A posteriori error analysis for time-dependent Ginzburg-Landau type equations

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1 A posteriori error analysis for time-dependent Ginzburg-Landau type equations Sören Bartels Department of Matematics, University of Maryland, College Park, MD 74, USA Abstract. Tis work presents an a posteriori error analysis for te finite element approximation of time-dependent Ginzburg-Landau type equations in two and tree space dimensions. Te solution of an elliptic, self-adjoint eigenvalue problem as a postprocessing procedure in eac time step of a finite element simulation leads to a fully computable upper bound for te error. Teoretical results for te stability of degree one vortices in Ginzburg-Landau equations and of generic interfaces in Allen-Can equations indicate tat te error estimate only depends on te inverse of a small parameter in a low order polynomial. Te actual dependence of te error estimate upon tis parameter is explicitly determined by te computed eigenvalues and can terefore be monitored witin an approximation sceme. Te error bound allows for te introduction of local refinement indicators wic may be used for adaptive mes and time step size refinement and coarsening. Numerical experiments underline te reliability of tis approac. Key Words. Ginzburg-Landau equations, Allen-Can equations, error analysis, eigenvalue approximation, finite element metod, adaptive mes refinement. AMS Subject Classification: 65M5, 65M6, 65M5. Date: April 5, 4.. INTRODUCTION Given a bounded Lipscitz domain Ω R n, n =, 3, a subset Γ D Ω wic is eiter empty or of positive surface measure, a positive integer m, a small number ε >, a parameter T >, initial data u ε H (Ω; R m ), and boundary data g ε = u ε ΓD H / (Γ D ; R m ), we aim to approximate te problem: Find u ε Z := H (, T ; Y ) L (, T ; H (Ω; R m )) suc tat for almost all t (, T ) and all v Y tere olds (P) u ε t ; v + ( uε ; v) + ε (f(u ε ); v) =, u ε ΓD = g ε, u ε () = u ε. Here, Y := {v H (Ω; R m ) : v ΓD = }, f(a) = ( a )a for a R m, ( ; ) stands for te scalar product in L (Ω; R m ), ; denotes te duality pairing of Y and Y, and u ε t is te time derivative of u ε. Trougout tis work a b denotes te scalar product of two vectors a, b R m.

2 Existence and uniqueness of a solution u ε for (P) follow from standard tecniques. Trougout tis work we abbreviate u = u ε, u = u ε, and g = g ε but stress tat te dependence of te error of numerical approximation scemes upon te parameter ε is te main focus of tis work. Te model problem (P) reduces to Allen-Can equations [] for te description of melting processes in binary alloys if m = and Γ D = and to (simplified) Ginzburg-Landau equations [3] as a matematical model for certain superconducting materials if n = m =, Γ D = Ω, and g(x). Since solutions for tese problems develop interfaces of tickness ε or evolve vortices wit large gradients of order ε in a small neigborood, optimal approximation scemes require locally refined meses wic resolve suc topological effects. Adaptive finite element metods based on a posteriori error estimates are known to allow for very efficient approximations wen features on small scales ave to be resolved. It is te aim of tis work to prove robust a posteriori error estimates wic lead to local refinement and coarsening indicators tat allow for an automatic mes and time-step size refinement and coarsening. Standard error estimates for te numerical approximation of (P) depend exponentially on ε and are useless if ε is small. We establis an a posteriori error estimate wic depends on ε only in a low order polynomial if no critical topological effects take place. Oterwise, te error estimate localizes critical times at wic suc effects occur. In particular, let U(t) S be te output of an approximation sceme and wic serves as an approximation of u(t) for some t (, T ). We ten approximate te self-adjoint, elliptic (after introduction of a constant sift) eigenvalue problem w + ε f (U(t))w = λw in Ω and te value λ determines te stability of te solution of (P) at time t and enters error estimates exponentially. Teoretical results in [5, 6] for Allen-Can equations and in [7] for Ginzburg- Landau equations indicate tat λ is bounded ε-independently from above if te zero level set of u is smoot and if zeros of u are of topological degree one, respectively. Our analysis is inspired by recent work by Feng and Prol [9, ] on te a priori error analysis for te approximation of Allen-Can and Can-Hilliard equations and by Kessler, Nocetto, and Scmidt [4, 5] on te a posteriori error analysis for te approximation of Allen-Can equations. Here, we do not restrict te analysis to initial conditions and parameters T > tat allow for an ε-independent upper bound for λ but propose its approximation in eac time step of a numerical simulation and tereby avoid te use of any a priori information. Numerical experiments indicate tat tis approac is reliable. We remark tat our analysis is still valid if outer body forces or inomogeneous Neumann boundary conditions on Ω \ Γ D are included in (P). Moreover, te function f may be replaced by any function tat satisfies te estimates of Lemma 3. below. For related numerical aspects of Allen-Can and Ginzburg-Landau equations we refer te reader to [7, 8] and references terein. Te outline of tis article is as follows. We present te main result in an abstract setting in Section employing a continuation argument from [4]. Te abstract framework allows for a posteriori error estimates tat are metod independent. Section 3 is devoted to te derivation of an error equation tat is needed in te proof of te main result. We ten specify notation in finite

3 element spaces in Section 4 and state estimates tat control te influence of approximated boundary data on te global approximation error. Error estimates for te approximation of eigenvalue problems are discussed in Section 5 and teir combination wit explicit bounds for te residual of a lowest order finite element approximation of u in Section 6 make te abstract result of Section practically applicable. Te fully computable error estimate ten motivates an adaptive mes and time step size coarsening and refining algoritm wic is stated in Section 7. Numerical experiments tat sow te necessity of approximating λ and tereby underline te relevance of te teoretical results of tis contribution are reported in Section DERIVATION OF THE MAIN RESULT Given an approximation U Z L (, T ; L (Ω; R m )) of te solution u of (P) suc tat U ΓD is time-independent, we define its residual R U L (, T ; Y ) for almost all t (, T ) and all v Y by (.) U t ; v + ( U; v) + ε (f(u); v) = R U ; v. We solve eigenvalue problems to obtain functions Λ and η EV suc tat for a constant c >, for almost all t (, T ), and κ:= max{, Λ + c η EV } tere olds (.) κ Λ c η EV inf v Y ( v; v) + ε (f (U(t))v; v). v L (Ω) One can ten sow (cf. Proposition 3.) tat for almost all t (, T ) and e:= u U tere olds (.3) d dt e L (Ω) + ε e L (Ω) ε R U Y + c ( + κ) e L (Ω) + d (e; w) dt + c 3 ε ( ) U L (Ω) + w L (Ω) e 3 L 3 (Ω) + c 4ηD, were w H (Ω; R m ) extends (te time-independent) e ΓD and η D = if e ΓD =. Integrating tis equation over (, t) for any t T, using a Sobolev inequality to estimate (.4) e 3 L 3 (Ω) e L (Ω) e L 4 (Ω) c ( 5 e L (Ω) e L (Ω) + e L (Ω)), and setting κ:= ess sup s (,T ) κ(s) yields (.5) t e(t) L (Ω) + ε e L (Ω) ds t t e() L (Ω) + ε R U Y ds + c ( + κ) e L (Ω) ds t t + c 4 ηd ds + 3 w L (Ω) ds + c 3 c 5 ε ( ) U L (,T ;L (Ω)) + w L (Ω) t ess sup s (,t) e L (Ω) (t ess sup s (,t) e L (Ω) + e L (Ω) ). ds We coose η and a constant c 6 > suc tat T (.6) e() L (Ω) + ε R U Y ds + c 4 T T ηd ds + 3 w L (Ω) ds c 6η.

4 4 Since te left-and side of (.5) depends continuously on t, we may conclude tat te set { t I := t [, T ] : ess sup s (,t) e(s) L (Ω) + ε e L (Ω) ds 4c 6η exp ( c ( + κ)t )} is non-empty and we aim to prove tat I = [, T ]. Let t := max I and suppose tat t < T. We use te definition of I and (.6) to derive from (.5) tat for all t t we ave e(t) L (Ω) + ε t were C T,κ := exp( c ( + κ)t ). If or equivalently e L (Ω) ds c 6η + c ( + κ) t + c 3 c 5 ε ( U L (,T ;L (Ω)) + w L (Ω) c 3 c 5 ε ( U L (,T ;L (Ω)) + w L (Ω) e L (Ω) ds ) 3 c 3/ 6 η 3 C 3 T,κ (T + ε ), ) 3 c 3/ 6 η 3 C 3 T,κ(T + ε ) c 6 η, ( (.7) U L (,T ;L (Ω)) + w L (Ω) = or η ε 4 U L (,T ;L (Ω)) + w L (Ω) 8 c 3 c 5 c / 6 (ε T + ) ten we ave for all t t tat e(t) L (Ω) + ε Gronwall s inequality yields t ess sup s (,t ) e(s) L (Ω) + ε e L (Ω) ds c 6η + c ( + κ) t t e L (Ω) ds. e L (Ω) ds c 6η exp ( c ( + κ)t ) ) C 3 T,κ, wic by continuity of te left-and side contradicts t < T and terefore proves I = [, T ]. Te argumentation leads to te following teorem. Teorem.. Suppose tat (.7) olds. Ten tere olds ess sup s (,T ) e(s) L (Ω) + ε T e L (Ω) ds 4c 6η exp ( c ( + κ)t ). In order to make te proof of te teorem complete we only need to prove (.3). In order to make it applicable we need to define a strategy to compute Λ and η EV in (.) and we need to establis a computable estimator η in (.6). Remark. Te error estimate of Teorem. is useful only if κ C for some ε-independent constant C >. Supposing tat u U L ((,T );L (Ω)) C ε for some ε-independent constant C > ten a uniform bound for κ may be deduced from [5] if m = and te zero level set of u is smoot, and from [7] if n = m = and te zeros of u are of topological degree one. 3. PROOF OF THE ERROR EQUATION (.3) We employ te following estimates wic relate te first equation in (P) to its linearization. Lemma 3.. For all a, b, c R m tere olds (f(a) f(b)) (a b c) f (b)(a b c) (a b c) f (b)(a b c) (a b c) a b c, + f (b)c (a b c) (5 b + c ) a b 3 b c 3,

5 5 were f : R m R m m denotes te total derivative of f. Proof. For all d R m tere olds f (b)(d, d) = 4(b d)d + d b and f (b)(d, d, d) = 6 d d and a Taylor expansion of te cubic function f sows f(a) f(b) = f (b)(a b) + (b (a b))(a b) + a b b + a b (a b). We tus ave (f(a) f(b)) (a b c) = f (b)(a b c) (a b c) + f (b)c (a b c) + (b (a b))(a b) (a b c) + a b b (a b c) + a b (a b) (a b c) f (b)(a b c) (a b c) + f (b)c (a b c) 3 b a b a b c a b 3 c. We employ te triangle inequality and Young s inequality to estimate 3 b a b a b c 3 b a b b a b c 5 b a b 3 + b c 3. Te combination of te last two estimates proves te first assertion. For all d R m we ave f (b)d d = ( b ) d + (b d) d and tis proves te second assertion of te lemma. If ε ten te following proposition proves (.3) wit c = 8, c 3 =, c 4 =, and η D = η D. Proposition 3.. Let w H (Ω; R m ) satisfy w ΓD = e ΓD for almost all t (, T ). For almost all t (, T ) tere olds d dt e L (Ω) + ε e L (Ω) ε R U Y + ( ε + ( ε )κ + 3 ) e L (Ω) + d (e; w) dt + ε (5 U L (Ω) + w L (Ω)) e 3 L 3 (Ω) + η D, were η D = ( ) ε + ( ε )κ (ε 4 /) f (U) L (Ω) w L (Ω) + ε U L (Ω) w 3 L 3 (Ω) + 3ε ( ( ε ) + 5/9 ) w L (Ω). Proof. Te governing equations in (P) for u and te definition of te residual R U in (.) lead to e t ; v + ( e; v) + ε (f(u) f(u); v) = R U ; v for all v Y. Te coice v = e w implies d dt e L (Ω) + e L (Ω) + ε (f(u) f(u); e w) = R U ; e w + e t ; w + ( e; w). We employ te first estimate of Lemma 3. wit a = u, b = U, and c = w to verify ε (f(u) f(u); e w) = ε (f(u) f(u); u U w) ε (f (U)(e w); e w) ε (f (U)w; e w) + ε (5 U L (Ω) + w L (Ω)) e 3 L 3 (Ω) + ε U L (Ω) w 3 L 3 (Ω).

6 6 Given any θ [, ] we split te first term on te rigt-and side into two contributions weigted by θ and θ and use te second estimate of Lemma 3. to deduce wile (.) yields θε (f (U)(e w); e w) θε e w L (Ω) ( θ)ε (f (U)(e w); e w) ( θ) (e w) L (Ω) + ( θ)κ e w L (Ω). Hölder inequalities allow us to derive ε (f (U)w; e w) ε 4 f (U) L (Ω) w L (Ω) + e w L (Ω). Young s inequality wit some positive α proves R U ; e w R U Y e w Y 4α R U Y + α e w L (Ω) + α (e w) L (Ω) and ( e; w) α e L (Ω) + α w L (Ω) Te combination of te estimates results in d dt e L (Ω) + e L (Ω) 4α R U Y + ( α + θε + ( θ)κ + / ) e w L (Ω) + ( θ + α) (e w) L (Ω) + e t; w + ε (5 U L (Ω) + w L (Ω)) e 3 L 3 (Ω) + ε 4 f (U) L (Ω) w L (Ω) + ε U L (Ω) w 3 L 3 (Ω) + α e L (Ω) + α w L (Ω). We abbreviate δ := θ α and set ϱ:= ( δ)/δ to deduce wit te elp of Young s inequality ( θ + α) (e w) L (Ω) = ( δ) (e w) L (Ω) ( δ) e L (Ω) + δ ϱ e L (Ω) + ϱ( δ) w L (Ω) + ( δ) w L (Ω) = ( δ/) e L (Ω) δ) + ( w L δ (Ω) + ( δ) w L (Ω) = ( θ/ + α/) e L (Ω) + δ (δ 3δ + ) w L (Ω). Tis, te coices θ = ε and α = ε /, and te identity e t ; w = d (e; w) lead to dt d dt e L (Ω) + ε e L (Ω) ε R U Y + ( ε + ( ε )κ + 3 ) e w L (Ω) + d (e; w) dt + ε (5 U L (Ω) + w L (Ω)) e 3 L 3 (Ω) + ε 4 f (U) L (Ω) w L (Ω) wic proves te proposition. + ε U L (Ω) w 3 L 3 (Ω) + 3ε ( ( ε ) + 5/9 ) w L (Ω) 4. FINITE ELEMENT SPACES AND BOUNDS FOR η D IN (.3) 4.. Notation in finite element spaces. We suppose tat Ω is polygonal or polyedral if n = or n = 3, respectively. Given a regular triangulation T of Ω into tetraedra and quadrilaterals or triangles and parallelograms if n = or n = 3, respectively, we let S (T ) m denote te discrete space of R m valued, continuous functions wic are T -elementwise affine or bilinear and we set SD (T )m := S (T ) m Y. We specify a T -elementwise constant function T L (Ω) by

7 requiring T S = S := diam(s) for all S T. Te set F consists of all faces of elements (edges of elements if n = ) and we assume tat Γ D is matced exactly by te union of all elements in F D := {F F : F Γ D }. Te function F L ( F) is defined by F F := diam(f ) for all F F. For a function ψ C(Γ D ; R m ) suc tat ψ F H (F ; R m ) for all F F we let D F ψ F denote te second weak derivative of ψ F wit respect to a proper coordinate system on F F. Te operator T satisfies T V S = (V S ) for all functions V S (T ) m and all S T. Given a T -elementwise smoot function φ L (Ω; R m n ) let [φ n F ] F := ( φ S φ S ) nf if F F is suc tat F = S S for distinct S, S T and n F R n is te unit vector wic is perpendicular to F and wic points from S into S. For F F suc tat F Ω we set { for F Γ D, [φ n F ] F := φ n Ω for F Ω \ Γ D, were n Ω denotes te outer unit normal to Ω on Ω. Given a function φ C(Ω; R m ) we let I T φ S (T ) m denote te nodal interpolant of φ on T Estimates for η D. Te construction of a function w H (Ω; R m ) wit w ΓD = e ΓD in [3] allows to estimate w L (Ω), w L (Ω), w L (Ω), and η D occurring in Proposition 3.. We tereby obtain a computable quantity (or an upper bound for) η D in te main result of Section. Lemma 4. ([3]). (i) Let T be a regular triangulation of Ω, assume g C(Γ D ; R m ) wit g F H (F ; R m ) for all F F D, and set G:= I T u ΓD. Tere exists w H (Ω; R m ) C(Ω; R m ) suc tat w ΓD = g G, supp w {S T : S Γ D }, and w L (Ω) = g G L (Γ D ) and w L (Ω) c D 3/ F D F g L (Γ D ), were c D > is an ( T, F )-independent constant. (ii) Under te assumptions in (i) and wit w as in (i) tere olds and w L (Ω) c D F L (Γ D ) 3/ F D F g L (Γ D ) w L 3 (Ω) c D F /3 L (Γ D ) 3/ F D F g L (Γ D ), were c D > is an ( T, F )-independent constant. Proof. A proof for (i) can be found in [3]. In order to verify (ii) we use Poincaré inequalities (on patces of elements) to estimate wit an ( T, F )-independent constant C > w L (Ω) = w L (S) C S w L (S) C F L (Γ D ) w L (Ω). S T, S Γ D S T, S Γ D A proof for te second estimate in (ii) ten follows from (.4) wit e replaced by w.

8 8 5. ERROR CONTROL FOR THE COMPUTATION OF Λ IN (.) Tis section discusses practical realizations of (.). For more details on te approximation of eigenvalue problems we refer te reader to [, 6]. Given t [, T ] we set ( v; v) + ε ( f (U(t))v; v ) (5.) λ(t):= inf. v Y v L (Ω) We remark tat a minimizing w in (5.) exists and satisfies for all v Y (5.) ( w; v) + ε ( f (U(t))w; v ) = λ(t)(w; v). Let P λ(t) denote te L projection onto te subspace of all w Y tat satisfy (5.). Te following lemma states an abstract version of (.) under te assumption tat an approximation of a minimizer in (5.) is not L -ortogonal to te exact eigenspace. Lemma 5. ([6]). Let (W, Λ) Y R satisfy (W ; P λ(t) W ) and let r W,Λ Y be suc tat r W ;Λ ; v = Λ(W ; v) ( W ; v) ε ( f (U(t))W ; v ) for all v Y. Ten tere olds Λ r W,Λ; P λ(t) W (W ; P λ(t) W ) = λ(t). Proof. Abbreviate p:= ε f (U(t)) and w := P λ(t) W. Tere olds (w; W ) ( λ(t) Λ ) = ( w; W ) (pw; W ) + ( W ; w) + (pw ; w) + r W,Λ ; w = r W,Λ ; w wic proves te lemma. We derive computable upper bounds for te residual r W,Λ in Lemma 5. for te case tat (W, Λ) is obtained from te following lowest order finite element sceme. { ( (t)) Let (W, Λ) S D (T ) m R satisfy W L EV (Ω) = and for all V SD (T )m ( W ; V ) + ε ( f (U(t))W ; V ) = Λ(W ; V ). ) can be recast as: Find (x, Λ) R l R suc tat Ax = ΛBx Te nonlinear problem ( EV (t) and x (Bx) =. Since we may introduce a constant sift, i.e., a term ε f (U(t)) L (Ω)(W ; V ) in te left-and side of te equation in ( EV (t) ), we may assume tat A and B are positive definite. A solution (x, Λ) can ten be obtained from classical vector iterations. Lemma 5. ([6]). Let (W, Λ) S D (T )m R solve ( EV (t) (5.3) W P λ(t) W L (Ω) /. For k =, set η (k) EV := k T ) and assume tat ( T W ε f (U(t))W + ΛW ) L (Ω) + k / F [ W n F ] L ( F). Let k {, } and suppose tat D φ L (Ω) c, φ L (Ω) for all φ H (Ω; R m ) Y if k =. Tere olds r W,Λ ; P λ(t) W (W ; P λ(t) W ) were c Y := inf v Y v L (Ω) v L (Ω) c EV η (k) EV { (ε f (U(t)) L (Ω) Λ) / for k =, (ε f (U(t)) L (Ω) + c Y ) for k =, and c EV > is an (ε, T, F )-independent constant.

9 9 Proof. Abbreviate p := ε f (U(t)) and w := P λ(t) W. Since r W,Λ ; V = for all V SD (T )m tere olds r W,Λ ; w = r W,Λ ; w V = Λ(W ; w V ) ( W ; (w V )) (pw ; w V ). A T -elementwise integration by parts and Hölder inequalities sow for k = and k =, r W,Λ ; w S T k T ( T W + pw + ΛW ) L (S) k T (w V ) L (S) + F F k / F [ W n F ] L (F ) / k F (w V ) L (F ). Let V be te Clément interpolant of w if k = and its nodal interpolant if k =. Standard estimates imply wit an (ε, T, F )-independent constant C > r W,Λ ; w C ( k T ( T W pw + ΛW ) L (Ω) + k / F [ W n F ] L (F)) D k w L (Ω). Notice tat W Y so tat λ(t) Λ. Since w L (Ω) W L (Ω) = we ave (5.4) Dw L (Ω) = w L (Ω) = λ(t)(w; w) (pw; w) Λ + p L (Ω). If k = ten we ave (5.5) D w L (Ω) c, w L (Ω) c, (λ + p)w L (Ω) c, ( λ + p L (Ω)). We estimate λ in te rigt-and side by noting tat p L (Ω) λ(t) c Y + p L (Ω). Since W w L (Ω) / and W L (Ω) = we ave (w; W ) = w L (Ω) + W L (Ω) w W L (Ω) /, and te combination of te estimates wit Lemma 5. concludes te proof of te lemma. Remark. If n =, Ω is convex, and Γ D = Ω ten tere exists a constant c, > suc tat D φ L (Ω) c, φ L (Ω) for all φ H (Ω; R m ) Y, cf. []. Te following proposition sows tat if U is piecewise affine in [, T ] ten it suffices to approximate λ at a finite number of times. Proposition 5.3. Let t j < t < t j+ T be suc tat t = t j + θ(t j+ t j ) for some θ (, ) and assume tat U(t) = ( θ)u(t j ) + θu(t j+ ). We ten ave ( θ)λ(t j ) θλ(t j+ ) 5 4 ε U(t j ) U(t j+ ) L (Ω) λ(t). Proof. For all v Y tere olds ( θ) ( ( v; v) + (f (U(t j ))v; v) ) + θ ( ( v; v) + (f (U(t j+ ))v; v) ) so tat = ( v; v) + ( f (U(t))v; v ) + ( [( θ)f (U(t j )) + θf (U(t j+ )) f (U(t))]v; v ) ( v; v) + ( f (U(t))v; v ) + ( θ)f (U(t j )) + θf (U(t j+ )) f (U(t)) L (Ω)(v; v) ( θ)( λ(t j )) + θ( λ(t j+ )) λ(t) + ( θ)f (U(t j )) + θf (U(t j+ )) f (U(t)) L (Ω).

10 Terefore, it suffices to verify tat ( θ)f (U(t j )) + θf (U(t j+ )) f (U(t)) L (Ω) 5 4 U(t j) U(t j+ ) L (Ω). Taylor expansions of te quadratic function f sow for a, b R m f (( θ)a + θb) = ( θ)f (a + θ(b a)) + θf (b ( θ)(b a)) = ( θ)f (a) + θf (b) + ( θ)θ(f (a) f (b))(b a)/ + ( θ)θ ( θf (a)(b a, b a) + ( θ)f (b)(b a, b a) ) /6. Tis, explicit formula for f and f, and te estimate θ( θ) /4 imply te assertion. Given any W SD (T )m wit W L (Ω) = tere olds W P λ(t) W L (Ω). Terefore, te assumption (5.3) does not seem to be restrictive. Since it is owever not clear ow to verify it in practice we include an explicit a priori error estimate for te difference λ(t) Λ in case tat te Laplace operator is H regular in Ω. We let c IT be te smallest constant suc tat for all φ H (Ω; R m ) Y tere olds were := T L (Ω). φ I T φ L (Ω) + (φ I T φ) L (Ω) c IT D φ L (Ω), Lemma 5.4. Suppose tat tere exists a constant c, > suc tat D φ L (Ω) c, φ L (Ω) for all v Y H (Ω; R m ). Let (W, Λ) S (T )m R solve ( EV (t) ) and assume tat Ten tere olds c IT c, (c Y + ε f (U(t)) L (Ω)) /. λ(t) Λ c 9 ( cy + ε f (U(t)) L (Ω)) 3/, were te constant c 9 > only depends on c IT and c,. Proof. Trougout tis proof C denotes a generic (ε, )-independent constant. Let w Y satisfy (5.) and w L (Ω) =. We abbreviate λ := λ(t) and p := ε f (U(t)) and define q := p + p L (Ω)I m were I m is te identity matrix in R m m. Ten q L (Ω; R m m ) is positive definite and symmetric almost everywere in Ω. Since W is minimal for V ( V ; V ) + (pv ; V ) among all V S D (T )m wit V L (Ω) = tere olds for all suc V λ Λ w L (Ω) p/ w L (Ω) + V L (Ω) + p/ V L (Ω) = w L (Ω) q/ w L (Ω) + V L (Ω) + q/ V L (Ω) ( V ; (V w) ) + (qv ; V w) ( V L (Ω) + q/ V L (Ω)) / ( (V w) L (Ω) + q/ (V w) L (Ω)) /. Let W := I T w. Te estimate (cf. (5.5)) D w L (Ω) c, w L (Ω) c, (c Y + p L (Ω)) =: c, γ and te assumption on imply W L (Ω) = w L (Ω) W L (Ω) w W L (Ω) c IT c, γ /

11 and ence W L (Ω) /. Set V := W / W L (Ω). Employing te estimate (cf. (5.4)) w L (Ω) γ / and using tat by assumption on tere olds γ / + γ C, we deduce tat W L (V w) L (Ω) (Ω) W W L (Ω) + ( W w) L (Ω) L (Ω) c IT c, γ W L (Ω) + ( W w) L (Ω) c IT c, γ w L (Ω) + ( + c IT c, γ ) ( W w) L (Ω) c IT c, γ 3/ + ( + c IT c, γ )c IT c, γ Cγ. and V L (Ω) (V w) L (Ω) + w L (Ω) Cγ + γ / Cγ /. Similarly, tere olds W q / L (V w) L (Ω) (Ω) W q / W L (Ω) + q / ( W w) L (Ω) L (Ω) c IT c, γ q / L (Ω) + q / L (Ω)c IT c, γ Cγ q / L (Ω) and q / V L (Ω) q / (V w) L (Ω) + q / w L (Ω) q / L (Ω)( Cγ +) C q / L (Ω). A combination of te estimates wit q / L (Ω) Cγ results in wic proves te lemma. λ Λ Cγ / ( q / L (Ω) + γ) Cγ3/ 6. APPROXIMATION OF (P) AND ESTIMATION OF R U Y IN (.6) In tis section we derive a computable upper bound for te residual R U for a semi-implicit finite difference in time and finite element in space discretization of (P) wit a linearized treatment of te nonlinear term. We follow te argumentation of [4]. (P τ, ) Given = t < t <... < t N = T, regular triangulations T, T,..., T N of Ω, U S (T ) m, set τ j := t j t j for j =,,..., N. For j =,,..., N and for all V S D (T j) m let U j S (T j ) m satisfy τ j (U j I Tj U j ; V ) + ( U j ; V ) +ε ( f(i Tj U j ) + f (I Tj U j )(U j I Tj U j ); V ) =, U j ΓD = U ΓD. Given j N and some U j S (T j ) m, existence of a unique solution U j S (T j ) m of te first equation in (P τ, ) is guaranteed if τ j ε f (I Tj U j ) L (Ω). A function U Z L (, T ; L (Ω; R m )) is defined troug a sequence (U j : j =,,..., N) H (Ω; R m ) by setting, for s [t j, t j ] for j N and θ [, ] suc tat s = t j + θ(t j t j ), (6.) U(s) := U(t j ) + θ ( U(t j ) U(t j ) ).

12 For an approximation of u resulting from te solution of (P τ, ) and a subsequent linear interpolation in time, te residual R U can be estimated by fully computable quantities. Te combination of te next lemma wit te estimates for η D and w L (Ω) in Lemma 4. leads to a computable η in (.6). Lemma 6.. Let U Z L (, T ; L (Ω; R m )) be defined troug a solution (U j : j =,,..., N) of (P τ, ) and (6.). Tere olds T were for j =,,..., N, R U Y ds N j= τ j ( ccl η (j) + η (j) t + η (j) c + η (j) ) l, η (j) := ( (τ Tj j + ε f (I Tj U j ) ) L (U j I Tj U j ) Tj U j + ε f(i Tj U j )) (Ω) + / F j [ U j n Fj ] L ( F j ), η (j) t := (U j U j ) L (Ω) + ε ( f (U j ) L (Ω) U j U j L (Ω) η (j) c + f (U j ) L (Ω) U j U j L 4 (Ω) + 6 f (U j ) L (Ω) U j U j 3 L 6 (Ω) := τ j I Tj U j U j L (Ω), ( η (j) l := ε f (I Tj U j ) L (Ω) U j I Tj U j L 4 (Ω) + 6 f (I Tj U j ) L (Ω) U j I Tj U j 3 L 6 (Ω) and c Cl > is an (ε, Tj, Fj )-independent constant. Proof. For almost all s (t j, t j ) and all v Y tere olds R U (s); v = τ j (U j U j ; v) + ( U(s); v) + ε (f(u); v) = τ j ( Uj I Tj U j ; v ) + ( U j ; v) + ε (f(i Tj U j ); v) + ε ( f (I Tj U j ))(U j I Tj U j ); v ) + ( (U(s) U j ); v ) + ε ( f(u(s)) f(u j ); v ) + τ j (I Tj U j U j ; v) + ε ( f(u j ) f(i Tj U j ) f (I Tj U j )(U j I Tj U j ); v). Let r ; v be defined by te first four terms, r t ; v by te fift and sixt term, r c ; v by te sevent term, and r l ; v by te last two terms on te rigt-and side of te equation. Te first equation in (P τ, ) allows to insert te Clément interpolant V S D (T )m of v in r ; v. An elementwise integration by parts and standard estimates for v V yield r ; v = r ; v V c Cl η (j) v L (Ω). Hölder inequalities, a Taylor expansion of f about U j, and linearity of U in s lead to r t ; v (U(s) U j ) L (Ω) v L (Ω) + ε ( f (U j ) L (Ω) U(s) U j L (Ω) + f (U j ) L (Ω) U(s) U j L 4 (Ω) + 6 f (U j ) L (Ω) U(s) U j 3 L 6 (Ω)) v L (Ω) η (j) t v Y. ), ),

13 3 Hölder s inequality proves r c ; v τ j A Taylor expansion of f about I Tj U j gives I Tj U j U j L (Ω) v L (Ω) = η c (j) v L (Ω). r l ; v ε ( f (I Tj U j ) L (Ω) U j I Tj U j L 4 (Ω) + 6 f (I Tj U j ) L (Ω) U j I Tj U j 3 L 6 (Ω) Te combination of te estimates proves te lemma. ) v L (Ω) = η (j) l v L (Ω). Remarks. (i) Te quantities η (j), η(j) t, η c (j), and η (j) l, j =,,..., N, represent space discretization, time discretization, coarsening, and linearization residuals, respectively. (ii) A sarper version of Teorem. can be deduced from te observation in [4] tat R U (s); v c Cl η (j) v L (Ω) + ( η (j) t + η c (j) + η (j) l ) v L (Ω) were s [t j, t j ] and v Y. Tis estimate avoids an additional factor ε for η (j) t η (j) l. (iii) A presumably smaller coarsening estimator η (j) c, η (j), and = τ j Tj (U j Π Tj U j ) L (Ω), were can Π Tj denotes te L projection onto S (T j ) m subject to certain boundary conditions, tan η c (j) be obtained if I Tj is replaced by Π Tj in (P τ, ). c 7. ADAPTIVE ALGORITHM Te error estimate of Teorem. and te local caracter of te computable quantities η D and η allows for te definition of local refinement and coarsening indicators in space and time. Te following algoritm follows ideas in [8] and aims to simultaneously solve (P τ, ) and automatically generate optimal time step sizes τ, τ,..., τ N and triangulations T, T,..., T N if Θ = and Θ t =. For Θ = or Θ t = te algoritm employs te same triangulation T j = T for all j =,,..., N or te same time-step step size τ j = τ for all j =,,..., N, respectively. Given S T j we set η (j) (S) := ( (τ Tj j + ε f (I Tj U j ) ) ) (U j I Tj U j ) Tj U j + ε f(i Tj U j ) + / F j [ U j n Fj ] L ( S). Adaptive Algoritm (A Θ,Θ t ). Input: A regular triangulation T of Ω, an initial time step size τ >, and a termination criterion δ >. (a) Set t :=, U := I T u, and j :=. (b) Set T j := T j and if Θ t = coarsen T j so tat η c (j) δ and T j ΓD = T j ΓD. (c) Set t j := min{t j + τ j, T } and compute U j S (T j ) m suc tat U j ΓD = U ΓD and τ j (U j I Tj U j ; V ) + ( U j ; V ) + ε ( f(i Tj U j ) + f (I Tj U j )(U j I Tj U j ); V ) = for all V SD (T j) m. (c) If η (j) t > δ and Θ t = set τ j := τ j and go to (b). L (S)

14 4 (d) If η (j) > ε δ and Θ = ten refine eac S T j for wic η (j) to obtain a refined regular triangulation T j and go to (c). (e) Stop if t j = T. (f) Coose a regular triangulation ˆT of Ω. (i) Compute (W, Λ (j) ) S D ( ˆT ) m R suc tat W L (Ω) = and ( W ; V ) + ε ( f (U(t j ))W ; V ) = Λ (j) (W ; V ) for all V S D ( ˆT ) m. (ii) If Θ t =, Θ = and η EV refine ˆT and go to (i). (g) Set j := j +, τ j := τ j, and go to (b). (S) max S T j η (j) (S ) If T and U are suc tat e() L (Ω) δ and w L (Ω) + η D δ, if te algoritm terminates, and if δ Cε 4 ten Teorem. implies ess sup s (,T ) e(s) L (Ω) + ε provided tat ( Λ (j)) is uniformly bounded from above. T e L (Ω) ds Cδ 8. NUMERICAL EXPERIMENTS In tis section we discuss te practical applicability of te error estimate of Teorem. by specifying (P) troug te following tree examples. Example (Allen-Can equations). Given ε > set n :=, m :=, Ω := (, ), Γ D :=, T :=, and for x Ω let u (x):= tan ( ( x )/ε ). Example (Ginzburg-Landau equations I). Given ε > set n = m:=, Ω:= (, ), Γ D := Ω, T := 5/, and u D := ũ D Ω for ũ D (x):= x/ x. Set a:= (, )/4 and for x Ω let u (x):= θ ( dist(x, Ω) ) ũ D (x) + ( θ ( dist(x, Ω) )) x a ( x a + ε ), / were θ(s) = 48s + 8s 3 for s /4 and θ(s) = for s /4. Example 3 (Ginzburg-Landau equations II). Given ε > set n = m :=, Ω := (, ), Γ D := Ω, and T :=. We identify R wit te complex plane, define for x Ω and set u D := u Ω. u (x):= x x + ε, Algoritm (A Θ,Θ t ) was implemented in Matlab wit a direct solution of linear systems of equations and an assemblation of stiffness matrices in C. Te adaptive refinement strategy was realized by standard bisection approaces and te mes coarsening was acieved as follows: given a (locally refined) triangulation T f, U S (T f ), δ >, and a coarse triangulation T c, locally refine T c until I T U U L (Ω) δ. Tis approac may be suboptimal but worked reliably in practice. In all te experiments reported below, te overall CPU time (on a node of a Compaq SC-Cluster

15 wit four Alpa-EV68 processors ( GHz, 8 MB Cace/CPU) and 3 GB RAM) of Algoritm (A Θ,Θ t ) was at most one week. 8.. Validity and failure of a uniform upper bound for Λ in (.). Te following numerical experiments reveal practical limitations of te error estimate of Teorem. in te sense tat Λ may not be uniformly bounded from above ε-independently if critical topological effects occur. We ran Algoritm (A, ) in Example for ε = /4, /8, /6, initial uniform triangulations T of Ω = (, ) wit maximal messizes = /8, /6, /3, respectively, and wit τ := ε 4. Figure sows te numerical solution U(t) for ε = /8 and t =.488,.93,.396, We observe tat te pases U and U are separated by a sarp interface, tat te region in wic U becomes smaller and tat te interface finally collapses for t.48. Figure displays for ε = /4, /8, /6 te eigenvalues Λ j computed in step (f) of Algoritm (A Θ,Θ t ) wit ˆT = T as functions of t [, ]. Te eigenvalues are uniformly bounded for t.3 and grow proportionally to ε for t (.3,.5). For t.5 we see tat Λ j is bounded from above by, corresponding to te stable solution u. Teorem. may terefore be employed for t.3 provided tat η Cε 4. In te region t (.3,.5) we would ave to ensure tat η C/ exp(c ε ), wic can only be expected to old if and τ are unrealistically small, i.e., satisfy, τ C/ exp(c ε ). We cose ε = /4, /8, /6, uniform triangulations T of Ω = (, ) wit maximal messizes = /8, /6, /3, respectively, and τ = ε 4 in order to approximate (P) specified troug Example wit Algoritm (A, ). We see in Figure 5 tat te vortex, initially located at (/, /), moves to te origin and te solution reaces a stable state. For all coices of ε we plotted te eigenvalues Λ j as functions of t [,.5] in Figure 4. As in te previous experiment, te same triangulation ˆT = T was used to compute Λ j in eac time step. In tis example te eigenvalues are uniformly bounded from above by 3. Tis is in good agreement wit teoretical results in [7] wic state tat degree-one vortices in Ginzburg-Landau equations are stable. Terefore, in tis example Teorem. can be used in practice to control te discretization error. A teoretical result in [4] states tat iger degree vortices in Ginzburg-Landau equations are unstable. Tis is numerically confirmed by simulations for Example 3 were te function u as a degree- vortex at te origin. Te snapsots of te numerical solutions generated by Algoritm (A, ) for ε = /8, = /6, and τ = ε 4, and displayed in Figure 5 for t =,.953,.396,.5859 sow tat te degree- vortex immediately splits into two degree one vortices wic repulse eac oter and reac a steady state for t. Te computed eigenvalues Λ j calculated in step (f) of Algoritm (A, ) in Example 3 wit ε = /4, /8, /6, uniform triangulations wit maximal mes-sizes = /8, /6, /3, respectively, and τ = ε 4, are sown in Figure 6. We observe tat te eigenvalues satisfy Λ(t) ε for t.3 and are uniformly bounded from above wen te two degree-one vortices are well-separated for t Performance of te automatic mes refinement and coarsening strategy. Figure 7 displays te triangulations T j for j =, 8, 6 automatically generated by Algoritm (A, ) wit an initial uniform triangulation T wit maximal mes-size = /6 and a (uniform) time-step size τ = ε 4 in Example wit ε = /8. Te algoritm efficiently resolves te interface of te numerical solution and employs a coarse mes in te remaining part of Ω. Moreover, te refined region moves togeter wit te interface towards te origin. 5

16 FIGURE. Numerical solution U(t j ) for j =,, 6, 9 in Example wit ε = /8. Te zero level set of U is a circle wic becomes smaller and vanises for t ε = /4 ε = /8 ε = /6 Λ t FIGURE. Computed eigenvalue Λ in Example. As te interface collapses Λ grows proportionally to ε.

17 FIGURE 3. Numerical solution U(t j ) for j =,, 3, in Example wit ε = /8. Te vortex is initially located at (/, /) and moves to te origin. 3 Λ ε = /4 ε = /8 ε = / t FIGURE 4. Computed eigenvalue Λ in Example. A uniform upper bound olds.

18 FIGURE 5. Numerical solution U(t j ) for j =, 5, 8, in Example 3 wit ε = /8. Te initial degree- vortex splits into two degree- vortices. 8 6 ε = /4 ε = /8 ε = /6 Λ t FIGURE 6. Computed eigenvalue Λ in Example 3. We observe a significant dependence on ε for t.3 wile a uniform upper bound for Λ olds once te two degree- vortices are well-separated.

19 FIGURE 7. Adaptively refined and coarsened triangulations T j for j =, 8, 6 in Example wit ε = /8. An automatic refinement towards te zero level set of U and a coarsening in te remaining part of te domain are observable FIGURE 8. Numerical solution and adaptively refined and coarsened triangulations T j for j =,,, 4 in Example wit ε = /8. Eac arrow corresponds to a vertex in te triangulation.

20 Te automatically refined and coarsened triangulations T j togeter wit te numerical solution U(t j ) for j =,,, 4 as outputs of Algoritm (A, ) in Example wit ε = /8, an initial uniform triangulation wit maximal mes-size = /3, and a uniform time step size τ = ε 4 are displayed in Figure 8. Te coarsened triangulation T sows a iger resolution in a boundary layer wic stems from a practically non-smoot u. In all displayed solutions we observe a smaller local mes-size around te moving vortex. For numerical evidence of improved experimental convergence rates obtained by a related automatic mes refining strategy we refer te reader to [4]. Acknowledgments. S.B. is tankful to G. Dolzmann and R.H. Nocetto for stimulating discussions. Tis work was supported by a fellowsip witin te Postdoc-Programme of te German Academic Excange Service (DAAD). REFERENCES [] S. ALLEN, J. W. CAHN: A microscopic teory for antipase boundary motion and its application to antipase domain coarsening. Acta. Metall. 7 (979), [] I. BABUŠKA, J. OSBORN: Eigenvalue problems. In Handbook of numerical analysis II, Nort-Holland (99), [3] S. BARTELS, C. CARSTENSEN, G. DOLZMANN: Inomogeneous Diriclet conditions in a priori and a posteriori finite element error analysis. Berictsreie des Matematiscen Seminars Kiel, Tecnical report -, Cristian Albrects Universit ät zu Kiel, Kiel (). [4] A. BEAULIEU: Some remarks on te linearized operator about te radial solution for te Ginzburg-Landau equation. Nonlinear Anal. 54 (3), [5] X. CHEN: Spectrum for te Allen-Can, Can-Hilliard, and pase-field equations for generic interfaces. Comm. Partial Differential Equations 9 (994), [6] P. DE MOTTONI, M. SCHATZMAN: Geometrical evolution of developed interfaces. Trans. Amer. Mat. Soc. 347 (995), [7] Q. DU, M. GUNZBURGER, J. PETERSON: Analysis and approximation of te Ginzburg-Landau model of superconductivity. SIAM Rev. 34 (99), [8] C. M. ELLIOTT: Approximation of curvature dependent interface motion. Te state of te art in numerical analysis, Inst. Mat. Appl. Conf. Ser. New Ser., 63, Oxford Univ. Press (997), [9] X. FENG, A. PROHL: Numerical analysis of te Allen-Can equation and approximation for mean curvature flows. Numer. Mat. 94 (3), [] X. FENG, A. PROHL: Numerical analysis of te Can-Hilliard equation and approximation of te Hele-Saw problem. Part I: Error analysis under minimum regularities. SIAM J. Numer. Anal. (to appear). [] X. FENG, A. PROHL: Analysis of a fully discrete finite element metod for te pase field model and approximation of its sarp interface limits. Mat. Comp. 73 (4), [] D. GILBARG, N. S. TRUDINGER: Elliptic partial differential equations of second order. Springer, Berlin (). [3] V. GINZBURG, L. LANDAU: On te teory of superconductivity. Z. Èksper. Teoret. Fiz. (95), In Men of Pysics: L.D. Landau, D. TER HAAR, ed. Pergamon, Oxford (965), [4] D. KESSLER, R. H. NOCHETTO, A. SCHMIDT: A posteriori error control for te Allen-Can problem: circumventing Gronwall s inequality. MAN (to appear) (4). [5] D. KESSLER, R. H. NOCHETTO, A. SCHMIDT: A duality approac to a posteriori error control for te Allen- Can problem. Preprint (3). [6] M. G. LARSON: A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems. SIAM J. Numer. Anal. 38 (), [7] E. H. LIEB, M. LOSS: Symmetry of te Ginzburg-Landau minimizer in a disc. Mat. Res. Lett. (994), [8] R. H. NOCHETTO, A. SCHMIDT, C. VERDI: A posteriori error estimation and adaptivity for degenerate parabolic problems. Mat. Comp. 69 (999), 4.