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1 Deutsce Forscungsgemeinscaft Priority Program 53 Optimization wit Partial Differential Equations Luise Blank, Martin Butz and Harald Garcke Solving te Can-Hilliard Variational Inequality wit a Semi-Smoot Newton Metod Marc 9 Preprint-Number SPP53-9- ttp://

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3 Solving te Can-Hilliard variational inequality wit a semi-smoot Newton metod Luise Blank, Martin Butz, Harald Garcke, Version vom Marc 6, 9 Abstract Te Can-Hilliard variational inequality is a non-standard parabolic variational inequality of fourt order for wic straigtforward numerical approaces cannot be applied. We propose a primal-dual active set metod wic can be interpreted as a semi-smoot Newton metod as solution tecnique for te discretized Can-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretization of splitting type is used in space leading to a discrete variational inequality of saddle point type in eac time step. In eac iteration of te primal-dual active set metod a linearized system resulting from te discretization of two coupled elliptic equations wic are defined on different sets as to be solved. We sow local superlinear convergence of te primal-dual active set metod and demonstrate its efficiency wit several numerical simulations. Key Words: Can-Hilliard equation, active-set metods, semi-smoot Newton metods, gradient flows, PDE-constraint optimization, saddle point structure. AMS subject classification: 35K55, 35K85, 9C33, 49N9, 8A, 8C6, 65M6 Introduction Te Can-Hilliard equation was initially introduced to model pase separation in binary alloys []. By now te Can-Hilliard equation as found many applications ranging from classical aspects in materials science [3, ] over image processing [6], fluid dynamics [9], topology optimization [37] up to te modelling of mineral growt [7] and galaxy structure formation [33]. Te Can-Hilliard equation can model interface motion in so called conserved systems, i.e. in systems were te concentration of a species or te volume occupied by a pase is conserved. In tese applications Can-Hilliard variational inequalities are frequently used. In tis paper we propose a numerical metod for solving Can-Hilliard variational inequalities and we eavily use te gradient flow structure of te Can- Hilliard model. We interpret te time discretized version of te gradient flow as a PDE-constraint optimization problem were in addition pointwise inequality constraints ave to old. Te PDE-constraint minimization problem wic we obtain is

4 non-standard as te objective functional obtains L -norm of gradients rater tan te L -norms of te involved functions itself. We propose to solve te fully discretized system wit a primal-dual active set metod wic can be reinterpreted as a semi-smoot Newton metod. It turns out tat tis approac is superior to earlier numerical metods for Can-Hilliard variational inequalities. Its efficiency in eac iteration step is comparable to a metod proposed by Gräser and Kornuber []. Tey introduce and analyse a Uzawa-multigrid algoritm. Tere in eac iteration step an intermediate primal-active set is determined wit te elp of obstacle problems wic are solved wit a monotone multigrid metod. In a second step a linear subproblem similar to ours were only te rigt and side differs as to be solved, to ten update te cemical potential by a damped gradient-type metod. After convergence te pase field is determined. In contrast we determine te active sets in a simple way using approximations of te primal and dual variables, we ave to solve te same linear subproblem were we solve for te cemical potential and te pase field simultaneously. Banas and Nürnberg [] extended te idea of [] and apply a monotone multigrid approac to te wole system of Can-Hilliard inequalities. Computational comparisons are not available up to now. Te outline of te paper is as follows. In te remainder of tis section we introduce te Can-Hilliard variational inequality. We will interpret te implicit time discretization of te Can-Hilliard variational inequality as a PDE-constraint optimization problem in Section. In Section 3 we introduce a primal-dual active set approac for te time discretized Can-Hilliard variational inequality and we formulate a finite element metod for a splitting formulation of te Can-Hilliard variational inequality in Section 4. We also sow local superlinear convergence. Finally, we numerically analyse te beaviour of te metod wit te elp of four examples of different type and sow some simulations in Section 5. Since te gradient flow perspective is important for wat follows we coose to derive te Can-Hilliard equations as a gradient flow. We remark tat our derivation will be formal. We now consider a vector space Z and an affine subspace U Z, i.e. tere exists a u Z and a linear space Y Z suc tat U = u + Y. Te gradient of a sufficiently smoot function E : U R depends on te inner product cosen for Z. We define te first variation of E at a point u U in a direction v Y by δe E(u + δv) E(u) (u)(v) := lim. δu δ δ We say grad Z E(u) Z is a gradient of E wit respect to te inner product (.,.) Z on Z if (grad Z E(u), v) Z = δe (u)(v) for all v Y. δu Now te gradient flow of E wit respect to te inner product (.,.) Z is given as t u(t) = grad Z E(u(t)). ()

5 Te energy decreases in time due to te inequality d dt E(u(t)) = (grad ZE(u(t)), t u(t)) Z = t u Z. In te following, in order to derive te Can-Hilliard equation, we introduce te Ginzburg-Landau energy E : H () R as E(u) = { γε u + ψ(u)}dx () ε were R d is a bounded domain wit Lipscitz boundary, γ > is a constant related to te interfacial energy density and ψ is a double well potential, e.g. ψ(u) = ( u ) or an obstacle potential, e.g. { } ψ (u) u [, ] ψ(u) = = ψ elsewere (u) + I [,] (u) (3) were ψ is smoot and I [,] is te indicator function, i.e. I [,] is set to infinity outside te interval [, ] and to on [, ]. In te following we will coose ψ (u) = ( u ) (4) wic is te typical coice in te literature, see e.g. [7], but oter non-convex functions are possible. In te Can-Hilliard model different pases correspond to te values u = ±. On an interface a solution rapidly canges from values close to to values close to and te tickness of tis interfacial region is proportional to te parameter ε in (), see e.g. Figure. If ψ is smoot te first variation of E in a direction v is given as δe δu (u)(v) = (γε u v + ε ψ (u)v). (5) Coosing Z = L (), U = Y = H () and u = we obtain grad L E(u) = γε u + ε ψ (u) (6) and te resulting gradient flow equation gives te so called Allen-Can equation. We remark ere tat for (6) we also need to require u = on were n is te n outer unit normal to. As mentioned above in te Can-Hilliard model te total concentration, i.e. u(x)dx is assumed to be conserved. Denoting by u te mean value of a function u, we now define for a given m (, ) te sets U := {u H () u = m}, Y := {u H () u = }. 3

6 In addition we introduce Z = H () = {u (H ()) u, = }, i.e. all bounded linear functionals on H () tat vanis on constant functions. Here and in wat follows.,. denotes te dual pairing. On Z = H () we define te H - inner product for v, v Z as (v, v ) H := ( ) v ( ) v (7) were y = ( ) v is te weak solution of y = v, =. We remark tat te solution to tis elliptic problem is only defined up to a constant and we always coose y suc tat y =. Te function space Y is canonically embedded into Z since u Y can be related to te linear functional y uy. For v, v Y we obtain y n (v, v ) H = (v, ( ) v ) L = (( ) v, v ) L. Tese identities also old more generally for functions v, v L () wit mean value zero. To compute te H -gradient of E we now need to find grad H E(u) Z suc tat (v, grad H E(u)) H = δe (u)(v) olds for all v Y. δu From te above we obtain (v, ( ) grad H E(u)) L = (v, grad L E(u)) L and ence grad H E(u) = ( )grad L E(u). (8) Ten, te Can-Hilliard equation is given as te H -gradient flow of te Ginzburg Landau energy E. If ψ is smoot we obtain te fourt order parabolic equation t u = grad H E(u) = ( γε u + ε ψ (u) ) (9) or equivalently introducing te so called cemical potential w te equation can be rewritten as a system as follows t u = w, () w = γε u + ε ψ (u). () In addition te boundary conditions u = w = on ave to old. Let us n n remark, tat in tis formulation we do not necessarily ave w =, i.e. in general w ( ) t u but bot functions only differ by an additive constant. It is also possible to derive te Can-Hilliard equation from te mass balance law and in tis case w is te mass flux were for simplicity a mobility coefficient was taken to be one, see e.g. Elliott [8] or Novick-Coen [3]. Te presentation so far is appropriate for smoot functions ψ. If te energy as te double obstacle form (3) we differentiate I [,] in te sense of subdifferentials, 4

7 i.e. for a u L () wit u we obtain tat µ L () is in te subdifferential of I [,] at u if and only if (, ] if u =, µ I [,] (u) = for u (, ), [, ) if u =, () is fulfilled pointwise almost everywere. complementarity form Tis can be rewritten in te following µ = µ + µ, µ +, µ, µ + (u ) =, µ (u + ) = (3) wic also as to old almost everywere. In tis case te H -gradient flow as te form t u = w, (4) w = γε u + ε (ψ (u) + µ) (5) wit µ I [,] (u), u and zero Neumann boundary conditions for u and w. Tis formulation can be restated in a variational inequality formulation, see e.g. Blowey and Elliott [7] or Kinderlerer and Stampaccia [6] and Friedman [] for oter obstacle problems, as follows: t u = w, (w, ξ u) L γε( u, (ξ u)) L + ε (ψ (u), ξ u) L ξ H (), ξ,(7) togeter wit u a.e.. For tis formulation it can be sown tat a unique solution u exists wic is H -regular in space. More precisely te following teorem, see [7], is true. Teorem. Assume is convex or C,, u H () wit u and u = m (, ). Ten tere exists a unique pair (u, w) suc tat u H (, T ; (H ()) ) L (, T ; H ()) L (, T ; H ()), u a.e. and w L (, T ; H ()) wic solves t u, η + ( w, η) L = for all η H () and t (, T ) a.e. togeter wit te variational inequality (7) and u(, ) = u. In particular µ = εw + γε u ψ (u) L ( T ). (6) 5

8 Can-Hilliard variational inequalities and PDEconstraint optimization Given discrete times t n = nτ, n N, were τ > is a given time step te backward Euler discretization of te gradient flow equation () is given as τ (un u n ) = grad H E(u n ). (8) Tis time discretization as a natural variational structure. In fact one can compute u n as te solution of te minimization problem min {E(u) + u u U τ un H }. (9) One ence tries to decrease te energy E but as to take into account tat deviations from te solution at te old time step costs were te cost depends on te norm on Z = H (). As Euler-Lagrange equation for (9) we obtain te backward Euler discretization (8). We remark ere tat (8) migt ave solutions wic are not necessarily global minimizers of te minimization problem (9). In te case of te Can-Hilliard model we need to minimize { γε u + ψ(u)} dx + u ε τ un H () under all u H () wit u = u n = m. In order to compute te H -norm of u u n we need to solve a Poisson problem and ence we obtain, in te case of te obstacle potential, te following PDE-constraint optimization problem { min γε } u + ψ ε (u) + τ v () suc tat τ v = u u n, () u, u = m, wit v = on and v =. n Tis formulation as te form of an optimal control problem were u is te control and v is te state. We now introduce Lagrange multipliers w H () for te weak formulation of (), and κ R for v = and define te Lagrangian L(u, v, w, λ) := γε u + ψ ε (u) + τ v τ w v (u u n )w + κ v. Wit tis Lagrangian te equality constraints are incorporated. In fact te equality constraints are obtained as te first variation of L wit respect to w and κ. In 6

9 particular, we obtain u = u n = m if we vary w by a constant. Tis implies tat w acts as a Lagrange multiplier simultaneously for te equality constraints () and u = m := u n. Introducing appropriately scaled Lagrange multipliers µ, namely wit, for te ε pointwise box-constraints we obtain te following version of te KKT-system were (3), (5) and (6) ave to be understood in its weak form. w τ (w v) = κ in, n = v on, (3) n v =, κ =, (4) τ (u un ) = v in, v n = on, (5) w + γε u ε ψ (u) µ = in, u = on, (6) ε n µ = µ + µ, µ +, µ, a.e. in, (7) µ + (u ) =, µ (u + ) =, a.e. in, (8) and u a.e. in. (9) Given (3)-(4) we obtain w w = v, i.e. v and w only differ by a constant. We can replace v by w in (5) and we ence obtain in particular a time discretization of (4), (5) using te complementary formulation (3). Te Lagrange multiplier w coincides wit te cemical potential, and te scaled Lagrange multiplier µ coincides wit te subdifferential of I [,]. Since te equations (3) and (4) are not needed we omit tem in te following. In te following we consider te coice of ψ (u) = ( u ) and sow tat te system (5)-(9) as a solution. Defining te admissible set U ad := {u H () u, u = m} () or respectively ()-() can be reformulated as min E(u) := { γε u + ψ u U ε (u) + ( ) (u u n ) }. (3) τ ad Since ( u ) is nonconvex te above minimization problem will in general omit more tan one solution. But one can sow tat for small τ > te problem is uniquely solvable. Tis is te content of te following lemma. Similar restrictions on te time step in order to obtain uniqueness ad to be imposed in a fully discrete situation in [3] and [8]. Lemma. Te minimization problem (3) as a solution. exists if τ (, 4γε 3 ). A unique solution 7

10 Proof: Since u we obtain tat ψ (u) = ( u ) is non-negative and tere exists a minimizing sequence (u k ) k N U ad for E, i.e. E(u k ) inf E(u) > for k. u U ad Given tat (E(u k )) k N is uniformly bounded we can conclude tat u k dx is uniformly bounded. Due to (u k m)dx = we can use Poincaré s inequality for functions wit mean value zero, see e.g. [], to obtain tat (u k ) k N is a bounded sequence in H (). Using te fact tat bounded sequences in H () ave weakly converging subsequences and applying Rellics teorem we obtain te existence of a subsequence suc tat u kj u in H (), u kj u in L () for j. Since te terms u dx and ( ) u dx are convex, we obtain tat tey are weakly lower semi-continuous in H (), see e.g. [9]. Since ψ (u kj ) converges strongly we conclude tat u is in fact a minimum of E in U ad. Te functional E is strictly convex on U if and only if F (η) := E(η + u n ) is strictly convex on Y. Since F is te sum of terms wic are constant or linear and of γε η η + ( ) η (3) ε τ we only need to sow tat (3) is strictly positive on Y \ {}. Using te definition of ( ) and Young s inequality we obtain for all η Y η = ( ( ) η) η δ ( ) η + η. ε ε 4ε 4δε Coosing δ = ε τ γε we obtain η η + ε τ ( ) η ( γε τ ) η. 8ε If τ < 4γε 3 we obtain uniqueness from te strict convexity of E. Lemma. A solution u U ad of (3) solves te variational inequality γε u (η u) u(η u) + ( ) (u u n )(η u) (3) ε τ for all η U ad. 8

11 Proof: Computing te first variation of te first two terms in (3) in a direction (η u) is standard. In addition we ave d dδ ( ) (u + δ(η u) u n ) δ= = ( ) (u u n ) ( ) (η u) = ( ) (u u n )(η u). Since te derivative of te functional in (3) as to be nonnegative in directions η u wit u U ad we obtain (3). Lemma.3 Let u U ad be a solution of te variational inequality (3). Ten tere exists a λ R suc tat for all η H () wit η te inequality γε u (η u) u(η u) + ( ) (u u n )(η u) (33) ε τ olds. λ (η u) Proof: We argue similar as in te proof of Proposition 3.3 in [7]. Let f = u ε τ ( ) (u u n ). Since u and u n are bounded by one in modulus we obtain from te teory of elliptic equations tat f is bounded. We now define for eac α R a function u α K := {u H () u } suc tat for all η K γε u λ (η u λ ) + u ε λ (η u λ ) f(η u λ ) + α (η u λ ). (34) Using standard teory of variational inequalities we deduce tat (34) as a unique solution, see e.g. [6]. We now introduce a function M : R R by M(α) := u α. For all η K and all α R we ave te pointwise inequalities ( ε f α)(η ) ( ε + f α)(η ) and ε f α)(η + ) ( ε f α)(η + ). Hence u is a solution of (34) if α + f ε and u is a solution of (34) if α ( + f ε ). We now obtain M(±( + f ε )) = ±. As in te proof of Proposition 3.3 in [7] we obtain tat M is monotone and continuous. Hence a λ R exists suc tat M(λ) = m. We now coose η = u λ in (3) and η = u in (34) wit λ = λ. Adding bot resulting terms leads to γε (u u λ ) + u ε λ u, 9

12 were we use te fact tat u = u λ. Hence u = u λ. Using tis result and te definition of f we conclude from (34) tat u fulfills (33). Using regularity teory for obstacle problems we obtain similar as in te proof of Lemma 3. in [7] Setting u W,p loc () for all p (, ), u C,α loc ( u u v = ( ) n τ µ + = ε(γε u + u + ε w)+, µ = ε(γε u + ε u + w), () for all α (, ). ), w = v + λ, µ = µ + µ we obtain similar as for oter optimization problems wit bilateral constraints (see e.g. [34]) te following result. Remark. Tere exists a solution (u, v, w, µ) of te KKT system (3)-(9). In particular tere is a Lagrange multiplier µ wit µ L (). Replacing ψ (u) in (6) by ψ (un ) gives a semi-implicite time discretization, see e.g. [8]. Arguing similar as above a solution to te semi-discrete version exists wic is in H (). In tis case te minimization problem related to (3) is always uniquely solvable, i.e. no restriction on te time step is necessary. 3 Primal-dual active set approac Te goal of tis section is to formulate a primal-dual active set metod in order to solve for a time step τ > a spatially discretized version of (u τ un ) = w in, w n = on (35) togeter wit (6)-(9). We now introduce for a c > te active sets { } { A + = x u(x) + µ(x) >, A = x u(x) + µ(x) c c } < and te inactive set I := \(A + A ). Te conditions (7)-(9) can be reformulated as u(x) = ± if x A ±, µ(x) = if x I. (36) Formally, tis leads to te following primal-dual active set strategy employing te primal variable u and te dual variable µ.

13 Primal-Dual Active Set Algoritm (PDAS-I):. Set k =, initialize A ± and define I = \ (A + A ).. Set u k = ± on A ± k and µ k = on I k. 3. Solve te coupled system of PDE s (35), (6) to obtain u k on I k, µ k on A + k A k and w k on. { } 4. Set A + k+ := x u k (x) + µ k(x) >, c A k+ := {x u k(x) + µ k(x) c < } and I k+ = \ (A + k+ A k+ ). 5. If A ± k+ = A± k stop, oterwise set k = k + and goto. Anoter reformulation of (7)-(9) is given wit te elp of a semi-smoot equation as follows H(u, µ) := µ (max(, µ + c(u )) + min(, µ + c(u + )) =. (37) A semi-smoot Newton metod applied in a formal way to (35), (6), (37) is equivalent to te above primal-dual active set metod, see e.g. [3] for a different context. It is known tat for obstacle problems te iterations in (PDAS-I) in general are not applicable in function space since te iterates µ k are only measures and not L -functions, see [4]. Te same is true for te non-standard obstacle problem (), as u k as obtained in te iterations of (PDAS-I) in general is only a measure. In te following section we introduce a finite element discretization of (5)-(9) and we sow tat for te discretized system local superlinear convergence olds. 4 Finite element discretization We now introduce a finite element approximation for te Can-Hilliard variational inequality using continuous, piecewise affine linear finite elements for u and w. In te following we assume for simplicity tat is a polyedral domain. Generalizations to curved domains are possible using boundary finite elements wit curved faces. Let {T } > be a triangulation of into disjoint open simplices. Furtermore, we define T to ave maximal element size := max T T {diam(t )} and we set J to be te set of nodes of T and p j J to be te coordinates of tese nodes. Te finite element space of piecewise affine linear, continuous finite elements associated to T is now given as S := {ϕ C () ϕ T P (T ) T T } H () were we denote by P (T ) te set of all affine linear functions on T. To eac p j J we associate te nodal basis function χ j S wit te property χ j (p i ) = δ ij. We replace te L -inner product (.,.) L at some places by a quadrature rule given by te lumped mass inner product (η, χ) = I (ηχ), were I := C () S is te

14 standard interpolation operator at te nodes. In te following, we consider eiter an implicit or an explicit discretization of te term ψ (u), i.e. we coose ψ (u ) were {n, n}. Ten, te spatial discretization of (35), (6)-(9) is given as: For n =,, 3,... and given u S find iteratively (u n, wn, µn ) S S S suc tat τ (un un, χ) + ( w n, χ) = χ S, (38) (w, n χ) γε( u n, χ) ε (ψ (u ), χ) ε (µn, χ) = χ S, (39) µ n = µ n,+ µ n,, µ n,+, µ n,, u n, (4) µ n,+(p j )(u n (p j ) ) = µ n, (p j )(u n (p j ) + ) = p j J. (4) Notice tat (4) does in general not imply (8) pointwise in all of. Coosing χ in (38) provides te mass conservation u n = u n = u. Te discretization of (7)-(9) can also be introduced as in Section 3 wit te elp of active nodes { } { } A n,+ = p j J u n (p j) + µn (p j) >, A n, c = p j J u n (p j) + µn (p j) < (4) c for any positive c. Ten we define te set of inactive nodes as I n = J \(A n,+ A n, ) and require u n (p j) = ± if p j A n,±, µ n (p j) = if p j I n. (43) As discussed in Sections and te equations (39), (4)-(43) can be rewritten as a variational inequality as follows. Introducing te space we searc u n K suc tat K := {η S η(x) for all x }. (w n, ξ u n ) γε( u n, (ξ u n )) + ε (ψ (u n ), ξ u n ) ξ K, (44) In order to compute (u n, wn, µn ) we now coose a discretized version of te primaldual active set metod (PDAS-I), were we iteratively update active sets A n,±,k for k =,,,.... We drop for convenience sometimes te indices n,. Te following discrete version of te primal-dual active set strategy is obtained using tat µ n (p j) = on I,k n in (39). Ten (39) reduces rougly spoken to a discretized PDE for un only on an interface determined by I,k n. For determined un, wn (39) determines µn on te active set. Here one as to use tat (, ) is a mass lumped L -inner product to uncouple (39) in active and inactive. For te precise formulation we introduce te notation S,k := { χ S χ(p j ) = if p j A n,+,k An,,k } = span{χ i p i I n,k}.

15 Primal-Dual Active Set Algoritm (PDAS-II):. Set k =, initialize A ± and define I = J \ (A + A ).. Solve for (u k, w k ) S S te system τ k u n, χ) + ( w k, χ) = χ S, (45) (w k, χ) γε( u k, χ) ε (ψ (u ), χ) = χ S,k, (46) 3. Define µ k S via u k (p j ) = ± if p j A ± k. (47) µ k (p j ) (, χ j ) = ε(w k, χ j ) γε ( u k, χ j ) (ψ (u ), χ j ) p j I k, (48) µ k (p j ) = p j I k. (49) 4. Set A + k+ := {p j J u k (p j ) + µ k(p j ) > }, (5) c A k+ := {p j J u k (p j ) + µ k(p j ) < } and I c k+ = J \ (A + k+ A k+ ). (5) 5. If A ± k+ = A± k stop, oterwise set k = k + and goto. Lemma 4. For all u n S and A ± k tere exists a unique solution (u k, w k ) S S of (45)-(47) wit = (n ), i.e. te semi-implicit case, provided tat I k = J \ (A + k A+ k ). Proof: Te idea of tis proof is to consider te discretized version of () and () under te constraint u = ± on A k and follow te existence proof as in Section 3. Hence, we define S,m := {χ S χ = m}, were m := u n, S I := {u S u(p j ) = if j A + k, u(p j) = if j A k }, and S,m I := SI S,m. Since I k we conclude S,m I. Te discrete inverse Laplacian ( ) : S, S,, η ( ) η is defined via ( (( ) η ), χ) = (η, χ) for all χ S,. (5) Since te omogeneous problem only as te trivial solution and S, is finite dimensional, te linear equation (5) as a unique solution. We define u k S I,m as te solution of te minimization problem min { ( ( η S,m I τ ) (η u n ), ( ) (η u n ))+ γε ( η, η)+ ε (ψ (u n ), η) } (53) wic exists uniquely since te Poincaré inequality similar as in te proof of Lemma. implies coerciveness. Computing te first variation of te minimisation problem 3

16 (53) gives for u k S I,m = ( ( τ ) (u k u n ), ( ) χ) + γε( u k, χ) + ε (ψ (u n ), χ) (54) for all χ S,k wit χ =. Now we define w k S as ( w k = ( ) uk u n ) + λ k (55) τ were λ k R is uniquely given by any nodal basis function χ j S wit p j I k by λ k = { (( τ ) (u k u n ), χ j ) + γε( u k, χ j ) + ε (ψ (u n ), χ j )}/(, χ j ). (56) Using te definition of te discrete inverse Laplacian, see (5) and te fact tat u k = u n now gives tat (45) olds. Furtermore (54), (5) and (55) imply tat (46) olds for all χ S,k wit χ =. For χ S,k wic do not satisfy te integral constraint χ = we set ˆχ := χ αχ j wit p j I k and α R suc tat ˆχ =. Wit tis coice of ˆχ as a test function in (54) we can conclude wit te elp of (5), (55) and (56) tat (46) olds for all χ S,k. Hence (45)-(47) as a solution. It remains to prove uniqueness. Let us assume tat (45)-(47) as two solutions (u k,, w k, ), (u k,, w k, ) S S. Ten we obtain for te differences v = u k, u k,, z = w k, w k, by testing (45), (46) for (u k,, w k, ) and (u k,, w k, ) wit v and z respectively, after taking differences: (v, z) + τ z L (z, v) + γε v L =. Since u k, = u k, = u n we obtain v in and ence u k, = u k,. Te identities (45), (46) imply tat necessarily te identities (55) and (56) ave to old. Tis implies tat also w k is unique. Now µ k is uniquely defined by (48), (49) and ence taking Lemma 4. into account we obtain tat a unique solution of (45)-(49) exists. In te following we require te condition I k = J \ (A + k A k ), wic guarantees tat tere is a u S wic can fulfill u = m. Oterwise (45) is not solvable. Remark 4. In order to solve (45)-(49) te main computational effort is to solve te system (45), (46) wic as a specific structure. Te discretized elliptic equation (45) for w is defined on te wole of wereas te elliptic equation (46) is defined only on te inactive set, see Figure. Te two equations are coupled in a way 4

17 u k n = ε u k = u k = + A k A + k I k u k n = Figure : Te system (45)-(46) leads to an equation for u k on te inactive set I k and for w k on te wole of. wic leads to an overall symmetric system wic will be used later wen we propose numerical algoritms. Te discretization of (7)-(9) can also be formulated wit te elp of te semismoot function H, see (37), as a nonlinear equation H(u n (p j ), µ n (p j )) = p j J. (57) Using te approac of [3] we can interpret (PDAS-II) as a semi-smoot Newton metod for te system (38), (39), (57) and te following local convergence result for te semi-implicit discretization. Teorem 4. Let (u, w, µ) S S S be a solution of (38), (39), (57) wit = (n ) suc tat {p j J u(p j ) < }. Ten te semi-smoot Newton metod for (38), (39), (57) and ence (PDAS-II) converges superlinearly in a neigborood of (u, w, µ). Proof: Sowing te existence of a solution to (38), (39), (57) is equivalent to te problem of finding a zero of te mapping G : S S S S S S were for (u, w, µ) S S S we define G = (G, G, G 3 ) via (G (u, w, µ), χ) := (u u n, χ) + τ( w, χ), (G (u, w, µ), χ) := (w, χ) γε( u, χ) ε (ψ (un ), χ) (µ, χ) ε, (G 3 (u, w, µ), χ) := (H(u, µ), χ). Te min-max-function H(u, µ) is slantly differentiable and a slanting function is given by DH(u, µ) = (, ) if u + µ and DH(u, µ) = ( c, ) oterwise, c 5

18 (see []). As a consequence G is slantly differentiable. Moreover similar as in [] we can derive tat te primal-dual active set metod (PDAS-II) is equivalent to a semi-smoot Newton metod for G. We now get local superlinear convergence of (PDAS-II) if we can sow tat te slanting function of G is invertible in a suitable neigbourood of (u, w, µ) and te inverses are uniformly bounded ([, 3]). Te semi-smoot derivative (slanting function) of G is invertible at (û, ŵ, ˆµ) S S S if and only if we can sow injectivity, i.e. tat a unique solution (u, w, µ) S S S of te following linear system exists (u, χ) + τ( w, χ) =, χ S, (58) (w, χ) γε( u, χ) (µ, χ) ε =, χ S. (59) u(p j ) = if p j {p  := j J û(p j) + ˆµ(p } j) c >, (6) µ(p j ) = if p j Î := J \ Â, (6) Testing (58) wit w, (59) wit u and using (µ, u) = we obtain τ( w, w) + γε( u, u) =. (6) Tis implies tat u and w are constant. Ten (58) gives u. Using te fact tat tere exists a p j J wit u(p j ) < and µ(p j ) = we can guarantee tat Î for (û, ŵ, ˆµ) out of a suitable neigborood of (u, w, µ). Hence testing in (59) wit χ j were p j J implies w and finally (6) and (59) yield µ. Te semi-smoot derivatives only differ if te active and inactive sets cange. Since only a finite number of different coices of tese sets are possible we obtain tat te inverses are uniformly bounded for all (û, ŵ, ˆµ) wit a non-vanising inactive set Î. Since we can find an open neigborood of (u, w, µ), were te condition Î, we proved te teorem. Remark 4. Let (u, w, µ) be a solution to (38), (39), (57). Te proof of Teorem 4. requires a neigborood of (u, w, µ), were te active sets do not vanis. Tis can limit te size of te neigborood in wic local superlinear convergence can be guaranteed. However in numerical simulations te mes size always as to be cosen suc tat at least eigt points lie across te interface. Hence te above mentioned condition never led to any problems in practice. Remark 4.3 Teorem 4. olds also for te implicit discretization if τ 4γε 3. Proof: Te proof follows align wit 4. if one can sow injectivity. Equation (59) canges to (w, χ) γε( u, χ) ε (µ, χ) + 3 (u, χ) = Te same testing as above leads to τ( w, w) + γε( u, u) (u, u) ε = and testing (58) wit u yields (u, u) = τ( w, u). Ten, if τ 4γε 3 we obtain wit Young s inequality injectivity. 6

19 5 Numerical results In tis section we discuss four test examples and numerically analyse te beaviour of te PDAS-algoritm. In te first test example we consider two concentric circles, were te exact sarp interface solution is known, and compare semi-implicit and implicit discretization. Te second example is a four-circle problem were concave as well as convex sections appear in te interface. Time evolution is given not only for u but also for w and te Lagrange multiplier µ. Wit tis example we compare te PDAS-metod wit a standard solver for te Can-Hilliard inequality, namely wit a projected block Gauss-Seidel metod [4]. It turns out tat te PDAS-metod is more efficient and reliable. In particular, te speed up is gained by a linear algebra solver wic is not based on a Gauss Seidel metod. Moreover, we ave not seen a difference in CPU-time between te semi-implicit and implicit discretization, and in te latter case ave not faced suc a severe restriction on te time step τ as indicated by te analysis. Te number of PDAS-iterations for eac time step were rater depending on τ tan on te mes size, never exceeded and after te interface settles 4 iterations were sufficient. For te number of iterations tere was nearly no difference between an adaptive and a uniform grid. But of course in CPU-time adaptivity was muc more efficient. In te tird test example we considered random initial data, i.e. a starting situation witout pure pases. Here and in our last example, a 3D simulation, we observed tat even wit large topological canges te maximum number of PDAS-iterations stayed always below. Before we present te examples we discuss some numerical issues as tere are mes generation, adaptivity in order to resolve te interface, coice of te parameter c, initialization of te active sets and te linear algebra solver. Also we describe sortly te mentioned projected Gauss Seidel type algoritm.. Mes generation and adaptivity For all te simulations presented in tis paper te finite element toolbox ALBERTA [35] was used for mes generation, te assembly of te matrices and administration. To generate te adaptive meses we used te mes adaption strategy of Barrett, Nürnberg, Styles [4]. Experiments sowed tat it is essential to ensure tat at least eigt vertices are present on te interfaces to avoid mes effects, see also [9]. We ence refine on te interface down to a level were eigt vertices are present and coarse in te areas were te concentration u is constant. For given parameters ε and γ tis results in an upper bound fine ε γ π, were 9 fine is te refinement level on te interface. Since we want to avoid too coarse meses we additionally define coarse := fine and coose a tolerance tol. Afterwards te mes adaption is done te following way: For eac element T T calculate te indicator η T := min u(x). Ten, a triangle is marked for refinement if it, or one of its x T neigboring elements, satisfies η T > tol and if T > fine. A triangle is 7

20 marked for coarsening if it satisfies η T < tol 3 and T < coarse.. Coice of te parameter c To determine te active sets we ave to coose te parameter c >. In te unilateral case te selection of c > as no influence on te iterates after te first iteration and can be cosen arbitrary, see [3]. However tis is no longer true in te case of bilateral bounds. Tis is discussed for obstacle problems in [6]. If c is cosen too small we observe cases in wic te iterates oscillated and te algoritm did not converge. Figure sows te values of u at various PDAS iterations in one time step of a simulation in one space dimension wit =, τ = 5 5, πε =. and c =.. In te 8t iteration te algoritm breaks down because all vertices are in u u 3 Figure : Oszillations in D if c is too small. te active set and te system no longer exibits a valid solution, compare Remark 4.. Redoing te simulation wit c =. fixed te problem and after two iterations te time step was completed wit only marginal canges to u since te initial data was close to a stationary solution. Te same penomenon was observed in iger space dimensions. A euristic approac sowed tat it is sufficient to ensure tat no vertex can cange from te positive active set A + to te negative inactive set A and vice versa in one iteration. Tis can be acieved by selecting a PDAS parameter c large enoug, depending on te magnitude of te Lagrange multiplier µ. In all te simulations a value of c = was sufficient wen te interfaces were already well developed and adequate initial guesses for te active sets were known. Terefore, if not mentioned oterwise c = is cosen in te calculation. In simulations wit distortions or jumps in te concentration u larger values depending on te mes size were necessary. Coosing te parameter c larger ad no discernible influence on te simulation. Wen using adaptive meses for te PDAS algoritm a coice as to be made for every newly created vertex if it sould belong to te active or inactive set. For now we restrict ourselves to te following: If all neigboring vertices are active ten tis vertex sould be active too. In every oter case we set te new vertex to inactive. 3. Initialization of active sets As mentioned previously te application of a PDAS-metod to te interface evolution as te advantage tat te good initialization due to te information of te u 5 u 7 8

21 previous time step leads to a large speedup. At te first time step n = te active set A n,± is initialized using te given initial data u. Since in te limit te active sets describe te sets were u is strictly active a good approximation of A,± is given by te active set of u. Hence we coose A,± = { p j S u (p j) 8}. For time steps n we can exploit in addition µ n n. Due to possible grid canges from time step n to time step n one may ave to apply additionally te standard interpolation I n to te new grid S n, i.e. wit u := I n u n n and µ := I n µ n n initialize te active set A n,± as in (5) and (5). However less time consuming to initialize te active set is te following way, wic is applied in tis paper: if an edge between two positive or two negative active vertices is bisected, coose te new vertex is set respectively active and oterwise set te new vertex as inactive. 4. Solver for te equation system (45)-(46) For moderate mes sizes direct solvers for sparse equation systems perform quite well. We use a Colesky decomposition of te system by means of te multifrontal metod by Duff and Reid [7] wic is realized in te software package te UMFpack [3], [4]. Tis metod is a generalization to te frontal metod of Irons [5]. One crucial point of tis metod is tat te decomposition steps of te matrix are computed but not directly applied to te system matrix. Furtermore an elimination tree and pivoting strategy are used wic make use of te sparsity of te system. For furter discussion of te metod refer to [8], [5]. We solved te wole reduced symmetric sparse system (45)-(46) wit (47) by UMFpack all at once. Up to now in our numerical experiments tis direct solver performed better tan e.g. a block SOR metod. Te application of oter metods like, for example a cg-metod for a preconditioned Scur-Complement or block multigrid metod, are currently under investigation. In te first tests, wic were of moderate size, UMFpack was still faster. Te limit of te UMFpack up to now was te available memory. However large 3D problems ave not been investigated yet torougly. 5. Gauss Seidel type algoritm for te variational inequality (psor) Te following Gauss Seidel type algoritm is often used for solving te Can-Hilliard variational inequality, see [3], and is implemented in te same ardware and software environment as our metod. Hence to obtain a comparison witin te same setting we use tis metod as a reference metod. Tis metod is based upon te discretization of te variational inequality (6)-(7) by a semi-implicit backwards Euler metod in time and by using continuous piecewise linear elements in space, compare [8]. Tis results in te same discretization as we introduced in tis paper, namely (38) and (44): For n =,, 3,... and given u K find (u n, wn ) K S suc tat τ (un un, χ) + ( w n, χ) = χ S, γε( u n, (ξ un )) (wn, ξ un ) + ε (ψ (un ), ξ u n ) ξ K. 9

22 As is known for obstacle problems (see e.g. [3]) one can apply for tis nonstandard variational inequality a projected block-gauss-seidel or a projected block-sor metod (psor). Tis as been studied in [4], were also convergence as been sown. Te blocks are determined by te blocks corresponding to te values (u n (p j), w n (p j)), wic are merged for eac vertex. In our numerical experiments we use te psor-metod wit overrelaxation using ω =.3 for comparison wit te PDAS-metod. As stopping criteria u k u k 7 and a maximum of 5 iterations is used. 5. Test cases Example : Two concentric circles Te distinction we made in Section 4 between te explicit and implicit discretization of te free energy term leads to differences in te evolution process. We use a radial symmetric situation were te exact sarp interface solution is known to compare tese two scemes. Te initial data are suc tat te interface is described by two concentric circles wit radii < r () =.5 < r () =.5 <. Tey bot will srink over time -te smaller faster tan te larger circle- until te smaller one vanises. Ten te solution remains stable due to mass conservation. In te limit ε te Can Hilliard model describes te evolution of te sarp interface Mullins Sekerka model, see [9], [3]. Te radial symmetric sarp interface solution is discussed in [36], []. In above situation te exact sarp interface solution can be calculated as solution of an ODE r = r σ r r r + r ln(r ) ln(r ), r = r σ r r r + r ln(r ) ln(r ) were σ = π. For te comparison between te semi-implicit and implicit discretization we used an equidistant mes. In Figure 3 we plotted te sum of bot radii 8 for fixed ε =.39 and varying time step widt τ. In te implicit case we can coose even larger τ and already gain a good approximation of te evolution unlike for te semi-implicit case were a smaller time step widt is necessary to acieve te desired result. Tis as been also observed wen considering te convergence wit respect to ε to te sarp interface wit a fixed time step. Te time step can be cosen larger in te implicit case. Example : 4-circles problem In tis example we realize a concave as well as a convex section of te interface, wic is a situation of large interest. Te initial data on = (, ) consist of four circular interfaces of widt επ. Te centres and radii are cosen in suc a way tat tree of circles intersect and one is detaced. Te values ± are connected by a sine profile wic is given as te lowest order term in an asymptotic expansion of te Can-Hilliard variational inequality, see e.g. [9]. In Figure 4 we sow te inital data

23 exact τ = 6 τ = 5.6 exact τ = 7 τ = 6 τ = Semi-implicit Implicit Figure 3: (r + r )(t) for different τ wit ε =.39. for two different interface widt. Te initial active sets sow a value of on eac inactive vertex and a positive resp. negative value on eac active vertex. We set u A,±, I u A,±, I Figure 4: Initial data for πε =. (left) and πε =.5 (rigt). T end =. for all te following simulations. In Figure 5 and Figure 6 te evolution of u, w and µ in time is plotted. Here we used a semi-implicit discretization wit an adaptive mes wit fine =. for πε =. and fine =.5 for πε =.5 respectively, te time step τ = 5. Simulations wit equidistant mes give te same results. Te columns from left to rigt sow te values of u, w, µ and te mes after 5, 5,, and time steps. Table sows tat for a small number of vertices te psor algoritm is still fast but wit an increasing number of vertices its performance quickly deteriorates. Using te corresponding block SOR-metod in combination wit te PDAS-metod, te resulting solver is even a bit slower for large time steps. Te direct solver on te oter and lowers te runtime extremly. Moreover we see tat tere is nearly no difference in CPU-time between semi-implicit and implicit discretization. Te severe restriction on te time step for te implicit case as stated in Lemma. as not been observed. Only for πε =.5 te coice τ = 4 failed even for very large parameter c =. Wen we compare te runtimes used on te fixed mes wit 664 vertices we notice tat te simulations wit πε =. used up almost double te time of te one wit πε =.. Te reason lies in te size of te inactive set, wic is rougly spoken te interface wit widt πε. Hence for πε =. te system (45)-(46) wic as to be solved is of larger dimension. In addition in Table te total, te averaged and te maximal number of PDASiterations are listed for te semi-implicit discretization. Te numbers for te implicit

24 n u 4. w µ Figure 5: Time evolution of Example wit πε =.. mes

25 n u 4. w µ Figure 6: Time evolution of Example wit πε =.5. 3 mes

26 πε τ CPU Time in seconds PDAS-iterations semi-impl. impl. (J ) psor PDAS PDAS total average max (45) (664) adaptive ( ) (664) (6649) adaptive ( 36) (6649) adaptive ( 7) Table : CPU Runtimes and iteration counts for Example. discretization are nearly te same except for te failures and ence not listed. Te average number of PDAS-iterations depend more on te time step tan on te mes. Tis is an expected beavior since wen we use larger time steps te active sets cange on a bigger scale tan wit smaller time steps. In most of te above simulations te maximum number of iterations was needed in te first time step. Te reason is tat te mean curvature of te interface is ig in te beginning of te time evolution, resulting in fast movement of te interface region. Even taking a rater large time step, like for example τ = 4 for πε =.5, te maximum number of necessary iterations per time step keeps low and never exceeded. Te averaged numbers of iterations are muc smaller since te time evolution of te interface becomes slow for larger t, resulting in only one or two PDAS-iterations. In Figure 7 we plot te time against te number of used PDAS-iterations per time step as well as against te number of canged vertices per time step for te above simulation wit πε =., τ = 5, in te semi-implicit and implicit case for 4

27 an adaptive mes wit fine =.55. In te first few time steps te evolution 6 6 PDAS it Canged vert. 6 6 PDAS it Canged vert Number of PDAS Iterations Number of canged vertices Number of PDAS Iterations Number of canged vertices Time t x Time t x semi-implicit PDAS implicit PDAS Figure 7: PDAS-iterations and vertices canging sets per time step. smootens te interfaces and te concave part is moving quickly. Tese two facts result in an increased number of neccessary PDAS-iterations. After tat typically two to four iterations are sufficient. Te steps were we only need two iterations are optimal in a way tat if tere is any cange in te active set we need at least tese two iterations. Only wen tere are no canges in te sets just one iteration is sufficient. Wat we can observe in tese plots is te expected rise in iteration numbers wen tere is a big cange in te active set. Te second peak is due to te disappearance of te bubble in te upper rigt quadrant. If we use an equidistant mes for te above example, te results and numbers of PDAS-iterations stay nearly te same, altoug in te adaptive case we ave to adapt te starting active set due to a grid cange in time (see 3.). However, instead of circa minutes CPU-time for an equidistant grid only 76 seconds CPU-time is needed in te adaptive case to determine a solution up to T =.. Furter speed up can be obtained as mentioned before by a different linear algebra solver. Example 3: Random initial data In applications one often as to consider initial data wic are a random perturbation of a equally distributed concentration u. Terefore we give also results on an equally distributed mass on = (, ) wit a stocastic distortion. As translation away from we coose. and.7 as te range of values created and define u (x) :=.5 σ(x) +., were σ : R [, ] denotes a random number generator. Consequently tere is no pure pase initially, i.e. all vertices are inactive, and te resulting equation system (45)-(47) is as large as possible. For tis simulation we used te implicit discretization and a uniform mes wit =.55, τ = 5, T end =.5, πε =.5 and c =. In eac timestep were a new active set emerges, we observe larger values in te Lagrangian multiplyier µ, namely max µ. Figure 8 sows u, w and µ after, 5, 5 and 5 time steps. Already after 5 time steps te pase separation can be clearly seen. In Figure 9 we see tat in te early stage of tis simulation one PDAS iteration is sufficient since tere is 5

28 387 n= n= n = 5 6 n = 5 7 u w Figure 8: D simulation wit random initial data 6 µ

29 no active set present and we just ave to solve te equation system. After tat a larger number of iterations is neccessary because tere are quite a few topological canges in te active set and a uge amount of vertices is canged from inactive to active. However tere ave never been more tan PDAS-iterations necessary. Afterwards wen te interfaces are well developed an average amount of -3 iterations is sufficient. Number of PDAS Iterations PDAS it Canged vert Number of canged vertices Time t x 3 Figure 9: PDAS-iterations and vertices canging sets per time step for random initial data. Example 4: Tree dimensional simulation Finally we give an example in 3D. Terefore we expand Example to initial data consisting of four balls in = (, ) 3. Figure sows te level sets of u of suc a simulation wit τ = 5, πε =. and c = after,, 5,, 3 and 7 time steps on an adaptive mes wit te semi-implicit primal-dual active set solver. Te simulation up to T end =.7, i.e. 7 time steps, were a coupled system n = n = n = 5 n = n = 3 n = 7 Figure : 3D simulation wit 4 speres as initial data 7

30 corresponding to rougly grid points as to be solved, took.8 ours wit a total of 84 PDAS-iterations. Tis is less tan alf te computation time used by te psor metod wic used 7.6 ours. Additional speed up -wic is not possible for te psor-metod- can be obtained by a different LA-solver. Even for tis tree dimensional problem wit te topological canges a maximal number of only four PDAS-iterations in eac time step is sufficient for te simulation. References [] Adams, R.A., Sobolev spaces, Pure and Applied Matematics 65. Academic Press New York-London (975). [] Banas, L. and Nürnberg, R., On multigrid metods for te Can-Hilliard equation wit obstacle potential, Computational Linear Algebra wit Applications, Harracov, Czec Republic, August 9-5, 7. [3] Barrett, J.W., Blowey, J.F. and Garcke, H., Finite element approximation of te Can Hilliard equation wit degenerate mobility, SIAM J. Numer. Anal. 37 (999), no., [4] Barrett, J.W., Nürnberg, R. and Styles, V., Finite element approximation of a void electromigration model, SIAM J. Numer. Anal. 4 (4), [5] Benzi. M., Golub., G.H. and Liesen, J., Numerical solution of saddle point problems, Acta Numerica (5), 37. [6] Blank, L., Garcke, H., Sarbu, L. and Styles, V., Primal-dual active set metods for Allen-Can variational inequalities wit non-local constraints, Preprint SPP [7] Blowey, J.F. and Elliott, C.M., Te Can-Hilliard gradient teory for pase separation wit nonsmoot free energy. I, Matematical analysis. European J. Appl. Mat., no.3 (99), [8] Blowey, J.F. and Elliott, C.M., Te Can-Hilliard gradient teory for pase separation wit nonsmoot free energy. II, Numerical analysis. European J. Appl. Mat. 3, no. (99), [9] Blowey, J.F. and Elliott, C.M., Curvature dependent pase boundary motion and parabolic double obstacle problems, IMA Vol. Mat. Appl. 47, Springer, New York (993), 9 6. [] Can, J.W. and Hilliard, J.E., Free energy of a nonuniform system. I, Interfacial energy, J. Cem. Pys. 8 (958),

31 [] Cen, X., Nased, Z., and Qi, L., Smooting metods and semismoot metods for nondifferentiable operator equations, SIAM J. Numer. Anal. 38, no.4 (), 6. [] Cen, X., Global asymptotic limit of solutions of te Can Hilliard equation, J. Differential Geom. 44, no. (996), 6 3. [3] Davis, T.A., Algoritm 83: UMFPACK, an unsymmetric-pattern multifrontal metod, ACM Trans. Matematical Software, 3() (3), [4] Davis, T.A., A column pre-ordering strategy for te unsymmetric-pattern multifrontal metod, ACM Trans. Matematical Software, 34() (3), [5] Davis, T.A. and Duff, I.S., An unsymmetric pattern multifrontal metod for sparse LU factorization, SIAM J. Matrix Anal. Appl. 8 (997), [6] Dolcetta, I.C., Vita, S.F. and Marc, R., Area-preserving curvesortening flows: From pase separation to image processing, Interfaces and Free Boundaries 4 (4) (), [7] Duff, I.S. and Reid, J.K. Te multifrontal solution of indefinite sparse symmetric linear, ACM Transactions on Matematical Software (TOMS) 9 (983), [8] Elliott, C.M., Te Can-Hilliard model for te kinetics of pase separation, Matematical models for pase cange problems, Internat. Ser. Numer. Mat. 88, Birkäuser, Basel (989). [9] Evans, L.C., Partial differential equations, Graduate Studies in Matematics 9. American Matematical Society, Providence, RI (998). [] Friedman, A., Variational principles and free-boundary problems, Wiley 98. [] Garcke, H., Mecanical effects in te Can-Hilliard model: A review on matematical results, in Matematical Metods and Models in pase transitions, ed.: Alain Miranvielle, Nova Science Publ. (5), [] Gräser, C. and Kornuber, R., On preconditioned Uzawa-type iterations for a saddle point problem wit inequality constraints, Domain decomposition metods in science and engineering XVI, 9, Lect. Notes Comput. Sci. Eng., 55, Springer, Berlin 7. [3] Hintermüller, M., Ito, K. and Kunisc, K. Te primal-dual active set strategy as a semismoot Newton metod, SIAM J. Optim. 3 (), no. 3, (electronic) (3) 9