,T] such that s t; 2. global stability: E(u(t), Γ(t)) E(v, J(v) Γ(t)) for all v A(g(t)); 3. energy balance: E(u(t), Γ(t)) E(u(s), Γ(s)) = 2 t s u(τ) ġ(τ)dxdτ. These three conditions should not be taken as the definition of a quasi-static crack evolution, as they underconstrain the system; however, they demonstrate that the limits of the time-discrete version of the model have several important properties, namely unilateral global stability and energy conservation. The ability to predict complicated crack paths is the greatest strength of the Francfort Marigo model and the reason for its popularity. In other respects, it may fall short of physical reality even on a qualitative level. For example, Francfort and Marigo acknowledged in their seminal work 25] that, from a mechanical point of view, it would be preferable to define an evolution by means of local minimizers. Unfortunately, this has proven to be a major challenge, which has not been resolved. In our numerical methods, which we describe in sections 2 5, we will be pragmatic regarding this point and will not make the distinction between local and global minimizers. In fact, we are unaware of an existing method that is able to compute global minimizers of highly nonconvex functionals such as the Ambrosio Tortorelli approximation, which we introduce next The Ambrosio Tortorelli approximation. Obtaining a numerical approximation of the time-discrete Francfort Marigo model is a nontrivial task. A direct discretization of the minimization problem (1.4) poses difficulties due to the irregularity of the energy functional and the need to accurately measure the surface area of the crack. A successful approach is to work instead with a regularization of E(u, Γ), which is able to represent the crack set in a manner more readily tractable by numerical methods. The regularized functional is chosen to approximate E(u, Γ) in the sense of Γ-convergence 12]. Consequently, minimizers of the approximating functional converge to minimizers of E(u, Γ) together with convergence of the minimized energy. One such approximation of E(u, Γ) is the Ambrosio Tortorelli functional I ε : H 1 (; R) H 1 (; 0, 1]) R, defined for 0 <η ε 1 as follows: ] 1 (1.5) I ε (u, v) := (v 2 + η) u 2 dx + κ 4ε (1 v)2 + ε v 2 dx. The family of functionals {I ε } ε>0 satisfies the following Γ-convergence result 11]. Let us define { Iε (û, ˆv) if û H G ε (û, ˆv) := 1 (), û = g(t) on D, ˆv H 1 (; 0, 1]), + otherwise, { E(û, J(û)) if û A(g(t)), ˆv = 1 a.e. on, G(û, ˆv) := + otherwise; then G ε Γ-converges to G in L 1 () L 1 () as ε 0. This is a modification of the original Ambrosio Tortorelli Γ-convergence result 4] for the approximation of the Mumford Shah functional 30]. The existence of minimizers to I ε has been shown in 5] for each ε, η>0. Further generalizations of the Γ-convergence result have since been established in 7, 23]. Using the Ambrosio Tortorelli functional, an approximation to the time-discrete Francfort Marigo model can be computed as follows. At time t = t 0, find (u ε (t 0 ),v ε (t 0 )) argmin{i ε (û, ˆv) :û H 1 (), û = g(t 0 )on D ;ˆv H 1 (; 0, 1])}.

6 984 SIOBHAN BURKE, CHRISTOPH ORTNER, AND ENDRE SÜLI At subsequent times t = t k, k =1,...,F, find (u ε (t k ),v ε (t k )) satisfying (1.6) (u ε (t k ),v ε (t k )) argmin{i ε (û, ˆv) :û H 1 (), û = g(t k )on D ;ˆv H 1 (), ˆv v ε (t k 1 )}. At the fixed time t = t k, the function u ε (t k ) is an approximation of the displacement of the body u(t k ), with u ε (t k ) u(t k )inl 1 () as ε 0. The auxiliary function v ε (t k ) is introduced in order to represent the crack Γ(t k ). A truncation argument shows that 0 v ε (x, t k ) 1 a.e. x. The crack is then approximated by the subset of the domain on which v ε (t k )takes values close to zero. Conversely, the unfractured part of the body is represented by the subset of the domain on which v ε (t k ) takes values close to one. The transition layer between the two regimes has thickness of order ε. In the limit ε 0, the minimization of I ε requires that v ε (t k ) 1 a.e. Consequently, the transition layer becomes infinitely thin, with v ε (t k ) 0 only on the (N 1)-dimensional surface representing the crack. It was shown by Giacomini 26] that an evolution satisfying (1.6) converges as Δt, ε 0 to an evolution satisfying the conditions of the time-continuous Francfort Marigo model. We will therefore restrict our consideration to the problem of minimizing the Ambrosio Tortorelli functional at the fixed moment in time t = t k and for fixed values of ε and η. The condition ˆv v ε (t k 1 ) in (1.6) enforces the irreversibility of the crack 26]. In practice, however, we choose to implement the irreversibility criterion through the following equality constraint introduced by Bourdin 8]. If at time t = t k 1, k {1,...,F}, theset (1.7) CR(t k 1 ):={x :v ε (x, t k 1 ) < CRTOL} is nonempty for some small specified tolerance CRTOL, thenfix v ε (x, t i )=0 x CR(t k 1 )and i such that k i F. Thus, if, at a particular time t = t k 1, v ε (x, t k 1 ) is close enough to zero to indicate that the point x lies on the crack path, then v ε issettobezeroatthatpointforall subsequent time steps. This simplifies the minimization over ˆv by allowing the use of an unconstrained minimization algorithm. Remark 1. We note that this modification of the irreversibility criterion could result in a crack-widening effect, since it is conceivable that setting v ε (x, t k 1 ) to zero causes v ε to be pulled below CRTOL at points in a neighborhood of x, which will then be set to zero at t = t k. In practice, however, we do not observe this effect to occur in the discrete case. As such, we continue to use this formulation of irreversibility here and refer the reader to 14] for an alternative implementation that uses a modified monotonicity condition. A numerical discretization of this minimization problem has been proposed and implemented by Bourdin, Francfort, and Marigo 11] and Bourdin 8, 9] in which a finite element method is used in conjunction with a minimization algorithm to compute minimizers of the Ambrosio Tortorelli functional. The method uses a fine mesh to discretize the whole domain. It does, however, seem apparent to us that the

7 AN ADAPTIVE FEM IN BRITTLE FRACTURE 985 behavior of minimizers can be sufficiently resolved using a mesh that is fine only in a layer around the crack, whilst remaining relatively coarse elsewhere. Naturally, we do not know the location of the crack a priori. Thus, we propose using an adaptive finite element method for the numerical computation of minimizers. One could argue that the refinement of only part of the mesh causes the crack to favor growth in this direction; we believe, however, that a carefully designed adaptive algorithm can circumvent this eventuality. We refer the reader to Remark 5 for a further discussion of this point Critical points. In the Ambrosio Tortorelli model, Lipschitz regularity of the domain is not required. Since, in practice, it is more convenient to model a pre-existing crack by a slit domain rather than an initial crack field v 0, we will, from now on, drop the assumption that is a Lipschitz domain. Motivated by the fact that we will need to partition for the purpose of defining a finite element approximation, we shall assume that is a polyhedral domain. By this, we simply mean that possesses a finite partition into nondegenerate N-simplexes: there exist open, disjoint, nondegenerate simplexes T 1,...,T K such that L N ( \ k T k )=0 (see also section 2). In that case, it is clear that the usual trace and embedding theorems for Sobolev spaces hold on. Since we consider the minimization of the Ambrosio Tortorelli functional at the fixed time t = t k, it is useful to define the following function spaces: (1.8) (1.9) (1.10) H 1 D () := {w H1 () : w =0on D }, H 1 g() := {w H 1 () : w = g(t k )on D }, H 1 c() := {w H 1 () : w =0onCR(t k 1 )}. We also fix ε and η, and for simplicity we set κ = 1. Accordingly we choose to relabel the Ambrosio Tortorelli functional as I :H 1 g () H1 c () R {+ }, where (1.11) I(u, v) := (v 2 + η) u 2 + α(1 v) 2 + ε v 2] dx, and α =1/4ε. It can be seen, using a truncation argument, that any local minimizer (u, v) ofi (in the H 1 () H 1 () topology) satisfies 0 v(x) 1 a.e. in. Thus all relevant trial functions for v lie in the space L (). As such, in the following discussion of differentiability of I, we work with trial functions for v from the space H 1 () L (). Proposition 1.1. I is Fréchet-differentiable in H 1 () (H 1 () L ()). Proof. Let (u, v) H 1 () (H 1 () L ()). A straightforward calculation yields that the directional derivative of I at (u, v) in the direction (ϕ, ψ) H 1 () (H 1 () L ()) is given by I (u, v; ϕ, ψ) =2 (v 2 + η) u ϕdx +2 vψ u 2 + α(v 1)ψ + ε v ψ ] dx =: 2a(v; u, ϕ)+2b(u; v, ψ). This is in fact the Fréchet derivative, since g(u, v; ϕ, ψ) :=I(u + ϕ, v + ψ) I(u, v) I (u, v; ϕ, ψ) = ψ 2 u 2 + ϕ 2( (v + ψ) 2 + η ) +(4vψ +2ψ 2 ) u ϕ + αψ 2 + ε ψ 2 ]dx

8 986 SIOBHAN BURKE, CHRISTOPH ORTNER, AND ENDRE SÜLI satisfies g(u, v; ϕ, ψ) ψ 2 L () u 2 L 2 () + (v + ψ)2 + η L () ϕ 2 L 2 () +4 v L () ψ L () u L 2 () ϕ L 2 () +2 ψ 2 L () u L 2 () ϕ L2 () + α ψ 2 L 2 () + ε ψ 2 L 2 (), and thus g(u, v; ϕ, ψ) 0 ϕ H 1 () + ψ H 1 () + ψ L () as ( ϕ H 1 () + ψ H 1 () + ψ L ()) 0. Remark 2. We note that I(u, v) is not finite for all (u, v) H 1 () H 1 (), and thus I is not Gâteaux-differentiable in H 1 () H 1 (). This motivates the following definition of a critical point. Definition 1.2. We say that (u, v) H 1 g() (H 1 c() L ()) is a critical point of I if I (u, v; ϕ, ψ) =0for all ϕ H 1 D () and ψ H1 c () L (). Proposition 1.3. If (u, v) H 1 g() (H 1 c() L ()) is a critical point of I, then 0 v(x) 1 for a.e. x. Proof. Suppose that (u, v) is a critical point of I and that there exist subsets A 1 and A 2 of with A 1 A 2 > 0 such that v>1ona 1 and v<0ona 2. Since (u, v) is a critical point, it follows that vψ u 2 + α(v 1)ψ + ε v ψ ] dx =0 ψ H 1 c () L (). Choosing 1 v(x) ifx A 1, ψ(x) = 0 ifx \(A 1 A 2 ), v(x) ifx A 2 yields v(1 v) u 2 α(1 v) 2 ε v 2] dx v 2 u 2 +αv(v 1)+ε v 2] dx =0, A 1 A 2 which is a contradiction, since the left-hand side is strictly negative. 2. Finite element approximation and minimization Finite element discretization. Since we assumed that is a polyhedral domain (see section 1.3), we may discretize it as follows. Let T h be a subdivision of inton-dimensional open simplexes such that = T Th T and T i T j = for T i, T j T h, with i j. The subdivision T h is chosen in such a way that the boundary of D is discretized as the union of faces of simplexes from T h. We define h := max T Th diam(t ), and each simplex is taken to be an affine transformation of the open unit simplex ˆT := {ˆx =(ˆx 1,...,ˆx N ):0< ˆx i,i=1,...,n, 0 < ˆx 1 + +ˆx N < 1}. Each simplex is called an element. We assume that the subdivision is conforming; that is, the intersection of the closure of any two elements either is empty

9 AN ADAPTIVE FEM IN BRITTLE FRACTURE 987 or is along an entire k-dimensional face, 0 k N 1. We also require that the subdivision be shape regular, i.e., h T sup K ρ T for some K (0, ), where h T := diam(t )andρ T is the diameter of the largest N-dimensional ball contained in T. Let N h N denote an index set for the vertices of T h. For a vertex with index i N h,letx i denote the position of the vertex and let ζ i be the continuous piecewise linear basis function such that ζ i (x j )=δ ij. Define N D,h := {i N h : x i D }. Let E h denote the set of (N 1)-dimensional open faces in the subdivision with E D,h := {e E h : e D }, E N,h := {e E h : e N }, and E I,h := {e E h \(E D,h E N,h )}. We define E h, E D,h, E N,h,andE I,h as the union of all faces in E h, E D,h, E N,h,andE I,h, respectively. For all i N h,letω i be the closure of the union of elements that have x i as the position of a vertex, that is, ω i := supp(ζ i ). For a face e E h and element, define ω e := i Nh :x i e ω i, ω T := i Nh :x ω i T i,andh e := diam(e). We now define the finite element spaces { } X h := λ i ζ i : λ i R, i N h { } X h,d := λ i ζ i : λ i R, λ i =0 i N D,h i N h and the finite element space at time t = t k { } Xh,g k := λ i ζ i : λ i R, λ i = g(t k,x i ) i N D,h. i N h In order to define a discrete function space setting for the phase-field variable v, we must first define the discrete analogue of CR(t k 1 ). For a given v h X h and a small specified tolerance CRTOL, we define Eh CR (t k 1) :={e E h : v h (x, t k 1 ) CRTOL x e} and CR h (t k 1 ):= e. We now define the finite element space Xh,c k := {w h X h : w h (x) =0 x CR h (t k 1 )}. e E CR h Since we consider a fixed time t k, we relabel Xh,g k and Xk h,c as X h,g and X h,c, respectively, for ease of notation. Also, for simplicity, we assume throughout that g lies in the finite element space X h. For our subsequent analysis, we require the discrete formulation to satisfy a maximum principle analogous to Proposition 1.3. In order to accomplish this, we use a mass lumping approximation 35, Chapter 11] for I together with an assumption on the stiffness matrix, which can be achieved through typical mesh regularity conditions. The mass lumping approximation of I is defined to be the functional ( (2.1) I h (u h,v h ):= Ph (vh)+η 2 ) u h 2 ( + αp h (vh 1) 2) + ε v h 2] dx, where P h :C() X h is the standard nodal interpolation operator 13, section 3.3].

10 988 SIOBHAN BURKE, CHRISTOPH ORTNER, AND ENDRE SÜLI Let K be the N h N h stiffness matrix with entries (k ij ) i,j Nh ; then we assume that (2.2) k ij := ζ i ζ j dx 0 i j N h. This condition has been studied in detail in two dimensions 17], 34, p. 78] and three dimensions 28]. We note that this assumption is made only for technical purposes, and in practice we do not believe that it is necessary for ensuring that the conclusion of Proposition 2.2 holds; see Remark 7. Definition 2.1. We say that (u h,v h ) X h,g X h,c is a critical point of I h if I h (u h,v h ; ϕ h,ψ h )=0for all ϕ h X h,d and ψ h X h,c,where I h(u h,v h ; ϕ h,ψ h )=2a h (v h ; u h,ϕ h )+2b h (u h ; v h,ψ h ), ( a h (v h ; u h,ϕ h ):= Ph (vh)+η 2 ) u h ϕ h dx, b h (u h ; v h,ψ h ):= Ph (v h ψ h ) u h 2 ( ) ] + αp h (vh 1)ψ h + ε vh ψ h dx. Proposition 2.2. Let (u h,v h ) X h,g X h,c be such that b h (u h ; v h,ψ h )=0for all ψ h X h,c ;then0 v h (x) 1 for all x. Proof. Let(u h,v h ) X h,g X h,c be such that b h (u h ; v h,ψ h ) = 0 for all ψ h X h,c. Let v h = j N h v j ζ j, and suppose, for contradiction, that there exist subsets J 1 and J 2 of N h with v j > 1 for all j J 1 and v j < 0 for all j J 2. First suppose that J 1 is nonempty and i J 1 is such that v i v j for all j N h. Consider the patch of elements ω i := supp(ζ i ), and define M i := {j N h : x j ω i }. On taking ψ h = ζ i,wehaveb h (u h ; v h,ζ i ) = 0, which implies that ε v h ζ i dx = P h (v h ζ i ) u h 2 dx α P h ((v h 1)ζ i )dx ω i ω i ω i (2.3) = 2v i N! < 0. u h 2 T T 2α N! (v i 1) ω i T ω i Since v h = j N h v j ζ j, on noting that the rows of the stiffness matrix sum to zero, we have v h ζ i dx = k ij v j = k ij (v j v i )+ k ij v i = k ij (v j v i ). ω i j M i j M i j M i j M i Thus, using the assumption that k ij 0fori j, it follows that ω i v h ζ i dx 0, which contradicts (2.3). A similar argument can be used to contradict the existence of vertices in J 2. In the finite element approximation, at time t = t k, we seek to find (2.4) (u h,v h ) argmin{i h (û h, ˆv h ):û h X h,g, ˆv h X h,c } The alternate minimization algorithm. The minimization of the functional I h is a nontrivial task, since the term P h (v 2 h ) u h 2 renders the functional nonconvex. A number of minimization schemes can be employed; we note, however, that none would in general be able to find the location of global minimizers, at least

11 AN ADAPTIVE FEM IN BRITTLE FRACTURE 989 not easily. Instead we must be satisfied with being able to locate local minimizers. (In any case, as we have previously noted, it is unclear to us that finding global minimizers of the Francfort Marigo model is physically justified.) The minimization will be achieved using an alternate minimization algorithm proposed by Bourdin, Francfort, and Marigo 11]. We state the algorithm for the minimization of I over the infinite-dimensional space H 1 g () H1 c () at time t = t k. The idea is as follows. Although the functional I is nonconvex with respect to the pair (u, v), I is convex with respect to u and v separately. Thus it is a straightforward computation to minimize with respect to one variable at a time. For some termination tolerance VTOL, the algorithm proceeds as follows: 1. Set v 1 =1ifk =0andv 1 = v(t k 1 )ifk>0. 2. For n =1, 2,..., u n =argmin{i(z,v n ):z H 1 g ()}; v n+1 =argmin{i(u n,w):w H 1 c()}; repeat until v n+1 v n L () < VTOL. 3. Set u(t k )=u n and v(t k )=v n+1. The algorithm has been successfully implemented by Bourdin, Francfort, and Marigo 11] and Bourdin 8, 9] for a range of computational examples. However, to date there does not exist a proof of convergence for the algorithm to a minimizer as n.it appears to us that the proof of 8, Theorem 1] is incomplete, even though the result is certainly correct. The missing arguments can be found in the present paper, in the proofs of Theorems 4.1 and 4.2 (in particular, Step 4 in the proof of Theorem 4.1 and Steps 3 and 4 in the proof of Theorem 4.2). For a proof of local convergence to isolated local minimizers, see 8, Theorem 2]. Remark 3. The alternate minimization algorithm can be understood as a Newtontype descent method with a special starting guess. To see this, let u 0 H 1 g () and v 1 = argmin H 1 c ()I(u 0, ). The first step of the alternate minimization algorithm, computing the pair (u 1,v 2 ), can be written as (2.5) u I(u 1,v 1 )=0 and v I(u 1,v 2 )=0, where u I and v I denote the partial derivatives of I. SinceI is quadratic in each of its two coordinates, with nonsingular partial derivatives uu I and vv I, the equalities (2.5) are equivalent to uu I(u 0,v 1 )(u 1 u 0 )= u I(u 0,v 1 ) and vv I(u 1,v 1 )(v 2 v 1 )= v I(u 1,v 1 ). It can now be easily seen that these two equations are equivalently written as ( )( ) ( ) uu I(u 0,v 1 ) 0 ū1/2 u 0 u I(u 0 vv I(u 0,v 1 ) v 1/2 v 1 = 0,v 1 ) v I(u 0,v 1 and ) ( )( ) ( ) uu I(ū 1/2, v 1/2 ) 0 ū1 ū 1/2 u I(ū 0 vv I(ū 1/2, v 1/2 ) v 1 v 1/2 = 1/2, v 1/2 ) v I(ū 1/2, v 1/2, ) where (ū 1/2, v 1/2 ):=(u 1,v 1 )and(ū 1, v 1 ):=(u 1,v 2 ). 3. Adaptive algorithm. The nature of solutions to the minimization problem strongly motivates the use of an adaptive finite element method for their computation. Such methods use a local refinement indicator, based on the computed solution, to identify those elements where mesh refinement would be most beneficial for improving

12 990 SIOBHAN BURKE, CHRISTOPH ORTNER, AND ENDRE SÜLI the accuracy of the solution. It is now a well-established technique to use residual estimates as refinement indicators 36]. For a critical point (u h,v h )ofi h from the finite-dimensional space X h,g X h,c, we use an estimate of the residual (3.1) I (u h,v h ; ϕ, ψ) for ϕ H 1 D (), ψ H1 c () in the ( H 1 D () H1 c ()) norm as a refinement indicator. Note that (3.1) is well defined, since u h, v h W 1, (). We derive an a posteriori upper bound on this residual in the first part of this section. In the second part, we propose two adaptive finite element algorithms based on this bound to determine mesh refinement Residual estimates. The following interpolation results will be needed for the subsequent residual estimate. Henceforth we use to denote C, wherethe positive constant C depends only on the shape-regularity parameter K of the mesh but not on the mesh size. We remind the reader that, for a node with position x i,afacee, oranelementt, ω i, ω e,andω T denote the closure of the union of all elements that touch the respective sets (or point). Here, the touching of two sets means that the intersection of their closure is nonempty. Let Δ i be the maximal set of the form B contained within ω i,whereb is a ball with center x i (cf. 37] for more detail). For χ H 1 (), define the quasi interpolants Π h,d χ X h,d and Π h,c χ X h,c as follows: ( Πh,D χ ) ( ) 1 (3.2) (x) := χ dx λ i (x), Δ i i N h \N Δ i h,d ( Πh,c χ ) (x) := ( ) 1 χ dx λ i (x). Δ i Δ i i N h x i CR h This quasi interpolant was defined by Verfürth 37] and was shown to satisfy the following approximation results. Let χ H 1 D (), and let Π h,d χ be as above; then, for all and e E h, (3.3) (3.4) Π h,d χ χ Hs (T ) h 1 s T χ L2 (ω T ), s {0, 1}, Π h,d χ χ L 2 (e) h 1/2 e χ L 2 (ω e). The same approximation result holds taking χ H 1 c () and replacing Π h,d χ with Π h,c χ. We also have the following approximation result for the nodal interpolant; see 13, section 4.4]. For all and w W s, (T ), (3.5) w P h w L (T ) h s T w W s, (T ), s {1, 2}. Before stating the main proposition of this section, it will be useful to introduce the following definition. For w h X h and all e E h, we define w h T1 w h T2 if e E h \, with e = T 1 T 2 w h ] e := for some T 1,T 2 T h, w h n e if e, where n is the outer unit normal vector to.

13 AN ADAPTIVE FEM IN BRITTLE FRACTURE 991 Proposition 3.1. Let u h X h,g, v h X h,c be such that I h (u h,v h ; ϕ h,ψ h )=0 for all ϕ h X h,d,ψ h X h,c ;then I (u h,v h ; ϕ, ψ) μ h ϕ L 2 () + ν h ψ L 2 () ϕ H 1 D(), ψ H 1 c(), where μ h,ν h are defined as follows: ] 1/2 ] 1/2 (3.6) μ h := μ T (u h,v h ) 2, ν h := ν T (u h,v h ) 2, where μ T (u h,v h ) 2 := h 4 T v h 4 L (T ) u h 2 L 2 (T ) + h2 T 2v h ( v h u h ) 2 L 2 (T ) + h e vh 2 + η 2 L 2 (e) u h ] 2 e, e T (E I,h E N,h ) ν T (u h,v h ) 2 := h 4 T u h 2 + α 2 L 2 (T ) v h 2 L (T ) + h2 T ( u h 2 + α)v h α 2 L 2 (T ) + ε 2 h e v h ] e 2 L 2 (e). e T Proof. SinceI h (u h,v h ; ϕ h,ψ h ) = 0 for all ϕ h X h,d, ψ X h,c, it follows that (3.7) a h (v h ; u h,ϕ h )=0 ϕ h X h,d and (3.8) b h (u h ; v h,ψ h )=0 ψ h X h,c. Let us fix ϕ H 1 D () and ψ H1 c(); then I (u h,v h ; ϕ, ψ) 2 a(v h ; u h,ϕ) +2 b(u h ; v h,ψ). Hence we shall examine the two functionals ϕ a(v h ; u h,ϕ)andψ b(u h ; v h,ψ) separately. We begin by considering the former. By (3.7), we have (3.9) a(v h ; u h,ϕ) a(v h ; u h,ϕ ϕ h ) + a(v h ; u h,ϕ h ) a h (v h ; u h,ϕ h ) ϕ h X h,d. Examining the first term in (3.9), a(v h ; u h,ϕ ϕ h ) = (vh 2 h (ϕ ϕ h )dx T T T h { = 2v h ( v h u h )(ϕ ϕ h )dx + T T (v 2 h + η) u h n (ϕ ϕ h )ds } 2v h ( v h u h ) L 2 (T ) ϕ ϕ h L 2 (T ) + u h ] e vh 2 + η ϕ ϕ h ds e (E I,h E N,h ) e 2v h ( v h u h ) L 2 (T ) ϕ ϕ h L 2 (T ) + u h ] e vh 2 + η L 2 (e) ϕ ϕ h L 2 (e). e (E I,h E N,h )

14 992 SIOBHAN BURKE, CHRISTOPH ORTNER, AND ENDRE SÜLI Since this inequality is true for all ϕ h X h,d,wechooseϕ h =Π h,d ϕ and use the approximation results (3.3) and (3.4) to obtain a(v h ; u h,ϕ ϕ h ) h T 2v h ( v h u h ) L 2 (T ) ϕ L 2 (ω T ) + h 1/2 e vh 2 + η L 2 (e) u h ] e ϕ L2 (ω e) e (E I,h E N,h ) ] 1/2 h 2 T 2v h ( v h u h ) 2 L 2 (T ) ϕ L2() 1/2 + h e vh 2 + η 2 L 2 (e) u h ] 2 e ϕ L 2 () e (E I,h E N,h ) h2 T 2v h ( v h u h ) 2 L 2 (T ) + e T (E I,h E N,h ) h e vh 2 + η 2 L 2 (e) u h ] 2 e 1/2 ϕ L2 (). Using the approximation result (3.5) for the nodal interpolant and (3.3) with s = 1, we can bound the second term in (3.9) as follows: (3.10) a(v h ; u h,ϕ h ) a h (v h ; u h,ϕ h ) = {vh 2 P h (vh)} u 2 h ϕ h dx vh 2 P h (vh) u 2 h ϕ h dx T vh 2 P h (vh) 2 L (T ) u h L 2 (T ) ϕ h L 2 (T ) h 2 T v h 2 L (T ) u h L 2 (T ) ϕ L 2 (ω T ) ] 1/2 h 4 T v h 4 L (T ) u h 2 L 2 (T ) ϕ L2(). Therefore, a(v h ; u h,ϕ) μ T (u h,v h ) 2 ] 1/2 ϕ L2(),

15 AN ADAPTIVE FEM IN BRITTLE FRACTURE 993 where μ T (u h,v h ) 2 = h 4 T v h 4 L (T ) u h 2 L 2 (T ) + h2 T 2v h ( v h u h ) 2 L 2 (T ) + h e vh 2 + η 2 L 2 (e) u h ] 2 e. e T (E I,h E N,h ) Now let us consider ψ b(u h ; v h,ψ). For all ψ h X h,c, it follows from (3.8) that (3.11) b(u h ; v h,ψ) b(u h ; v h,ψ ψ h ) + b(u h ; v h,ψ h ) b h (u h ; v h,ψ h ). Considering the first term in (3.11), b(u h ; v h,ψ ψ h ) = { ( uh 2 + α)v h α}(ψ ψ h )+ε v h (ψ ψ h ) ] dx T T T h ( u h 2 + α)v h α L2 (T ) ψ ψ h L2 (T ) + ε (ψ ψ h ) v h n ds T T T h ( u h 2 + α)v h α L 2 (T ) ψ ψ h L 2 (T ) + ε ψ ψ h L 2 (e) v h ] e L 2 (e). e E h This is true for all ψ h X h,c, so taking ψ h =Π h,c ψ it follows that b(u h ; v h,ψ ψ h ) h T ( u h 2 + α)v h α L 2 (T ) ψ L 2 (ω T ) +ε e v h ] e L2 (e) ψ L2 (ω e) e E h h 1/2 h 2 T ( u h 2 + α)v h α 2 L 2 (T ) +ε 1/2 h e v h ] e 2 L 2 (e) ψ L2 () e E h h2 T ( u h 2 + α)v h α 2 L 2 (T ) + ε 2 e T h e v h ] e 2 L 2 (e) 1/2 ] 1/2 ψ L2() ψ L2 (). Finally we bound the second term in (3.11), noting that we have now fixed ψ h =

16 994 SIOBHAN BURKE, CHRISTOPH ORTNER, AND ENDRE SÜLI Π h,c ψ: (3.12) b(u h ; v h,ψ h ) b h (u h ; v h,ψ h ) = (v h ψ h P h (v h ψ h ))( u h 2 + α)dx v h ψ h P h (v h ψ h ) ( u h 2 + α)dx T T T h T u h 2 + α L 2 (T ) v h ψ h P h (v h ψ h ) L (T ) h N/2 T u h 2 + α L2 (T ) v h ψ h W 2, (T ) h (N/2)+2 T u h 2 + α L 2 (T ) v h L (T ) ψ h L (T ). h (N/2)+2 Using the equivalence of norms on a finite-dimensional space and the shape regularity of T h,wehave Therefore, where b(u h ; v h,ψ h ) b h (u h ; v h,ψ h ) T u h 2 + α L2 (T ) v h L (T ) ψ h L2 (T ) h (N/2)+2 (N/2) h 4 T u h 2 + α 2 L 2 (T ) v h 2 L (T ) b(u h ; v h,ψ) ν T (u h,v h ) 2 ] 1/2 ψ L2(). ] 1/2 ψ L2(), ν T (u h,v h ) 2 = h 4 T u h 2 + α 2 L 2 (T ) v h 2 L (T ) + h2 T ( u h 2 + α)v h α 2 L 2 (T ) + ε 2 h e v h ] e 2 L 2 (e). e T We use μ T (u h,v h ) as a local refinement indicator for u h,useν T (u h,v h ) as a local refinement indicator for v h, and define (3.13) E T (u h,v h ):= μ T (u h,v h ) 2 + ν T (u h,v h ) 2] 1/2. Remark 4. We are in fact estimating the ( H 1 D () H1 c() ) norm of the gradient of I, since I (u h,v h ) (H 1 D () H 1 c ()) = sup ϕ H 1 D (), ψ H 1 c () I (u h,v h ; ϕ, ψ) ϕ 2 H 1 () + ψ 2 H 1 () ] 1/2. E T (u h,v h ) 2 ] 1/2

17 AN ADAPTIVE FEM IN BRITTLE FRACTURE Adaptive algorithm. We now propose two adaptive algorithms for computing local minimizers of I. The difference between the two algorithms is the stage at which the adaptive refinement of the mesh takes place. In Algorithm 1, this occurs after the alternate minimization algorithm has converged, whilst in Algorithm 2 the mesh is refined at each step of the alternate minimization algorithm. There are two user-specified tolerances associated with the algorithms: VTOL is the tolerance which determines when to halt the alternate minimization loop, whilst REFTOL is the tolerance determining when to halt the refinement loop. The marking parameter θ is a fixed number lying in the interval (0, 1]. In Algorithm 1, we denote the mesh at the jth level of refinement by T j with h j := max T Tj diam(t )forj N. In Algorithm 2, the mesh is also dependent on the alternate minimization step; accordingly, within each alternate minimization step n {m/2 :m N}, wedenotethemeshatthejth level of refinement by T n j h n j := max T T n j diam(t )forj N. with Algorithm Input: Initial crack field v 0 and initial mesh T For j =1, 2,..., (a) set vj 1 = v j 1; (b) for n =1, 2,..., u n j := argmin {I h j (z,vj n):z X h j,g}; vj n+1 := argmin {I hj (u n j,z):z X h j,c}; repeat until vj n+1 vj n L () < VTOL; (c) set v j = v n+1 j, u j = u n j ; (d) if T T j E T (u j,v j ) 2] 1/2 > REFTOL, find the smallest set M j T j such that T M j E T (u j,v j ) 2 θ T T j E T (u j,v j ) 2 ; refine elements in M j to obtain the new mesh T j+1 ; (e) repeat until T T j E T (u j,v j ) ] 2 1/2 REFTOL. 3. Set u h (t k )=u j and v h (t k )=v j. Algorithm Input: Initial crack field v 1 and initial mesh T 1/2. 2. For n =1, 2,..., (a) set T1 n = T n 1/2 ; (b) for j =1, 2,..., compute u n j := argmin {I h n(z,vn ):z X j h n j,g}; if T T μ j n T (u n j,vn ) ] 2 1/2 > REFTOL/ 2, find the smallest set M j Tj n such that T M j μ T (u n j,v j) 2 θ T T μ j n T (u n j,v j) 2 ; refine elements in M j to obtain the new mesh T n repeat until T T μ j n T (u n j,vn ) ] 2 1/2 REFTOL/ 2; j+1 ;

18 996 SIOBHAN BURKE, CHRISTOPH ORTNER, AND ENDRE SÜLI (c) set u n = u n j, T n = Tj n n+1/2,andt1 = T n ; (d) for j =1, 2,..., compute v n+1 j if T T n+1/2 j := argmin {I n+1/2 h (u n,z):z X n+1/2 j h j,c }; ν T (u n,v n+1 j ) 2 ] 1/2 > REFTOL/ 2, find the smallest set M j T n+1/2 j such that T M j ν T (u n,vj n+1 ) 2 θ T T n+1/2 j ν T (u n,v n+1 j ) 2 ; refine elements in M j to obtain the new mesh T n+1/2 j+1 ; repeat until ν T T n+1/2 T (u n,v n+1 j ) ] 2 1/2 REFTOL/ 2; j (e) set v n+1 = v n+1 j and T n+1/2 = T n+1/2 j ; (f) repeat until v n+1 v n L () < VTOL. 3. Set v h (t k )=v n+1 and u h (t k )=u n. The marking strategy used to identify elements for refinement is due to Dörfler 21]. In our implementation, the refinement of the mesh is achieved using the newest node bisection method 29], which guarantees a uniform bound on the shape-regularity parameters of the generated family of meshes. Remark 5. We return to address a possible criticism, briefly mentioned at the end of section 1.2, that a locally refined mesh may favor certain crack paths over others. For example, for Algorithm 1, it is conceivable that an ill-chosen initial mesh may have this effect. Algorithm 2, however, is designed so as to ensure that local adaptive refinement does not influence the formation of cracks beyond the usual numerical perturbation. Namely, at each alternate minimization step, the error between the numerical and the exact solutions is controlled by the refinement tolerance, in the spirit of standard adaptive finite element algorithms for linear elliptic problems. Therefore, steps (b) and (d) compute the exact solution up to a specified tolerance. For example, if the exact solution for one alternate minimization step initiates a new crack in a region of the domain where the mesh is coarse, then the adaptive mesh refinement algorithm of step (b) or (d) will force mesh refinement in that region, provided the refinement tolerance is set sufficiently small. In fact, local mesh refinement allows us to compute with a smaller regularization parameter ɛ than one could conceivably use on a uniform mesh whose mesh size is equal to the minimum mesh size in an adaptively refined mesh, and this, in turn, results in a more reliable computation of crack paths. Finally, we present a modification of Algorithm 2, which is useful for theoretical purposes. Algorithm 2. In step n of Algorithm 2, we replace REFTOL by REFTOL n,and we require that REFTOL n 0asn. Furthermore, we remove the termination condition (f). Remark 6. In order for Algorithms 2 and 2 to be meaningful, steps (b) and (d) in Algorithm 2 need to terminate after a finite number of iterations. Without the mass lumpingapproximation, thiswouldfollowimmediatelyfromstandardconvergenceresults for adaptive finite element methods for linear elliptic problems 15]. It is beyond the scope of this paper to prove that the same results remain true in the presence of mass lumping; however, we do not expect serious difficulties in extending existing results. Instead, in Theorem 4.2 we shall make the following assumption: (A) Steps (b) and (d) in Algorithm 2 terminate in a finite number of iterations.

19 AN ADAPTIVE FEM IN BRITTLE FRACTURE 997 We now state some properties of the sequences generated by the preceding algorithms, which will prove useful later in the convergence analysis. Lemma 3.2. The sequence ((u j,v j )) j=1 X h j,g X hj,c generated by Algorithm 1 satisfies the following properties: 1. 0 v j (x) 1 on for all j N; 2. ((u j,v j )) j=1 is bounded in H1 () H 1 (). Proof. The first property follows from Proposition 2.2. In order to prove the second property, note that u j := u n j, v j := v n+1 j,where (3.14) (3.15) a hj (vj n ; un j,ϕ j)=0 ϕ j X hj,d, b hj (u n j ; vn+1 j,ψ j )=0 ψ j X hj,c. Taking ϕ j = u n j P h j g X hj,d in (3.14), we have (P hj ((vj n ) 2 )+η) u n j (u n j P hj g)dx =0. Hence, η u n j 2 L 2 () andsincewehaveassumedthatg X hj,g, (P hj ((v n j ) 2 )+η) u n j P hj g dx (1 + η) u n j L 2 () P hj g L 2 (), u n j L 2 () 1+η g L2 (). η Therefore, ( u j L 2 ()) j=1 is a bounded sequence, which by a variant of the Friedrichs inequality implies that ( u j L2 ()) j=1 is bounded as well; to see this, note that u j P hj g H 1 D (), and so, since P h j g = g, u j L 2 () u j g L 2 () + g L 2 () c (u j g) L 2 () + g L 2 () c u j L2 () +(c 2 +1)1/2 g H1 (), where c = c () > 0 is the constant in the Friedrichs inequality. This implies that (u j ) j=1 is bounded in H1 (). Taking ψ j = v n+1 j X hj,c in (3.15), we have P hj ((v n+1 j ) 2 ) u n j 2 dx + α v n+1 j (v n+1 j 1) dx + ε v n+1 j 2 dx =0, which implies that ε v n+1 j 2 L 2 () α (1 v n+1 j )v n+1 j dx α 4. Hence ( v j ) j=1 is bounded in L2 (), and since (v j ) j=1 is bounded in L (), it is also bounded in L 2 (). Thus ((u j,v j )) j=1 is bounded in H1 () H 1 (). Lemma 3.3. The sequence ((u n,v n )) n=1 X hn,g X hn,c generated by Algorithm 2 satisfies the following properties: 1. 0 v n (x) 1 on for all n N;

20 998 SIOBHAN BURKE, CHRISTOPH ORTNER, AND ENDRE SÜLI 2. ((u n,v n )) n=1 is a bounded sequence in H1 () H 1 (); 3. I h n(u n,v n ) I h n 1/2(u n 1,v n ) I h n 1(u n 1,v n 1 ) for all n N, n 2; 4. lim n I(u n,v n+1 ) I h n+1/2(u n,v n+1 ) =0; 5. lim n I(u n,v n ) I h n(u n,v n ) =0; 6. lim inf n I(u n,v n ) lim inf n I(u n 1,v n ) lim inf n I(u n 1,v n 1 ). Proof. Properties 1 and 2 follow in a manner similar to that of Lemma 3.2. To show property 3, recall that u n = argmin {I h n(z,v n ):z X hn,g}; therefore I h n(u n,v n ) I h n(u n 1,v n ). Note that on each element n 1/2 the function (v n ) 2 is convex. Note also that on each element n 1/2 the function P h n 1/2((v n ) 2 ) is linear and interpolates (v n ) 2, whilst P h n((v n ) 2 ) is a piecewise linear function that interpolates (v n ) 2 at a greater or equal number of points than P h n 1/2((v n ) 2 ) does. It therefore follows that (3.16) (v n ) 2 P h n((v n ) 2 ) P h n 1/2((v n ) 2 ). Hence, I h n(u n 1,v n ) I h n 1/2(u n 1,v n ), which proves the left-hand inequality. The right-hand inequality can be shown similarly. In order to show property 4, note that I(u n,v n+1 ) I h n+1/2(u n,v n+1 ) = (v n+1 ) 2 P h n+1/2((v n+1 ) 2 ) ] ( u n 2 + α)dx = b(u n ; v n+1,v n+1 ) b h n+1/2(u n ; v n+1,v n+1 ), which is one of the terms bounded in the residual estimate ψ b(u n ; v n+1,ψ), with ψ = v n+1 (see (3.11)). Therefore, I(u n,v n+1 ) I h n+1/2(u n,v n+1 ) ν h n+1/2(u n,v n+1 ) v n+1 L 2 (). Since lim n REFTOL n = 0, it follows that lim n ν h n+1/2(u n,v n+1 ) = 0; therefore lim n I(un,v n+1 ) I h n+1/2(u n,v n+1 ) =0. The proof of property 5 uses the same ideas but is slightly more involved. We have I(u n,v n ) I h n(u n,v n ) = (v n ) 2 P h n((v n ) 2 ) ] ( u n 2 + α)dx (v n ) 2 P h n((v n ) 2 ) ] u n (u n u n 1 )dx + (v n ) 2 P h n((v n ) 2 ) ] ( u n u n 1 + α)dx = a(v n ; u n,u n u n 1 ) a h n(v n ; u n,u n u n 1 ) + (v n ) 2 P h n((v n ) 2 ) ] ( u n u n 1 + α)dx.

21 AN ADAPTIVE FEM IN BRITTLE FRACTURE 999 Note that (v n ) 2 P h n((v n ) 2 ) ] ( u n u n 1 + α)dx 1 (v n ) 2 P h n((v n ) 2 ) ( u n 2 + α)dx (v n ) 2 P h n((v n ) 2 ) ( u n α)dx 2 1 (v n ) 2 P h n((v n ) 2 ) ( u n 2 + α)dx (v n ) 2 P 2 h n 1/2((v n ) 2 ) ( u n α)dx = 1 2 I(un,v n ) I h n(u n,v n ) b(un 1 ; v n,v n ) b h n 1/2(u n 1 ; v n,v n ), wherewehaveused(3.16)inthethirdline. Therefore, I(u n,v n ) I h n(u n,v n ) 2 a(v n ; u n,u n u n 1 ) a h n(v n ; u n,u n u n 1 ) + b(u n 1 ; v n,v n ) b h n 1/2(u n 1 ; v n,v n ). Noting that the terms on the right-hand side appear in the bounds for the residual estimates ϕ a(v n ; u n,ϕ), with ϕ = u n u n 1 (see (3.9)) and ψ b(u n 1 ; v n,ψ), with ψ = v n (see (3.11)) we have I(u n,v n ) I h n(u n,v n ) 2μ h n(u n,v n ) (u n u n 1 ) L 2 () + ν h n 1/2(u n 1,v n ) v n L 2 (). Since lim n REFTOL n = 0, it follows that lim n μ h n(u n,v n ) = 0 and lim n ν h n 1/2(u n 1,v n ) = 0; therefore Finally, we show property 6; we have lim n I(un,v n ) I h n(u n,v n ) =0. I(u n,v n ) I h n(u n,v n )+ I(u n,v n ) I h n(u n,v n ) I h n 1/2(u n 1,v n )+ I(u n,v n ) I h n(u n,v n ) I(u n 1,v n )+ I(u n 1,v n ) I h n 1/2(u n 1,v n ) + I(u n,v n ) I h n(u n,v n ), where we have used property 3 in the second line. properties 4 and 5, On letting n and using lim inf n I(un,v n ) lim inf n I(un 1,v n ). The right-hand inequality follows in a similar manner.

22 1000 SIOBHAN BURKE, CHRISTOPH ORTNER, AND ENDRE SÜLI 4. Convergence analysis. In this section, we state and prove two results that support the use of the adaptive algorithms proposed in the previous section. Provided that Algorithm 1 terminates for any given input (we are in fact unable to prove this at present), then Theorem 4.1 demonstrates convergence of the numerical solutions for decreasing tolerance REFTOL to a critical point of I. Theorem 4.1. Assume that is an open bounded domain in R N. Suppose that there exists a sequence ((u j,v j )) j=1 H1 g() H 1 c(), 0 v j (x) 1 a.e. in such that (4.1) a(v j ; u j,ϕ) μ j ϕ L2 () ϕ H 1 D (), (4.2) b(u j ; v j,ψ) ν j ψ H1() ψ H 1 c() L (), for some μ j,ν j R 0,withμ j, ν j 0 as j. Suppose also that ((u j,v j )) j=1 is a bounded sequence in H 1 () H 1 (). Then, there exists a subsequence of ((u j,v j )) j=1 (not relabelled) and (u, v) H 1 g () H1 c (), with0 v(x) 1 a.e. in, such that u j u strongly in H 1 () and v j v strongly in H 1 () as j. Moreover, u and v satisfy (4.3) (4.4) a(v; u, ϕ) =0 b(u; v, ψ) =0 ϕ H 1 D(), ψ H 1 c () L (); that is, (u, v) is a critical point of I in H 1 g () (H1 c () L ()). Theorem 4.2 proves that the sequence ((u n,v n )) n=1 computed by Algorithm 2 (which is designed without a termination criterion) converges to a critical point of I and thus, subject to a justification of assumption (A), establishes the convergence of Algorithm 2. To the best of our knowledge, this is the first convergence result for an adaptive finite element algorithm for a nonconvex minimization problem. Theorem 4.2. Assume that is an open bounded domain in R N. Let ((u n, v n )) n=1 H 1 g() H 1 c() be the sequence generated by Algorithm 2 under assumption (A). Then, there exists a subsequence ((u nj,v nj )) j=1 of ((u n,v n )) n=1 and (u, v) H 1 g() H 1 c(), with0 v(x) 1 a.e. in, such that u nj u strongly in H 1 () and v nj v strongly in H 1 () as j. In addition, u and v satisfy (4.5) a(v; u, ϕ) =0 ϕ H 1 D (), (4.6) b(u; v, ψ) =0 ψ H 1 c() L (); that is, (u, v) is a critical point of I in H 1 g() (H 1 c() L ()). Proof of Theorem 4.1. Step 1. Existence of a convergent subsequence of ((u j,v j )) j=1 : The sequence ((u j,v j )) j=1 is bounded in H1 () H 1 (). Since H 1 () is a Hilbert space, there exists a subsequence of ((u j,v j )) j=1 (not relabelled) such that (u j,v j ) (u, v) in H 1 () H 1 () as j. As H 1 g() is a convex and closed subset of H 1 (), it is also weakly closed (cf. Proposition 2.5 on p. 35 of 6]). Hence, we have that u H 1 g(). Similarly, as the set K:={w H 1 c () : 0 w(x) 1 a.e. x } is a convex closed subset of H1 (), and since v j K for all j N, it follows that 0 v(x) 1 a.e. in. The compact embedding of H 1 () in L 2 () also implies that u j u and v j v strongly in L 2 () as j.

23 AN ADAPTIVE FEM IN BRITTLE FRACTURE 1001 Step 2. lim j v v j w 2 dx =0forallw H 1 (): (Recall that we have not relabelled the convergent subsequence.) This result will prove to be repeatedly useful in the subsequent analysis. Fix w H 1 (), and let (v jk ) k=1 be a subsequence of (v j ) j=1 such that lim k v j k v w 2 dx = lim sup j v j v w 2 dx, and v jk v a.e. in. Using Lebesgue s dominated convergence theorem 38, section 5.2], we have lim k v j k v w 2 dx = 0 and hence lim sup j v j v w 2 dx = 0. It thus follows that lim j v j v w 2 dx =0. Step 3. a(v; u, ϕ) = 0 for all ϕ H 1 D (): Fixing ϕ H1 D (), we have a(v; u, ϕ) = (v 2 + η) u ϕdx = (v 2 + η) (u u j ) ϕdx + (vj 2 + η) u j ϕdx + (v 2 vj 2 ) u j ϕdx =: R j + S j + T j. Our aim is to show that R j,s j,t j 0asj. First, consider R j = (v 2 + η) ϕ (u u j )dx. We know that (v 2 + η) ϕ (L 2 ()) N ; so, by the weak convergence of ( u j ) j=1 to u in (L 2 ()) N, we obtain that R j 0asj. Turning our attention to S j, (4.1) implies that Finally, we estimate T j by T j S j μ j ϕ L 2 () 0 as j. v + v j v v j u j ϕ dx ( 1/2 2 v v j ϕ dx) 2 u j L2 (), whereweusedthefact v + v j 2and v v j 1. Since ( u j ) j=1 is bounded in (L 2 ()) N, and using Step 2 with w = ϕ, wehavet j 0asj. Thus we conclude that a(v; u, ϕ) = 0 for all ϕ H 1 D (). Step 4. u j u strongly in (L 2 ()) N :Notethat η u u j 2 L 2 () (vj 2 + η) (u u j ) (u u j )dx = (vj 2 + η) u j (u u j )dx + (vj 2 + η) u (u u j )dx μ j (u u j ) L2 () + (vj 2 + η) u (u u j)dx. Since u u j H 1 D (), we have a(v; u, u u j) = 0, and it follows that η u u j 2 L 2 () μ j (u u j ) L2 () + (vj 2 v2 ) u (u u j )dx μ j (u u j ) L 2 () + (v 2 j v2 ) u L 2 () (u u j ) L 2 ().