AN OPTIMAL ERROR BOUND FOR A FINITE ELEMENT APPROXIMATION OF A MODEL FOR PHASE SEPARATION OF A MULTI-COMPONENT ALLOY WITH NON-SMOOTH FREE ENERGY

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1 Mathematical Modelling and Numerical Analysis MAN, Vol. 33, N o 5, 999, p Modélisation Mathématique et Analyse Numérique AN OPTIMA ERROR BOUND FOR A FINITE EEMENT APPROXIMATION OF A MODE FOR PHASE SEPARATION OF A MUTI-COMPONENT AOY WITH NON-SMOOTH FREE ENERGY John W. Barrett and James F. Blowey Abstract. Using the approach in 5 for analysing time discretization error and assuming more regularity on the initial data, we improve on the error bound derived in for a fully practical piecewise linear finite element approximation with a backward Euler time discretization of a model for phase separation of a multi-component alloy with non-smooth free energy. AMS Subject Classification. 65M6, 65M5, 35K55, 35K35. Received: March 6, Introduction In, we proved an error bound for a fully practical finite element approximation of the following deep quench Cahn-Hilliard model: P) Find {u,t), w,t),ξ,t)} K Y Ω) such that u, ) = u ) andfora.e. t,t), η + w, η) = η H Ω),.a) γ u, η u)) I )Au, η u) w + ξ, η u) η K.b) where Y := {η H Ω) : ηx) = fora.e. x Ω},.a) K := {η H Ω) : ηx) for a.e. x Ω}.b) Here Ω is a bounded domain in R d d 3) with a ipschitz boundary Ω. In the above {u} n is the fractional concentration of the n th component of the alloy, and so the following assumptions are made on the initial data a) u x) and b) N u x) = x Ω..3) Keywords and phrases. Finite elements, error analysis, Cahn-Hilliard, phase separation. Department of Mathematics, Imperial College, ondon, SW7 BZ, United Kingdom. j.barrett@ic.ac.uk Department of Mathematical Sciences, South Road, Durham, DH 3E, United Kingdom. c EDP Sciences, SMAI 999

2 97 J.W. BARRETT AND J.F. BOWEY In.a,b) γ is a positive constant and A and are symmetric constant N N matrices. It is further assumed that has a one dimensional kernel such that = and is positive semi-definite..4a).4b) From physical considerations A must have at least one positive eigenvalue, and the analysis simplifies if this were not the case. et λ Amax be the largest positive eigenvalue of A. We define R N by {} n := for n = N. Here and throughout we write ζ n for the n th component of ζ R N and set ζ := N ζ n. N For later purposes, we introduce for any µ R n= Mµ) :={ξ R N : N ξ = µ}.5) Finally, we introduce η := ηx)dx Ω Ω η Ω). The system P) models the isothermal phase separation of a multi-component ideal mixture with N components in the deep quench limit, see and the references cited therein. The well-posedness of P), see Theorem. in, is proved under the following assumptions on the initial data u : D) u H Ω) such that.3) holds and u >δ for some δ, /N). We note that the integral constraint above only excludes the degenerate case when one or more components of u are not present, in which case the system can be modelled with a smaller value of N. The finite element approximation of P) was studied in under the following assumptions: A) et Ω be convex polyhedral and T h be a regular partitioning of Ω into disjoint open simplices κ with h κ := diamκ) andh := max κ T h h κ,sothatω κ T hκ. In this paper we strengthen these assumptions to A) In addition to A) let T h be a quasi-uniform partitioning of Ω. Associated with T h is the continuous piecewise linear finite element space S h := {χ CΩ) : χ τ is linear τ T h } H Ω). We extend these definitions to vector functions, i.e. χ S h χ n S h,n= N. et π h : CΩ) S h be the interpolation operator such that π h ηx m )=ηx m )m = M), where {x m } M m= is the set of nodes of T h. Throughout, ) denotes the standard inner product over Ω, naturally extended to vector and matrix functions, e.g. for I J matrices Cx) anddx), with entries in Ω) C, D) := I i= j= J C ij,d ij ):= I i= j= J C ij x)d ij x)dx..6) Ω

3 FINITE EEMENT APPROXIMATION OF MUTI-COMPONENT PHASE SEPARATION 973 Also, denotes the duality pairing between H Ω)) and H Ω), which is extended to vector functions in the standard way. We now introduce the corresponding approximations of.a,b): A discrete semi-inner product on CΩ) is then defined by Y h := {χ S h : χx m )=, m = M},.7a) K h := {χ S h : χx m ), m = M}.7b) η,η ) h := Ω π h η x) η x)) dx M β m η x m ) η x m ).8) where β m >. Once again, this is naturally extended to vector and matrix functions as in.6). Given K, a positive integer, let := T/K denote the time step and t k := k, k = K; Barrett and Blowey considered the following fully practical piecewise linear finite element approximation, based on a backward Euler time discretization, of P): P h, ) For k = K find {U k, W k, Ξ k } K h Y h S h such that h U k U k ), χ) + W k, χ = χ S h,.9a) ) γ U k, χ U k ) I )AU k + W k +Ξ k, χ U k) h χ K h.9b) where U = Q h i u for i = or. Here m= i) Q h : Ω) S h is such that {Q h η} n Q h η n and Q h : Ω) S h is defined by Q h η, χ)h =η, χ) χ S h..) ii) Q h : H Ω) S h is such that {Q h η} n Q h η n and Q h : H Ω) S h is defined by γ I Q h )η, χ)+i Qh )η, χ) = χ Sh..) et the assumptions D) and A) hold. et U = Q h u. Then for all h>andall<4γ/λ Amax ), Barrett and Blowey proved the well-posedness of P h, ) on assuming that U C. Moreover, they proved that u Û,T ;H Ω)) + u U,T ;H Ω)) ) C h + + h4..) Here we have adopted the notation: for k and U,t):= t t k U k )+ t k t U k ) t t k,t k.3a) Û,t):=U k ) t t k,t k..3b) In the above and throughout the paper, operating on matrices is that induced by the Euclidean vector norm, i.e. the spectral radius for symmetric matrices. We note that the assumption U C holds under

4 974 J.W. BARRETT AND J.F. BOWEY the stronger mesh assumptions A). It follows immediately from.) with the choice of C h < 4γ/λ Amax ) that u Û,T ;H Ω)) + u U,T ;H Ω)) ) Ch..4) It is the purpose of this paper to improve on the error bound.) using the approach developed by Rulla 5 for proving an optimal time discretization error for the backward Euler method applied to subgradient flows without requiring bounds on the second order time derivatives, which do not exist for the variational inequality system P). This approach does require the following stronger assumptions on the initial data: D) u H 3 Ω) such that.3b) holds, / ν =on Ω, where ν is normal to Ω, and u x) δ, x Ωforsomeδ, /N. With D, A) replaced by D, A), U = Q h u and the restriction h h ; we prove in this paper that the term + h 4 / on the right-hand side of.) can be replaced by ), yielding an optimal error bound. Hence the bound.4) can be achieved by choosing larger time steps; C h<4γ/λ Amax ). Remark.. In the case N =, assuming that A = A, = = /, defining u := u u, w := w w and θ c = A A we obtain that {u, w} satisfies the system w =, w γ u θ cu + I, u).5) where I, ) is the subdifferential of the indicator function of the set,. This is the Cahn-Hilliard equation with an obstacle free energy. The corresponding finite element approximation of this problem has been studied by Blowey and Elliott 4. Obviously the results in this paper are easily adapted to improve on the error bound derived there in an analogous way. Notation and auxiliary results We adopt the standard notation for Sobolev spaces, denoting the norm of W m,p Ω) m N, p, ) by m,p and the semi-norm by m,p. We extend these norms and semi-norms in the natural way to the corresponding spaces of vector functions W m,p Ω) := {W m,p Ω)} N. For p =,W m, Ω) will be denoted by H m Ω), with the associated norm and semi-norm written as, respectively, m and m. Furthermore, we define Ω T ):=,T; Ω)). For η H Ω), η denotes the N d matrix with entries { η} ij := η i / x j and then η/ ν := η) ν. Below we recall some well-known results concerning S h under the assumptions A): The inverse inequality for p p and m =or For m =orandp χ m,p Ch dp p ) p p χ m,p χ S h..6) χ χ h := χ, χ) h d +) χ χ S h,.7) χ,χ ) χ,χ ) h Ch +m χ m χ χ,χ S h,.8) where the last result follows immediately from.6,.9,.). I π h )η m,p Ch m d p ) η η H Ω),.9) I Q h )η m Ch m η η H Ω),.) I Q h )η m,p Ch m d p ) η η H Ω).)

5 FINITE EEMENT APPROXIMATION OF MUTI-COMPONENT PHASE SEPARATION 975 Below we recall the following inverse aplacian operators introduced in : a) G : F V is such that Gv, η) = v, η η H Ω),.) where F := { v H Ω)) : v, = } and V := {v H Ω) : v, ) = }. b) G : F V is defined by {Gv} n := Gv n,where and F := {v : v n F,n= N, and v =}.3) V := {v : v n V, n = N, and v =}.4) c) Noting.4a,.5), it follows that M) M) is invertible. Hence we can define G : F V by G M) G;thatis d) G h : F V h := {v h S h :v h, ) = } is such that G v, η) = v, η η H Ω)..5) G h v, χ) = v, χ χ S h..6) e) G h : F V h is defined by {G h v} n := G h v n,where V h := { v h : v h n V h,n= N, and v h =} V..7) f) G h : F V h is such that G h M) Gh ;thatis, G h v, χ) = v, χ χ S h..8) g) Ĝ h : F c V h is defined by Ĝ h v, χ) =v, χ) h χ S h,.9) where F c := {v CΩ) : v, ) h =}. h) Ĝ h : F c V h is defined by {Ĝ h v} n := Ĝ h v n,where F c := { v : v n F c,n= N, and v =} V h..3) i) Ĝ h : F c V h is defined by Ĝ h M)Ĝh ;thatis, On noting the Poincaré inequality Ĝ h v, χ) =v, χ)h χ S h..3) η C η + η, ) ) η H Ω),.3)

6 976 J.W. BARRETT AND J.F. BOWEY the well-posedness of G, G, G h, G h follows. In addition, on noting.7) we deduce the well-posedness of Ĝ h and Ĝ h. Finally, as M) M) is invertible, or equivalently noting that λ min v v, v) v V.33) where λ min is the smallest positive eigenvalue of ; yields the well-posedness of G, G h and Ĝ h. Noting.5) one can then define a norm on F by It follows from.33,.34) that v := / G v G v, G v) / v, G v / v F..34) In addition it follows from.5,.34,.35) that λ min G v v v F..35) λ min v Gv v v F..36) Hence is equivalent to the standard H Ω)) norm on F. Similarly, one can define norms on F and F c by and v h := / G h v / / G h v, G h v) v, G h v v F.37) / v := / Ĝ h v Ĝ h v, Ĝh v) v, Ĝ h / v)h v F c,.38) respectively. It follows from.8,.3,.37,.38) that for all v h V h and for all α> It is well-known that v h / G h v h, / v h ) v h h / Ĝ h vh, / v h ) α vh h + α vh, α vh + α vh..39a).39b) G G h )η Ch m G m η m, η H m Ω)) F, m =or..4) Hence, it follows from.5,.8,.36,.4) that G G h )η λ min G Gh )η Ch m G m η m Ch m η m It is easily deduced from.8), e.g. see 4, that Hence it follows from.8,.3,.4) that η H m Ω)) F, m =or..4) G h Ĝ h )v h Ch v h v h V h..4) G h Ĝ h )vh λ min Gh Ĝ h )v h Ch v h, v h V h..43)

7 FINITE EEMENT APPROXIMATION OF MUTI-COMPONENT PHASE SEPARATION 977 Next we note that C h v h C h v h v h h v h C 3 v h h v h V h..44) The first inequality on the left is just an inverse inequality, recalling that the partitioning is quasi-uniform. The second follows from the first and.39a). The third follows from noting that / G h vh / G v h.the final inequality follows from noting.4) with m = and the second inequality above. Finally, we have an analogue of.44) h v h C h v h h C v h C 3 v h h C 4 v h v h V h..45) The first inequality on the left is just an inverse inequality on noting.7). The second follows from the first and.39b). The third and fourth follow from.43) and noting the first two inequalities in.44) and.45), respectively.. The continuous problem It is easily established, see for details, that P) can be rewritten as: Find {u,t), λt),ξ,t)} K m M) Ω) such that u, ) = u ) andfora.e. t,t) where γ u, η u)) + G I )Au λ ξ, η u) η K.) K := {η K and N ηx) =fora.e. x Ω},.a) K m := {η K and η = m := u }.b) In the above we have eliminated w Y by noting from.a,.5,.33,.3) that w G + λ,.3) where λt) M) can be viewed as an unknown agrange multiplier. Theorem.. et the assumptions D) hold. et Ω be convex polyhedral or Ω C,. Then there exists a unique solution {u,t), λt),ξ,t)} {u,t), w,t),ξ,t)}) to P) such that the following stability bounds hold: u W,,T ;H Ω)) ) + u H,T ;H Ω)) + u,t ;H Ω)) + λ,t ) In addition we have for a.e. t a,t b with <t a <t b <T that,t b),t a) + w,t ;H Ω)) + w,t ;H Ω)) + ξ,t ; Ω)) C..4) + C t b t a )..5) Proof. Existence, uniqueness and the bounds.4) are proved in Theorem. of 3 for a concentration dependent mobility matrix. We note that the bounds.4) hold for any T> for the present case of a constant mobility matrix.

8 978 J.W. BARRETT AND J.F. BOWEY For a.e. t δt, T) andforallδt >, on choosing η u,t δt) Km in.) and η u,t) Km in.) at t = t δt, adding, using.5), a Young s inequality and.34) it follows that γ u,t) u,t δt) + d dt u,t) u,t δt) Au,t) u,t δt)), u,t) u,t δt)) γ u,t) u,t δt) +Cγ,λ Amax ) u,t) u,t δt)..6) Integrating.6) over t a,t b ) δt, T), dividing through by δt), taking the limit as δt and noting.4) yields that tb γ t a Hence the desired result.5). dt +,t b),t a),t a) tb + Cγ,λ Amax ) t a + C t b t a )..7) dt For later purposes, we note that J : H Ω) R defined by Jη) := γ η Aη, η) η H Ω).8) is a yapunov functional for P). To see this, we fix δt > then it follows for a.e. t δt, T), on choosing η u,t δt) Km in.), that γ u,t), u,t) u,t δt))) + δt Noting the identity u,t) u,t δt) G,t), δt ) Au,t), u,t) u,t δt))..9) a b)b = b a +a b) a, b R,.) it follows from.9) and.8) that for a.e. t δt, T) andforallδt > ) u,t) u,t δt) Ju,t)) Ju,t δt)) + δt G,t), δt) δt λ Amax u,t) u,t δt) δt..) Dividing.) by δt, integrating from t = δt to t k, taking the limit δt and noting.4),.34) and D) yields for k = K that Ju,t k )) +,t) dt Ju ) C..)

9 FINITE EEMENT APPROXIMATION OF MUTI-COMPONENT PHASE SEPARATION Finite element approximation Firstly, we note the following results concerning Q h i. It follows immediately from.,.) and the assumptions D, A) that for i =and Q h i u = u, N Q h i u x) =, x Ω, 3.a) and Q h i u C u C. 3.b) Under the same assumptions it follows that u Q h u Ch u Ch and Q h u x) x Ω, 3.) see for details. Under the assumptions D, A) it follows from.,.3,.34,.35) that u Q h u C u Q h u Ch4 u Ch4 3.3a) and in addition from.) with m =andp = that for h h Q h u x) x Ω. 3.3b) We now consider the finite element approximation P h, ), see.9a, b), to P). et K h := {χ K h and N χx m )=,m= M}, 3.4a) K h m := {χ K h and χ = m := u } 3.4b) Similarly to.), on noting 3.a), it is easily established, see for details, that P h, ) can be rewritten as: For k = K, find {U k, Λ k, Ξ k } K h m M) S h such that ) γ U k, χ U k ) + Ĝ h U k U k I ) h )AU k, χ U k Λ k +Ξ k, χ U k) h χ K h, where U = Q h i u, i = or. In the above, similarly to.3), we have eliminated W k Y h by noting from.9a,.3.33) that W k Ĝ h U k U k + Λ k k = K. 3.6) Theorem 3.. et the assumptions D) and A) hold. et U = Q h u. Then for all h h and for all <4γ/λ Amax ), there exists a solution {U k, Λ k, Ξ k } K k= {U k, W k, Ξ k } K k= ) to Ph, ). Moreover {U k } K k= is unique and the following stability bounds hold 3.5) U,T ;H Ω)) + U H,T ;H Ω)) ) +) U H,T ;H Ω)) + Ŵ,T ;H Ω)) + ˆΛ,T ) + ˆΞ,T ; Ω)) C, 3.7) where U and Û are defined as in.3a, b) with Ŵ, ˆΛ and ˆΞ being similarly defined. Furthermore we have that U H,T ;H Ω)) + U W,,T ;H Ω)) ) + ˆΛ,T ) C. 3.8)

10 98 J.W. BARRETT AND J.F. BOWEY Proof. Existence, uniqueness and the bounds 3.7) are proved in Theorem 3. of with the assumption D) replaced by D) and the projection Q h replaced by Q h under no constraint on h. It is a simple matter to adapt these proofs to the projection Q h with the mesh constraint on noting 3.a, b) and 3.3a, b). Therefore we need only prove 3.8). For the purposes of the analysis, it is convenient to introduce U such that U U V h and U γ U, χ) AU, χ) h + Ĝ h U )h, χ = χ V h. 3.9) For m, it follows from adding 3.5) with k = m and χ U m to 3.5) with k = m andχ U m 3.9) if m = with χ U U that γ U m U m ) AU m U m ), U m U m ) h + Ĝ h U m U m U m U m ) h, U m U m. 3.) Summing 3.) for m = k, noting.38,.39b,.) yields for k = K that γ k U m U m + U k U k m= U U k + A U m U m, U m U m )h m= U U k + Cγ) U m U m m=. 3.) Choosing χ U U in 3.9) and noting.,.5,.34,.38), assumption D),.3,.44,.45, 3.b) yields that U U = γ = U, U U )) + γ u +Q h u u ), U U AU, U U )h ) + AU, U U C u 3 C. 3.) Hence combining 3., 3.) and noting.3,.44,.45, 3.7) yields the first two bounds in 3.8). The final bound in 3.8) follows from the second and recalling from ) of that Λ k C U k U k +, k = K. 3.3) )h In Theorem 3.3 below we adapt the technique in Rulla 5 to improve on the temporal discretization error bound in for the scheme P h, ). In the next lemma we bound a key term required in the proof of this theorem.

11 FINITE EEMENT APPROXIMATION OF MUTI-COMPONENT PHASE SEPARATION 98 emma 3.. et the assumptions of Theorem 3. hold. Then for k = K, we have that where J h : S h R is defined by dt J h U k ) J h Q h u )+C + h, 3.4) J h χ) := γ χ Aχ, χ)h χ S h. 3.5) Proof. Choosing χ = U m in 3.5) with k = m ifm andχ = U U in 3.9), noting., 3.5) yields for m = K that J h U m ) J h U m )+ γ U m U m U m Ĝ h U m, U m U m )h + λ Amax U m U m h. 3.6) It follows from.39b,.44,.45, 3.8, 3.6) that for m = K J h U m ) J h U m ) Ĝ h Noting.38,., 3.7) we have for m = K that U m U m + J h U m ) J h U m ) U m U m U m U m, U m U m ) h + C). 3.7) Summing 3.8) and noting 3.), then yields for k = K that U m U m dt J h U k )+J h Q h u ) U U + C ). 3.8) + C C. 3.9) The desired result 3.4) then follows from 3.9) on noting.8,.3,.34,.38,.4,.43, 3.8). Theorem 3.3. et the assumptions D, A) hold. et U = Q h u. Then for all h h and for all < 4γ/λ Amax ) we have that u Û,T ;H Ω)) + u U,T ;H Ω)) ) + u U H,T ;H Ω)) ) C h +). 3.) Proof. Using the notation.3a,b), 3.5) can be restated as: Find U H,T; K h m) such that U, ) = Q h u ) andfora.e. t,t) ) h γ Û, χ Û)) + Ĝ h AÛ ˆΛ, χ Û χ K h. 3.)

12 98 J.W. BARRETT AND J.F. BOWEY We set e := u U, ê := u Û, e A := u π h u, e h := π h u U and ê h := π h u Û. Note that e = e A + e h =, ê = e A + ê h = and e A = e h = ê h =. Fora.e. t,t)wehavethat ê ê, êh )+ ê e A. 3.) Introducing µ t) := t k t t t k,t k, k = K 3.3) we have that e,t) ê,t)=µ t),t), t t k,t k ), k = K. 3.4) It follows from., 3.4) that ) e G, ê = d dt e µ e G, ) = d dt e + e µ ) Next we note that ) ) ) h γ ê, ê h e )+ G, ê = γ u, ê h )+ G, ê γ Û, ê h )+ Ĝ h, êh + Ĝ h G ) ) ) ) h ), êh G, ea + Ĝ h, êh Ĝ h, êh. 3.6) From.), with η Û, and.4) it follows that ) γ u, ê h )+ G, ê Au + λ, ê h ) γ u, e A ) Au + λ, e A ) = Au + λ, ê h ) γ u Au λ, e A ). 3.7) Choosing χ π h u in 3.) yields that ) h γ Û, ê h )+ Ĝ h, êh AÛ + ˆΛ, ê h ) h. 3.8)

13 FINITE EEMENT APPROXIMATION OF MUTI-COMPONENT PHASE SEPARATION 983 Combining 3., ) and noting.3,.34,.35,.4,.43), a Young s inequality,.8, 3.4) and that λ ˆΛ, ê) = yields γ ê + d dt e + µ e C u + + λ e A + G Ĝ h ) ê h +Aê, ê h ) + AÛ, ê h ) AÛ, ê h ) h λ ˆΛ, e A )+Ch Ĝ h ê h + µ + γ ê e A γ ê u + C + + λ + Û + ˆΛ e A + Ch + Û ê h + µ + C e +) + e A. 3.9) Integrating 3.9), and noting that e, ) = I Q h )u ), / is constant over t k,t k ),.9,.4, 3.3a, 3.7) yields that for k = K e,t k ) + γ ê +µ e dt { I Q h )u + C e + µ } dt + C e A,T ;H Ω)) +),T ;H Ω)) ) + Ch + Û,T ;H Ω)) ê h,t ;H Ω)),T ;H Ω)) + C u,t ;H Ω)) + + λ,t ) + Û Ω T ),T ;H Ω)) ) + ˆΛ,T ) e A Ω T ) { } C e + µ dt + C h +). 3.3) Setting ū,t):=u,t+/) and defining ē := ū U, it follows in an analogous manner to 3.3) that for k = K ē,t k ) + µ ē dt ē, ) + C h +) { + C ē + µ ū } 3.3) dt.

14 984 J.W. BARRETT AND J.F. BOWEY Next we note for k = K that µ t) ū,t) dt µ t ),t) + dt + µ t t k ),t) dt µ t ),t) dt, 3.3) t where t k+ := t k + t k+ ), k = K, andµ t) :=µ t +) fort,. Noting for t,tthat µ t)+µ t )= + µ t), 3.33).4,.5) we have that µ t)+µ t ),t) = dt dt + k dt + 6 m= k m m= m) µ t),t), m ) ) dt m, ) + C dt + C. 3.34) Furthermore as µ t) / fort t m,t m, m = K, it follows that k m= tm t m e dt µ e dt. 3.35) Similarly to 3.35), on noting that / is constant on t m,t m ), m = K, wehavethat k m= tm t m e dt k m= tm t m ē dt µ ē dt. 3.36)

15 FINITE EEMENT APPROXIMATION OF MUTI-COMPONENT PHASE SEPARATION 985 Combining.4, 3.3a, , ) yields for k = K that e,t k ) + ē,t k) + γ ê + e dt u, ) u, ) + I Qh )u, ) + C e + ē dt + µ t)+µ t ),t),t) dt + + µ t t k ),t) dt + C h +) C e + ē dt + dt + C h +). 3.37) We now bound the second integral on the right-hand side of 3.37). Combining., 3.4) we have for k = K that dt J h U k ) Ju,t k )) J h Q h u )+Ju )+C + h. 3.38) From.8,.,.8, 3.5) and D) it follows that From.8,.9,.4,.8, 3.5) it follows for k = K that Ju ) J h Q h u ) Ch. 3.39) Ju,t k )) J h π h u,t k )) Ch. 3.4) It follows from.7,.9,.5,.3,.44,.45,.4,., 3.5, 3.8, 3.5) and a Young s inequality that for k = K and for all α> J h U k ) J h π h u,t k )) γ U k, e h,t k ))) AU k, e h,t k )) h + λ Amax e h,t k ) h h Ĝ h U k Uk )+Λ k, e h,t k )) + C e h,t k ) U k U k C I )e A,t k ) + e,t k ) + C Λ k e A,t k ) + e,t k ) + e A,t k ) α ) e,t k ) + C + h. 3.4) Combining ) for α sufficiently small yields for k = K that e,t k ) + ē,t k) + γ ê + e dt C e + ē dt + C h +). 3.4)

16 986 J.W. BARRETT AND J.F. BOWEY 3 4 u u u u u u Next we note for k = K that k e dt m= Figure. u,t)fort =and.5. tm t m k m= k m= t e e,t m ) + t m s ds dt k tm ) t e,t m ) + e t m s ds dt m= t m e,t m ) +) e dt. 3.43) The desired result 3.) then follows from combining 3.4, 3.43) and a similar bound with e replaced by ē, applying a discrete Gronwall inequality and noting.3,.4, 3.8). 4. Numerical experiment We chose N =3, /3 /3 /3 /3 /3 /3 and A =. 4.) /3 /3 /3 We note that the eigenvalues of and A are respectively,, and,,. As no exact time dependent solution to P) is known with a free boundary, a comparison between the solutions of P h, ) on a coarse mesh, U, with that on a fine mesh, u, was made. The data used in the experiment on the coarse meshes were Ω, ), γ =.5, T =.5, =.6h and h =/M ) where M = p + p =5, 6, 7, 8). The data were the same for the fine mesh except that M = +. As λ Amax =and = the condition in Theorem 3.3 on is that <4γ =.. The initial data u was taken to be the clamped complete) cubic spline with u taking the values {s, s, s, s/,s/8,s/4,s/,s/,s/} at the equally spaced points i/8, i = 8; u x) =u x) andu 3 x) = u x) u x). In the above we chose s = 4/779, so that u n /3, n= 3; see Figure, where we plot u, ) and u,.5). Note that u H 3 Ω) \ H 4 Ω), / ν = and u δ for δ =.4 3. Hence u satisfies the assumptions D). This choice of initial data also ensured that there was a free boundary for U on all of the coarse meshes. In addition for all choices of h, the discrete initial data Q h u satisfied 3.3b). We used the iterative method discussed in to solve for U k at each time level in P h, ) with the same stopping criterion: maximum difference of the successive iterates was less than 7.

17 FINITE EEMENT APPROXIMATION OF MUTI-COMPONENT PHASE SEPARATION 987 We computed the quantity ζ n := K / K π h u n,k) Un ) k n =,, 3 k= and obtained the following table of values to three significant figures: M ζ ζ ζ ζ We see that the ratio of consecutive ζ is approximately 5.7, 4.8 and 4. which are around 4., the rate of convergence proved in Theorem 3.3. References J.W. Barrett and J.F. Blowey, An error bound for the finite element approximation of a model for phase separation of a multi-component alloy. IMA J. Numer. Anal ) J.W. Barrett and J.F. Blowey, Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy. Numer. Math ) J.W. Barrett and J.F. Blowey, Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy and a concentration dependent mobility matrix. M 3 AS 9 999) J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy, part II: Numerical analysis. Eur. J. Appl. Math. 3 99) J. Rulla, Error analysis for implicit approximations to solutions to Cauchy problems. SIAM J. Numer. Anal )