CONVERGENT DISCRETIZATION OF HEAT AND WAVE MAP FLOWS TO SPHERES USING APPROXIMATE DISCRETE LAGRANGE MULTIPLIERS

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1 CONVERGENT DISCRETIZATION OF HEAT AND WAVE MAP FLOWS TO SPHERES USING APPROXIMATE DISCRETE LAGRANGE MULTIPLIERS SÖREN BARTELS, CHRISTIAN LUBICH, AND ANDREAS PROHL Abstract. We propose fully discrete scemes to approximate te armonic map eat flow, and te wave map into speres. Te proposed finite-element based scemes preserve a unit lengt constraint at te nodes by means of approximate discrete Lagrange multipliers, satisfy a discrete energy law, and iterates are sown to converge to weak solutions of te continuous problem. Comparative computational studies are included to motivate finite-time blow-up beavior in bot cases.. Introduction Let Ω R d for d be a bounded domain, and S m R m for m te unit spere. Te energy of a map w : Ω S m is defined as. Ew = w dx. Ω Critical points are called weakly armonic maps into te spere ], wic are of interest in more extended models in micromagnetics 5], liquid crystal teory ], color image denoising, 3, 5], or in generalized form in general relativity 8]. Related prototype nonstationary problems for solutions u : Ω T S m are A te L -gradient flow for.,..3.4 u t u = u u in Ω T :=,T Ω, u n = on Ω T :=,T Ω, u, = u in Ω, and B te wave map flow into te spere S m, u tt u = u u t u in Ω T, u n = on Ω T, u, = u, t u, = v in Ω. In bot cases, static solutions are armonic maps to te spere; clearly, evolution is different in A and B. For Problem A, existence of weak solutions can be found in ]; te development of singularities in finite time, i.e., lim sup t T ut, W, =, is sown in, 9] for equivariant initial data. As to Problem B, stable self-similar finite-time blow-up beavior of existent weak solutions cf. 7, 7, 4] is known in te 3+-dimensional case using equivariant initial data, and existence of k-equivariant solutions wit winding number k 4 in a +-dimensional Minkowski space-time setting is known wic sow finite-time blow-up 6, 8]. Numerical analysis of Problems A and B is nontrivial for te following reasons: Date: Marc, 7.

2 In order to approximate or construct in te limit weak solutions in bot cases A and B by using fully practical scemes based on finite elements, we cannot benefit from regularity properties of solutions in bot cases, but need to verify crucial stability properties, like discrete spere constraint and discrete energy identity. Straigtforward spatio-temporal discretizations, togeter wit standard finite elements violate te spere constraint, and lack a discrete energy law; see e.g. 3, 3]. 3 Convergent penalization strategies of Problems A and B wic use a Ginzburg-Landau penalty wit parameter ε > to approximate te spere constraint allow for convergent discretizations for every ε > ; owever, to specify tis parameter in terms of discretization parameters is a nontrivial task, in particular in te context of blow-up beavior of weak solutions to tese problems; cf. 6] for computational evidence for Problem B. In 3, 4], a fully practical implicit sceme is given to solve Problem A, wic is based on a reformulation of. using cross products m = 3, u t + u u u = in Ω T. Togeter wit reduced spatial integration, midpoint formula, as well as projected discrete Laplacian, a lowest order conforming finite element discretization enjoys a discrete spere constraint and energy law, and solutions unconditionally sub-converge to weak solutions; moreover, a simple fixed point strategy is proposed to successively solve linear problems, wose solutions still satisfy te discrete spere constraint, and conditionally converge to weak solutions by a contraction argument. Unfortunately, tis strategy is not successful for Problem B; instead, an explicit time-splitting sceme was proposed for Problem B in 6], and conditional convergence towards weak solutions is verified; ere te idea, wic originally was given in ] in a different context, is to discretize te following reformulation of.5, wic uses special test-functions w C,T Ω, R m wic satisfy u,w = a.e. in Ω T, suc tat t u,wdt + u, wdt =. In tis work, we construct different convergent discretizations for bot problems A and B, wic use approximate discrete Lagrange multipliers: To motivate te approac, recall tat to describe te gradient flow for. requires mappings u : Ω T R m, and a Lagrange multiplier λ : Ω T R +, suc tat u t u = λu and u = in Ω T. In fact, λ = u is easy to verify in te present case, were te target manifold is te spere. Te following sceme uses an approximate discrete Lagrange multiplier to enforce bot te discrete spere constraint, i.e., unit lengt of iterates of finite element functions at nodes of a triangulation T, and a discrete energy law. We employ some notation wic is furter detailed in Section. Let V W, Ω be te lowest order conforming finite element space subordinate to a triangulation T of Ω, and V = V ] m. By N, we denote te set of all nodes associated wit T. Below,, denotes te discrete version reduced integration of te inner product in L Ω, R m, and we use d t ϕ n := k ϕ n ϕ n, and ϕ n+/ := ϕn+ + ϕ n wit n, for a sequence {ϕ n } n, and for an equidistant time-step size k >. Ten, te approximation sceme for Problem A reads as follows. Algoritm A. For n, let U n V be given, and find U n+,λ n+ V V, suc tat.8.9 d t U n+,φ + U n+/, Φ = λ n+ U n+/,φ Φ V, U n+ z = z N.

3 As will be sown in Section 3, an explicit formula to compute λ n+ = λ n+ U n+/ is available; owever, in contrast to te continuous Lagrange parameter above, its computation at a single node z N requires to consider values of U n+/ at neigboring ones wic accounts for finite sizes k, > in te discretization sceme. Conditional solvability of Algoritm A olds by Lemma 3.. Te metod is devised suc tat a discrete energy identity olds: coose Φ = d t U n+ in.8, and use reduced integration for te first and last term, togeter wit U n+ z U n z =, for all z N, to obtain d t U n+ + U n+/, d t U n+ =, and after summation over all iteration steps n N, N. EU N+ + k d t U n+ = EU. n= As is worked out in Section 3, tis discrete energy law is ten crucial to verify subsequence convergence of iterates from Algoritm A to weak solutions of..4; see Teorem 3.. Remark.. Note tat λ n+ V for n is not te Lagrange multiplier associated to te discrete spere constraint U n+ z = for all z N, since te rigt-and side of.8 is modified from λ n+ U n+,φ to λ n+ U n+/,φ to obtain te discrete energy law.. A similar program is now evident for te wave map problem.5.7, were.5 is of te form u tt u = λu and u = in Ω T, wit λ = u u t in te present form. In Section 4, we sow conditional convergence of te following implicit discretization, wic again uses approximate discrete Lagrange multipliers. Algoritm B. For n, let U n,u n V be given, and find U n+,λ n+ V V, suc tat.. d t Un+,Φ + U n+/, Φ = λ n+ U n+/,φ Φ V, U n+ z = z N. Te second starting value is cosen as U = U + kv, were V is te initial velocity. As for Problem A, a discrete energy law can be sown, using V n,w n = V n + Wn ],.3 dt U N+,U N+ + k N d tu n+ = V,U N n= and conditional solvability; cf. Lemma 4.. As is evident from te second term in.3, Algoritm B uses numerical dissipation, wile a symmetric conservative discretization of.5.7 is discussed in Remark 4.. However, in te analysis for Algoritm B in Section 4, its dissipative caracter is needed to conclude convergence of iterates towards weak solutions of.5.7; see Teorem 4.. Te main result for te eat flow of armonic maps is Teorem 3., wic verifies subsequence convergence of iterates from te implicit Algoritm A towards weak solutions of Problem A, provided tat k C ; tis unexpected mes constraint is sufficient for solvability for finite k, >, wereas bot, te discrete energy law and te spere constraint do not require mes constraints. We remark tat no constraint is required in 3] for te special case were te target is te two-dimensional spere. For te wave-map equation, conditional convergence towards weak solutions of Problem B is verified in Teorem 4.; again, only Brouwer s fixed-point argument to verify existence of iterates requires a mes-constraint k min{d/,}, wile discrete energy law and 3

4 spere constraint old unconditionally. For comparison, to validate a sligtly perturbed discrete energy law and eventually conclude convergence for te splitting-based algoritm in 6] requires te more restrictive mes-constraint k o 4+d 3 to old. Interestingly, te fixed point iterations employed to solve te nonlinear systems of equations in te numerical experiments reported in Section 5 seem to converge exactly under tese constraints, wic indicates tat our results may be sarp. Te remainder is organized as follows: Section collects some notations wic ared used trougout te paper. Section 3 verifies convergence of Algoritm A to obtain weak solutions of Problem A in te limit k,. Section 4 correspondingly sows convergence of Algoritm B towards Problem B. Computational experiments to motivate possible blow-up for bot Problems are reported in Section 5, and are compared wit corresponding studies in 3] armonic map eat flow, and 6] wave map equation.. Preliminaries Standard notation is adopted trougout tis paper., denotes te standard inner product of te Euclidean space R d,, :=, Ω is te standard L -inner product over te domain Ω R d. W l,p Ω, R m denotes te l,p-sobolev space of vector-valued functions, and W l,p its norm. Trougout tis paper, C > is used as a generic positive k,-independent constant wic may take different values at different locations. We also introduce u t := t u, u := x u,.., xd u, D := t,, and define te nonlinear Sobolev space W, Ω, S m = { u W, Ω, R m ; u S m a.e. in Ω }. were boldface letters are used for vector-valued quantities. For simplicity, let Ω be a bounded polygonal wen d = or polyedral wen d = 3 domain. Let T denote a quasiuniform triangulation of Ω into triangles or tetraedra wit mes size > for n = or n = 3, respectively. For a domain K R d, let P K stand for te set of all affine functions on K. We define te Lagrange finite element spaces V := { w CΩ; w K P K K T }, V := V ] m. Let N denote te set of all nodes associated wit te finite element space V, and { } ϕ z ; z N te nodal basis for V ; we define te following nodal interpolation operator I : CΩ V by I w := wzϕ z w CΩ. z N For any two functions v,w CΩ, R m, we define a discrete L -inner product by v,w := I v,w dx = β z vz,wz, Ω z N were β z = Ω ϕ z dx, for all z N. We also define w := w,w. It is easy to ceck tat tere olds for all v,w V, w L w d + w L, v,w v,w C v L w L. 4

5 3. Harmonic map eat flow to te spere We numerically approximate weak solutions of..4 in te sense of ]. Definition 3.. Given u W, Ω, S m, a function u W, Ω T, R m is called a weak solution of..4 if for all T > tere old i u, = u W, Ω, S m in te sense of traces, ii u = almost everywere in Ω T, iii for almost all T,T tere olds 3. Ω ut,x dx + t ut, dt Ω u x dx, and iv for all φ C Ω T, R m tere olds 3. t u,u φ dxdt + u, u φ dxdt =. Ω T Ω T For m = 3 we ere ave te usual wedge or cross product in R 3. For m > 3, an expression u,v w is to be interpreted as te 3-volume of te parallelepiped in te vector space spanned by u,v,w. Since tis 3-volume still turns out to depend linearly on eac of u,v,w, te usual rules for differentiation,... as for te cross product in R 3 apply also in te iger-dimensional case. We recapitulate Algoritm A were te approximate discrete Lagrange multiplier is specified to study necessary conditions for well-posedness for finite k,, and convergence beavior for k,. Algoritm A. Let U V, wit U z = for all z N. For n =,,,.., find U n+,λ n+ V V, suc tat for all Φ V, and all z N tere olds d t U n+,φ + U n+/, Φ = λ n+ U n+/,φ, { if U n+/ z =, λ n+ z = U n+/,u n+/ z ϕ z β z U n+/ z Next, we verify solvability for Algoritm A for restricted coices k = O, and quasiuniform meses T. Te proof uses a regularization in a first step to apply Brouwer s fixed-point teorem; ten, solutions are sown to satisfy , and discrete versions of te spere constraint and te energy law. In te following, we use te notation for te energy given in.. Lemma 3.. Let T be a quasiuniform triangulation of Ω R d, and U V suc tat U z = for all z N. For sufficiently small C = CΩ, T > independent of k, > suc tat k C, tere exists U n+ V wic satisfies , enjoys U n+ z = for all z N, and N 3.5 EU N+ + k d t U n+ = EU N. n= Proof. Step. Fix n. For every ε >, and all Φ V, define te continuous mapping F ε : V V, were 3.6 Fε W,Φ := k {W Un },Φ W,Wz ϕz + W, Φ β z max{ Wz,ε} ϕ zw,φ. z N We compute β z W,Wz ϕ z max{ Wz ϕ z, W,ε} 3.7 = else. Wz W,Wz ϕz max{ Wz,ε} W, Wzϕ z ] supp ϕ z C W, Wz supp ϕz. 5

6 For all W = Φ suc tat W U n, and values k C for some existing < C CΩ, on using Young s inequality, and te fact tat te number of nodes y N suc tat ϕ y, ϕ z is bounded independently of >, Fε W,W W k U n,w + W C W W k W Ck W U n + W, and a corollary to Brouwer s fixed-point teorem 9, p. 37] ten implies existence of U n+/ V, suc tat F ε U n+/ =. Step. We sow tat U n+/ V solves F U n+/ =. For tis purpose, it suffices to sow for all z N parallelogram identity 3.8 U n+/ z = + U n+ z k d tu n+ z = k d tu n+ z! >. Tanks to 3.6, te iterate U n+ := U n+/ U n satisfies for all Φ V, 3.9 dt U n+,φ + Un+/, Φ = U n+/,u n+/ z ϕ z β z max{ U n+/ z ϕ z U n+/,φ,ε}. z N On putting Φ = d t U n+ zϕ z for z N, and using properties of reduced integration wit U n+ z U n z =, and inverse estimates, we ave β z d t U n+ z U n+/,d t U n+ z ϕ z d t U n+ z U n+/ L ϕ z L dt U n+ z C U n+/ L ϕ z L. We may ten conclude tat dt U n+ z C. Hence, assertion 3.8 is valid values k C, for some C CΩ >. Convergence beavior of iterates {U n } of Algoritm A towards weak solutions of..4 for k, is verified below. In te sequel, we define U k, : Ω T S m, were for all t,x t n,t n+ ] Ω, U k, t,x := t t n k U n+ x + t n+ t U n x, k and U + k, t,x := Un+ x resp. U k, t,x := U n+/ x for all t,x t n,t n+ Ω. Teorem 3.. Let te assumptions of Lemma 3. be valid, Eu <, and U u W, Ω, S m for. Tere exists a subsequence of {U k, } wic for k, converges weakly in W, Ω T, R m to a weak solution of..4. Proof. Step. Te bounds of Lemma 3. yield te existence of convergent subsequences {U k, }, and u W, Ω T, R m suc tat for k C d, and, U k,,u + k,,u k, u in L,T;W, Ω, R m, U k,,u + k,,u k, u in L Ω T, R m, t U k, u t in L Ω T, R m. 6

7 Since U + k, = for all z N and all t,t], tere olds I U + k, ] = for all t,x,t] Ω, and for all K T U + k, L K C U + k, ] L K C U + k, T U + k, L K C U + k, L K. As a consequence, U + k, almost everywere in Ω T, and ence u = almost everywere. We use weak lower semicontinuity of norms and U u in W, Ω, R m to conclude from 3.5 tat u W, Ω, S m satisfies 3.. Since te trace operator is bounded and linear, it is weakly continuous as an operator from W, Ω T into L Ω, and we deduce u, = u in te sense of traces. Step. It remains to verify property 3. for u. For tis purpose, we rewrite 3.3 as 3. t U k, t,,φ + U k, t,, Φ = Uk, t,,u k, t,z ϕ z ϕ β z U k, z z U k, t,,φ z N for Φ V, and all t,t. Let Ψ C Ω T, R m ; tanks to a b,a =, te coice Φ = I Uk, t, Ψt, ] ten leads to 3. t U k, t,,u k, t, Ψt, + U k, t,, I Uk, t, Ψt, ] =. We compute t U k,,u k, Ψ t u,u Ψ 3. = t U k,, I Uk, Ψ ] t U k,, I Uk, Ψ ] + t U k,, I Uk, Ψ ] U k, Ψ + t U k,,u k, u] Ψ + t U k, u],u Ψ. Te properties of,, W, Ω-stability of I, and U k, L yield t U k,, I Uk, Ψ ] t U k,, I Uk, Ψ ] Similarly, C t U k, I Uk, Ψ ] C t U k, U k, + Ψ W,. t U k,, I Uk, Ψ ] U k, Ψ C t U k, U k, + Ψ W,. Convergence towards zero of te last two terms in 3. follows from U k, u in L Ω T, R m, and t U k, t u in L Ω T, R m, and 3., 3.5. Summing up, we find for k C, 3.3 lim k, tu k,,u k, Ψ = t u,u Ψ Ψ C Ω T, R m. Next, we wis to verify for te second term in 3. tat in case k C, 3.4 lim Uk,, I Uk, Ψ ] = u, u Ψ] Ψ C Ω T, R m. k, Terefore, on using te identities U k,, {U k, Ψ} = U k,,u k, Ψ, and u, {u Ψ} = u,u Ψ almost everywere, Uk,, I Uk, Ψ ] u, u Ψ] = U k,, {I Uk, Ψ ] U k, Ψ} + U k,,u k, u] Ψ + U k, u],u Ψ =: I + II + III. We compute I C U k, U k, + Ψ W,, by an interpolation estimate, using D U k, K = for all K T. For te terms II resp. III, we use U k, u in L Ω T, R m, and U k, u 7

8 in L Ω T, R m, respectively to conclude tat II,III, for k C, and k,. Terefore, te limit u : Ω T R m satisfies Wave map to te spere We recall te notion of weak solutions to.5.7. Below, let E v,w := v + w ]. Definition 4.. Given T > and u,v W, Ω, S m L Ω, R m wit u,v = a.e. in Ω, we call u : Ω T R m a weak solution of.5.7 if i Du L Ω T, R m, ii u = almost everywere in Ω T, iii T ut u,φ t dt + u u, φ dt = v u,φ, iv E u t t,,ut, Ev,u t, φ C,T;W, Ω T, R m, v ut, u in W, Ω; R m, u t t, v in L Ω, R m t. We numerically approximate weak solutions of.5.7. Next, we give an explicit formula for λ n+ in Algoritm B to study well-posedness. Algoritm B. Given U n,u n V, find U n+,λ n+ V V, suc tat for all Φ V and all z N, tere olds d tu n+,φ + U n+/, Φ = λ n+ U n+/,φ, { for U n+/ z =, λ n+ z = dtun z + d tu n z,d tu n+ z ]+ U n+/ z,u n+/ z ϕ z β z U n+/ z else. Below, te discrete energy is denoted as V n,w n = V n + Wn ]. In te following, restricted coices k = O min{d/,}, and quasiuniform meses T are sufficient to verify solvability of Algoritm B. Te proof uses a regularization in a first step to apply Brouwer s teorem; ten, solutions are sown to satisfy discrete versions of te spere constraint and te energy law in te case of a mes constraint, and converge to weak solutions of.5.7. In te following, set U = U +kv wit te given initial velocity V, and let V n := d t U n for n. Lemma 4.. Let T be a quasiuniform triangulation of Ω R d, and U,V V V wit U z = and U z,v z = for all z N. For n, for sufficiently small C = CΩ, T > independent of k, > suc tat k C min{d/,}, tere exists U n+ V, wic satisfies Algoritm B, as U n+ z = for all z N, and 4. V N+,U N+ + k N d t V n+ = V,U N. n= A verification of tis lemma follows te steps of te proof of Lemma 3. in te present case. Proof. Te discrete energy law 4. follows from te first equation in Algoritm B, on coosing Φ = d t U n+. Step. Fix n. For every ε >, and all Φ V, define te continuous function F ε : V V, were Fε W,Φ := k W + U n / U n,φ + W, Φ ] Iz ε W;Φ + IIε z W;Φ, 8 z N

9 for We compute and 4. I ε zw;φ := II ε zw;φ := z N I ε z dt U n z + k d tu n z,wz U n z ] β z max{ W,ε} W,Wz ϕz β z max{ W,ε} ϕ zw,φ W;Wzϕz Wz IIz ε W;Wzϕz max{ Wz,ε}. C d t U n + 4k W Un, = ϕ z W,Φ C d t U n + ] 4k W Un Wz W,Wz ϕz max{ Wz,ε} W, Wzϕ z ] supp ϕ z C W, Wz supp ϕz. For all W = Φ suc tat W G { U n i W,} i=,, d t U n > sufficiently large, we ten find for sufficiently small, positive C CΩ, and values k C, using Young s inequality, and te fact tat te number of nodes y N suc tat ϕ y, ϕ z is bounded independent from >, Fε W,W k W 4 U n /,W U n,w + W C d t U n / 4k W Un C W W k W Ck W 4 U n / 4 U n + W. and Brouwer s fixed-point teorem ten implies existence of U n+/ V, suc tat F ε U n+/ =. Step. We sow tat U n+/ V solves F U n+/ =. For tis purpose, it suffices to sow for all z N parallelogram identity 4.3 U n+/ z = + U n+ z k d tu n+ z k d tu n+ z! >. Te iterate U n+ := U n+/ U n V satisfies for all Φ V, d 4.4 t U n+,φ + U n+/, Φ = IzU ε n+/ ;Φ + IIzU ε n+/ ;Φ. z N On putting Φ = d t U n+, and using U n+ z = U n z, for all z N yields d t d t U n+ + Un+ ] + k d tu n+ =. By inverse estimate, for values k C, k d tu n+ L Ck d d t U n+ Ck d d t U n + Un ]. Hence, assertion 4.3 is valid for values k C d/, tanks to 4.. 9

10 Similarly, we may derive k C as anoter sufficient bound to verify 4.3: on setting Φ = d t U n+ zϕ z for z N in 4.4, using U n+ z = U n z, and inverse estimates, we obtain β z d t d t U n+ z + k d tu n+ z ] U n+/,d t U n+ z ϕ z d t U n+ z C U n+/ L β z. By binomial formula, d t ϕ n ] = ϕ n + ϕ n ] d t ϕ n for n, and a given sequence {ϕ n } n, and we ten conclude d t d t U n+ z C z N. Ten, given T >, by discrete Gronwall s lemma tere exists C CΩ,T >, suc tat for k C tere olds k d t U n+ z. Step 3. Te last step deletes te first possibility in 4., provided k C min{d/,}. In order to compute λ n+ z, for every z N, we put Φ = U n+/ zϕ z. For te leading term, we find d t Un+ z,u n+/ z = ] 4.5 d t U n z + d t U n z,d t U n+ z, tanks to d t U n+ z = for all z N. Convergence beavior of iterates {U n } V of Algoritm B towards weak solutions of.5.7 for k, is verified below. In te sequel, given {Φ n } V, we define Φ k, : Ω T R m, were for all t,x t n,t n+ Ω, Φ k, t,x := t t n Φ n+ x + t n+ t Φ n x, k k Φ + k, t,x := Φn+ x, Φ k, t,x := Φ n+/ x. Subsequently, we drop sub-indices k, and use U +,U,U resp. λ +,λ, and V +,V to stand for U + k,,u k,,u k,, resp. λ + k,,λ k,, and V + k,,v k,. For Φ C,T;V, on putting V n+ = d t U n+, we may rewrite te first equation of Algoritm B as follows, 4.6 V t,φ + U, Φ λ + U,Φ ] dt =. We first derive te following reformulation of Algoritm B. Lemma 4.. Suppose tat te assumptions of Lemma 4. are valid. Tere olds for all Ψ,T;C Ω, R m, C 4.7 U t,u Ψ] t + U, I U Ψ] ] dt V,U, Ψ, V + V,U Ψ] t dt + Ck / V,U Ψ L Ω T, were U, = k V + U. Proof. Let Ψ C,T;C Ω, R m, and take Φ = I ] U Ψ. We use te following identity in Ω, 4.8 Ut, = Ut, + t n+ + t n t ] V + t, t t n,t n+

11 to conclude V t,u U] Ψ k V t V + Ψ L, and obtain te following bound by 4., and Young s inequality after integration in time, Vt,U U] Ψ / Ψ L dt Ck/ k V t t, dt Ω T C k V,U Ψ L Ω T. For te remaining term, we use integration by parts, and V + t, = U t t, in Ω, for all t t n,t n+, T V,U Ψ] t V dt = V + ] U t,u Ψ] t dt. Putting tings togeter, 4.6 ten implies te assertion of te Lemma. Te first term in 4.7 may be restated as a controllable perturbation of Ut,U Ψ t dt: we restate 4.8 as Ut, = Ut, + k t tn k V+ t,, for t t n,t n+, to find U t, U + k t t n ]V + ] Ψ k t U t, U Ψ ] 4.9 t dt k U t, t t n V V + ] Ψ k t + U t, t t n V Ψ k t dt Ck / / / Ψ L U t dt k V t dt Ω T + Ck U t L L V L L Ψ t L L Ck / Ψ L Ω T + k / Ψ t L L ], were again we use additional control over V t as is given in Lemma 4.. Effects like numerical integration, interpolation, and combination of successive iterates in 4.7 are considered next to establis convergence of iterates of Algoritm B to weak solutions of.5.7. Teorem 4.. Let te assumptions of Lemma 4. be valid, and U u W, Ω, R m resp. V v L Ω, R m, for. Tere exist u L,T,W, Ω, R m W,,T,L Ω, R m, and a subsequence {U k, } suc tat for k, U k, u in L,T;W, Ω, R m, U k, t ut in L,T;L Ω, R m. Moreover, u : Ω T R m is a weak solution of.5.7. Trougout te proof, we again drop sub-indices. Proof. Step. Te bounds of Lemma 4. yield te existence of u L,T;W, Ω, R m W,,T;L Ω, R m, suc tat for k C min{d/,} and, 4. U,U +,U u in L,T;W, Ω, R m, U,U +,U u in L Ω T, R m, U t,v,v + ut in L,T;L Ω, R m. Since U + z = for all z N and all t,t], tere olds I U + ] = for all t,x Ω T, and for all K T U + L K C U + L K. Consequently, U + almost everywere in Ω T, and ence u = almost everywere.

12 Step. We compute U t,u Ψ t u t,u Ψ t Ut, I U Ψ t ] U t, I U Ψ t ] 4. + U t, I U Ψ t ] U Ψ t + Ut,U u] Ψ t + ut U t,u Ψ t =: I IV. We use properties of,, W, Ω-stability of I, and U L to find I C U t I U Ψ t ] C U t U + Ψ t W,. Te term II can be bounded correspondingly. Convergence towards zero of te terms III and IV follows from U u in L Ω T, R m, and U t u t in L Ω T, R m. Hence, for k C min{d/,}, 4. lim k, U t,u Ψ t dt = u t,u Ψ t dt Ψ C Next, we verify tat te limit for te second term in 4.7 for k C min{d/,} is 4.3 lim k, U, I U Ψ ] dt = u, u Ψ]dt Ψ C,T;C Ω, R m.,t;c Ω, R m. For tis purpose, since U, U Ψ] = U,U Ψ, and u, u Ψ] = u,u Ψ almost everywere, U, I U Ψ ] u, u Ψ] = U, {I U Ψ ] U Ψ} + U,U u] Ψ + U u],u Ψ =: I + II + III. We compute I C U U + Ψ W,, by an interpolation estimate, using D U K = for all K T. For te terms II resp. III, we use U u in L Ω T, R m, and U u in L Ω T, R md, respectively to conclude tat II dt, III dt, for k C min{d/,}, and. Terefore, assertion 4.3 is valid, and ence te limit u : Ω T R m satisfies 4.. Convergence V,U, Ψ, v,u Ψ,, for k C min{d/,}, and follows from properties of,, and V v in L Ω, R m, resp. U u in W, Ω, R m. Finally, since U, we conclude similar to 4.9 V + T / V,U Ψ] t dt Ck/ k V t T / dt U t dt + Ψ W, Ω T, for te last term in 4.7. Terefore, u : Ω T R m satisfies assertion iii of Definition 4.. Step 3. We verify assertion iv of Definition 4.. u = lim t lim k, Ut, in L Ω, R m follows from 4.. It remains to sow U t t, v in L Ω, R m as t. Terefore, multiply u u = wit Ψ C,T;C Ω, R m, integrate by parts on Ω T, and subtract te resulting equation from 4.7. We find for te limit k,, ut, v,u Ψ = Ψ C,T;C Ω, R m. On noting v x,u x = u t,x,u x = for almost every x Ω, it follows from te vector identity v = u,v u u u v wit v = u t, v and u = u tat u t t, v in L Ω, R m as t. We also need to sow u t t, v in L Ω, R m as t. By weak lower semicontinuity of L -norm and Fatou s lemma Dut, L lim inf k, DUt, L t.

13 Hence, for all t, because of properties of,, and assumptions on initial data, Terefore, E u t t,,ut, lim inf E U t t,,ut, k, = lim inf E Ut t,,ut, E v,u. k, lim sup u t t, L v L, t lim sup ut, L u L, t and te weak convergence u t t, v in L Ω, R m and ut, u in L Ω, R md implies strong convergence u t t, v and ut, u in L Ω, R md as t. Consequently, u : Ω T R m attains prescribed initial data continuously in W, Ω, R m L Ω, R m. Since all requirements of Definition 4. are verified, ence, te map u : Ω T R m is a weak solution to.5.7. Te proof is complete. Remark 4.. A symmetric variant of Algoritm B is: For n =,,..., find U n+,λ n+ V V, suc tat for all Φ V, and all z N tere olds d tu n+,φ + λ n z = U n+ + U n ], Φ = λ n U n+ + U n ],Φ for U n+ + U n ]z =, d tu n z,d tu n+ z + Un+ +U n ]z U n+ +U n ],U n+ +U n ]z ϕ z + 4 β z Un+ +U n ]z else. Tis coice of λ n again ensures tat U n+ z = for all z N. Te discrete energy is denoted as Ẽ V n, {U n j } j= = V n + U n + U n ], for V n := d t U n. Again, existence of solutions U n+ V in te case k C min{d/,} can be sown, and Ẽ V N+, {U N+ j } j= = V + ] U + U, N. However, convergence for k, is not clear because of te absence of te second term on te left-and side of 4., wic gives enoug control over temporal variations of {d t U n } to pass to te limit k, in every term in Computational Studies In tis section, we report on practical performance of Algoritms A and B. Te nonlinear systems of equations in eac time step were approximately solved using fixed-point iterations wic utilize te old λ n+,l defined troug te actual iterate U n+,l to determine te update U n+,l+. Bot algoritms were implemented in MATLAB wit a direct solution of linear systems of equations. Te initial data tat we employ for bot, te armonic map eat flow as well as te wave map evolution, in te following two subsections are defined in te following example. Example 5.. Given w >, let Ω :=, + w,. Wit r j r j x := x p j and a j = a j x := r j x 4 for x Ω, j =,, and p := /,/ and p := 3/,/, we 3

14 define for x = x,x Ω,,, x, and r /, u x := x /a,x /a,a r a +r, x, and r /,,sinx π/w,cosx π/w, x, + w,,, x + w, + w and r /, x /a,x /a,a r a +r, x + w, + w and r /. A projection of te nodal interpolant of te vector field u onto te xy-plane is sown in te top plot of Figure 5. For te simulation of te wave flow we also define te initial velocity v :=. All employed triangulations were obtained from uniform refinements of te triangulation T of Ω wic consists of 6 triangles wic are all alved squares if w =. Te discrete initial data is obtained by nodal interpolation of u. Unless oterwise stated, we set w =. 5.. Experimental results for te wave map problem. We run Algoritm B for Example 5. wit te triangulation T 3 obtained from tree uniform red- refinements of T and wit k = /8, were = 3. We stopped te time stepping at t = /4, i.e., after 3 time steps, and replaced Vt, by Vt, at t = /4 to reverse te evolution and run anoter 3 time steps. Te numerical results for t = /4 and te almost recovered initial data are sown in te second and tird from top plot of Figure 5. Owing to an instability related to occurrence of large maximal gradients wic motivates finite-time blow-up, we can not approximately recover te initial data wen we reverse te evolution at t = /, wic is beyond te instability; cf. te fourt and fift plot in Figure 5. Tis beavior does not improve wen we significantly decrease te stopping criterion for te fixed point iteration, i.e., wen we solve te nonlinear systems of equations almost exactly. In order to compare te performance of Algoritm B to te projection strategy proposed in 6] we display in Figure te total energy of te approximations obtained wit te two scemes on uniform triangulations wit mes-size = 3, 4, and for time-step sizes k =,/. We observe tat te total energy is for all pairs of discretization parameters decreasing for te numerical approximation obtained wit Algoritm B. Tis is not te case for te approximations computed wit te explicit projection sceme of 6]; in fact, te energy rapidly increases for k = wic indicates strong numerical instabilities. Neverteless, for k = / all results are qualitatively comparable. Te total energy, te kinetic energy E kin Vt, := Vt,, and te W, Ω- semi-norm as functions of time t,/ for te two different scemes on a triangulation wit = 5 and k = / are displayed in Figure. We observe tat large gradients occur and tat energy is lost wen a large cange of te W, Ω-norms takes place. Te experimental results are sligtly different wen te symmetric but teoretically unjustified sceme of Remark 4. is used to compute numerical approximations. As opposed to Algoritm B and te projection sceme of 6], te evolution can always be reversed. Te reported irreversibility in Example 5. wic is related to a numerical instability wen te vectors at /,/ or 3/,/ canges its direction witin a small time interval is different ere and te vectors remain fixed wen te symmetric sceme is used. Also, tere is no loss of te modified energy as can be seen in Figure 3. Neverteless, we empasize tat te symmetric sceme is not known to converge to a weak solution. 5.. Experimental results for te eat flow problem. Our numerical experiments for te armonic map eat flow problem based on Algoritm A do not sow significant advantages of te proposed sceme over te algoritms developed in 5, 3]. Te reason for tis is tat te fixed point iteration requires in all of our numerical studies tat k /5, and terefore does not improve existing results. However, for tis coice of te time-step size we obtain reasonable results for te 4

15 4 V,U Alg. B, = 3, k = V,U Alg. B, = 3, k = / V,U Alg. B, = 4, k = V,U Alg. B, = 4, k = / V,U Proj., = 3, k = 8 V,U Proj., = 3, k = / V,U Proj., = 4, k = 6 V,U Proj., = 4, k = / t Figure. Total energy for numerical approximations obtained wit Algoritm B and wit te projection sceme of 6] for various discretization parameters in te wave map problem defined wit initial data from Example 5.. V,U Alg. B, = 5, k = / kin V Alg. B, = 5, k = / U, Alg. B, = 5, k = / V,U Proj., = 5, k = / 8 kin V Proj., = 5, k = / U, Proj., = 5, k = / t Figure. Total energy, kinetic energy, and W, Ω-semi-norm for numerical approximations obtained wit Algoritm B and wit te projection sceme of 6] for fixed discretization parameters in te wave map problem defined wit initial data from Example 5.. 5

16 V,U Symm. Sceme, = 5, k = / kin V Symm. Sceme, = 5, k = / U, Symm. Sceme, = 5, k = / t Figure 3. Total energy, kinetic energy, and W, Ω-semi-norm for numerical approximations obtained wit te symmetric sceme from Remark 4. for fixed discretization parameters in te wave map problem defined wit initial data from Example U = 4, k = /5 5 U, " U = 5, k = /5 4 U, " U = 6, k = /5 U, " t Figure 4. Energy and W, semi-norm for numerical approximations of te armonic map eat flow problem obtained wit Algoritm A for various discretizations parameters and wit initial data defined in Example 5.. 6

17 Figure 5. Numerical solutions obtained wit Algorim B in Example 5.. Initial data top plot, numerical approximations at t = /4 and t = / second and fourt plot from top, and approximations at t = wen te evolution is reversed 7 at t = /4 and t = / tird and fift plot. All vectors are scaled by te factor /8 for grapical purposes.

18 evolution defined by te initial data specified in Example 5.. Figure 6 displays snapsots of te numerical solution for t =,.875, 375,.565,.75. Large maximal gradients occur for t.5 and afterwards te solution appears to be smoot and converges to a steady uniformly constant state. We remark tat te numerical results do not cange qualitatively wen we employ oter values for te parameter w. Rapid decay of te energy accompanied by occurrence of large gradients on eac mes wen k /5 are te main conclusions of te practical experience wit Algoritm A Effect of different winding numbers. As is detailed in ], weak solutions of te Diriclet problem for te armonic map eat flow for l-equivariant maps, wit l =, do not blow up. Tis is in contrast to te wave-map flow, were finite-time blow-up beavior is still expected; cf. 6, Remark.6]. Te following example reports on corresponding numerical studies in an equivariant setting but restricted to a square centered around te origin. Te initial data tat we use are defined as follows. Example 5.. Set Ω :=, and for l = or l = define sin χr,lθ sinlθ u r,θ := sin χr,lθ cos lθ v r,θ, cos χr,lθ were r,θ denotes polar coordinates in R and χr,lθ := r 3 /4exp 4r / 4. We ran Algoritm B in Example 5. on uniform triangulations of Ω wit = 4 5 and = 4 6 and time-step size defined troug k = /5. Figure 7 displays te total energies and W, seminorm as functions of t,4]. We observe tat large gradients occur for l = wile tey do not occur for l =. References ] F. Alouges, A new algoritm for computing liquid crystal stable configurations: te armonic mapping case, SIAM J. Numer. Anal. 34, pp ] F. Alouges, P. Jaisson Convergence of a finite elements discretization for te Landau Lifsitz equations, Mat. Models Metods Appl. Sci. 6, pp ] S. Bartels, A. Prol, Constraint preserving implicit finite element discretization of armonic map flow into speres, Mat. Comp. accepted, 7. 4] S. Bartels, A. Prol, Convergence of an implicit finite element metod for te Landau-Lifsitz equation, SIAM J. Numer. Anal. 44, pp ] J.W. Barrett, S. Bartels, X. Feng, A. Prol, A convergent and constraint-preserving finite element metod for te p-armonic flow into speres, SIAM J. Numer. Anal. accepted, 6. 6] S. Bartels, X. Feng, A. Prol, Finite element approximations of wave maps into speres submitted, downloadable at: ttp://na.uni-tuebingen.de/preprints.stml. 7] P. Bizoń, T. Cmaj, Z. Tabor, Dispersion and collapse of wave maps, Nonlinearity 3, pp ] P. Bizoń, T. Cmaj, Z. Tabor, Formation of singularities for equivariant + -dimensional wave maps into te -spere, Nonlinearity 4, p ] K.C. Cang, W.Y. Ding, R. Ye, Finite-time blow-up of te eat flow of armonic maps from surfaces, J. Diff. Geom. 36, pp ] Y.M. Cen, M. Struwe, Existence and partial regularity results for te eat flow for armonic maps, Mat. Z., pp ] J.-M. Coron, J.-M. Gidaglia, Explosion en temps fini pour le flot des applications armoniques, CR. Acad. Sci., Paris, Ser. I 38, pp ] J.F. Grotowski, J. Sata, A note on geometric eat flows in critical dimensions, Preprint 6, downloadable at: ttp://mat.nyu.edu/faculty/sata/preprints/gs6.pdf. 3] E. Hairer, C. Lubic, G. Wanner, Geometric numerical integration: structure-preserving algoritms for ordinary differential equations, nd edition, Springer 6. 8

19 Figure 6. Snapsots of te numerical solutions for evolution governed by armonic map eat flow in Example 5. and simulated wit Algoritm A. Displayed solutions correspond to t =,.875, 375, 9.565,.75 from top to bottom. All vectors are scaled by te factor /8 for grapical purposes.

20 E U = 5, L = U, " U = 6, L = U ", E U = 5, L = U ", U = 6, L = U ", t Figure 7. Energy and W, semi-norm for numerical approximations of te wave map problem obtained wit Algoritm B for various discretizations parameters and wit initial data defined in Example 5.. 4] J. Krieger, W. Sclag, D. Tataru, Renormalization and blow up for carge one equivariant critical wave maps, preprint arxiv-series, 6. 5] M. Kruzik, A. Prol, Recent developments in te modeling, analysis, and numerics of ferromagnetism, SIAM Review 48, pp ] I. Rodnianski, J. Sterbenz, On te formation of singularities in te critical O3 σ-model, preprint arxiv-series, 6. 7] J. Sata, Weak solutions and development of singularities in te SU σ model, Comm. Pure Appl. Mat. 4, pp ] J. Sata, M. Struwe, Geometric wave equations, New York University, Courant Institute of Matematical Sciences, New York ] R.E. Sowalter, Monotone operators in Banac space and nonlinear partial differential equations, AMS 997. ] M. Struwe, Geometric evolution problems, IAS/Park City Mat. Series, vol., pp ] M. Struwe, On te evolution of armonic maps of Riemannian surfaces, Mat. Helv. 6, pp ] B. Tang, G. Sapiro, V. Caselles, Diffusion of generated data on non-flat manifolds via armonic maps teory: te direction diffusion case. Int. J. Comput. Vision 36, pp ] B. Tang, G. Sapiro, V. Caselles, Color image enancement via cromaticity diffusion, IEEE Trans. Image Proc., pp ] D. Tataru, Te wave maps equation, Bull. Amer. Mat. Soc. 4, pp ] L.A. Vese, S.J. Oser, Numerical metods for p-armonic flows and applications to image processing, SIAM J. Numer. Anal. 4, pp

21 Department of Matematics, Humboldt-Universität zu Berlin, Unter den Linden 6, D-99 Berlin, Germany address: Matematisces Institut, Universität Tübingen, Auf der Morgenstelle, D-776 Tübingen address: Matematisces Institut, Universität Tübingen, Auf der Morgenstelle, D-776 Tübingen address: