Stability and convergence of finite element approximation schemes for harmonic maps

Transcription

1 Stability and convergence of finite element approximation scemes for armonic maps Sören Bartels Department of Matematics, Humboldt-Universität zu Berlin, Unter den Linden 6, D-199 Berlin, Germany Abstract Tis article discusses stability and convergence of approximation scemes for armonic maps A finite element discretization of an iterative algoritm due to F Alouges is introduced and sown to be stable and convergent in general only on acute type triangulations An a posteriori criterion is proposed wic allows to monitor sufficient conditions for weak convergence to a armonic map on general triangulations and for adaptive mes refinement Numerical experiments sow tat an adaptive strategy automatically refines triangulations in neigboroods of typical point singularities and tereby underline its efficiency Key Words Harmonic maps, iterative algoritm, finite element metod, weak convergence, liquid crystals, adaptive refinement AMS Subject Classification: 35A4, 65C2, 65N3 Date: Marc 11, 25 1 INTRODUCTION A variational model in te teory of nematic liquid crystals due to Oseen and Frank [31, 12, 24, 14] leads to a minimization of te energy functional I(v):= 1 k 1 divv 2 +k 2 v curlv 2 +k 3 v curl v 2 +(k 2 +k 4 ) ( tr [(Dv) 2 ] (divv) 2) dx 2 over a space of admissible configurations v A(u D ) := {v H 1 (; R 3 ) : v = u D, v = 1 ae in } Here, R 3 is a bounded Lipscitz domain and represents te pysical domain in wic te liquid crystal is embedded, u D H 1/2 ( ; R 3 ) wit u D = 1 almost everywere on are given boundary data, and k 1, k 2, k 3, k 4 are material and temperature dependent constants A vector field v A(u D ) locally represents te mean direction of te Supported by Deutsce Forscungsgemeinscaft troug te DFG Researc Center MATHEON Matematics for key tecnologies in Berlin 1

2 2 molecules wic constitute te liquid crystal and a local minimizer of I in A(u D ) defines a stable configuration of te liquid crystal Te pointwise constraint v = 1 models te pysically motivated assumption tat in te liquid crystal pase te molecules are rod-like wit a fixed lengt Existence of (global) minimizers of I in A(u D ) can be establised if A(u D ) [15] Sufficient for A(u D ) is tat u D is Lipscitz continuous on [15] Owing to te non-convex constraint v = 1 uniqueness and iger regularity of solutions cannot be expected [15, 16, 27, 28, 29, 3, 31] Typically, stable points of I in A(u D ) are not continuous and ave point singularities wic correspond to defects in te nematic material In addition to non-uniqueness and existence of singularities, te non-convex nature of te problem makes it extremely difficult to numerically approximate stationary points Te crux in te design of numerical scemes lies in a stable realization of te constraint v = 1 In order to make te main ideas for te approximation of te constraint more clear we will only investigate te pysically relevant one-constant approximation of I, wic assumes k 1 = k 2 = k 3 = 1 and k 4 = and reduces te minimization problem to te problem of finding armonic maps: Find u A(u D ) wic is a local minimizer for (P) E : A(u D ) R, v 1 Dv 2 dx 2 Solutions of (P) will be called armonic maps Tey satisfy te Euler-Lagrange equations u = Du 2 u, u = 1 in Iterative algoritms for te approximation of armonic maps ave been proposed in [1, 21, 9] and successfully been tested numerically Convergence of an iterative sceme on a continuous level and stability of a related finite difference discretization ave been proved in [1] Te goal of tis work is to analyze finite element discretizations of tat algoritm wic allow for local mes refinement and tereby a more efficient resolution of point singularities of solutions We prove tat, in general, finite element discretizations cannot be expected to be stable and are convergent only on structured triangulations Sufficient for stability and convergence is tat te te underlying triangulation is of acute type (cf Lemma 32 for details) We provide an a posteriori criterion tat allows to monitor reliability of te algoritm on general triangulations and gives rise to automatic local mes refinement Numerical experiments indicate tat adaptive strategies are more efficient wen compared to scemes on uniform triangulations Wile we restrict te analysis to te one-constant approximation of I we stress tat te ideas can be carried over to te full model and refer te reader to [2] for related ideas An alternative approac to approximating local minimizers of I consists in regularizing te problem by introducing a penalty term ε 2 v in I wit < ε 1 in L 2 () order to approximate te constraint v = 1 Difficulties in analyzing suc an approac stem from te lack of regularity of minimizers of I and a reliable discretization of te gradient flow of te penalized formulation generally requires very small time step sizes

3 wic limit te practical use For related approaces and te numerical analysis of more sopisticated models we refer te reader to [3, 4, 22, 23, 25, 13] Te rest of tis article is organized as follows We briefly recall te definition and te main properties of te iterative algoritm of [1] in Section 2 Section 3 discusses finite element discretizations of tat algoritm and gives sufficient a priori conditions for its convergence A few numerical experiments sow te efficiency of te discrete algoritm and are presented in Section 4 Section 5 is devoted to an a posteriori analysis and introduces local refinement indicators Te efficient performance of te resulting adaptive strategy is illustrated by some numerical experiments in Section ALOUGES ALGORITHM For an initial u () A(u D ) Alouges algoritm computes a sequence ( u (j) : j N ) H 1 (; R 3 ) by iterating te following two steps: (A 1 ) Let w (j) K u (j) satisfy E(u (j) w (j) ) E(u (j) v) for all v K u (j), were K u (j) := {v H 1 (; R 3 ) : v =, v u (j) = ae in } (A 2 ) Set u (j+1) := u(j) w (j) u (j) w (j) Given any u (j) H 1 (; R 3 ) step (A 1 ) consists in minimizing an elliptic functional on a subspace of H 1 (; R 3 ) and admits a unique solution w (j) K u (j) Supposing tat u (j) = 1 almost everywere in, te definition of K u (j) yields u (j) w (j) 2 = u (j) 2 + w (j) 2 1 almost everywere in Hence, u (j+1) in step (A 2 ) is well defined and satisfies u (j+1) = 1 almost everywere in It is not difficult to verify tat for a function v H 1 (; R 3 ) satisfying v 1 almost everywere in tere olds ( v ) (21) E E(v), v in particular v/ v H 1 (; R 3 ) and tus u (j+1) A(u D ) Noting tat v K u (j) it tus follows tat ( u E(u (j+1) (j) w (j) ) ) = E E(u (j) w (j) ) E(u (j) ) u (j) w (j) Tis is te energy decreasing property of Alouges algoritm Te main features of te iteration are summarized in te following teorem Teorem 21 ([1]) Let u () A(u D ) Suppose tat te sequences ( u (j) : j N ) and ( w (j) : j N ) are generated by iterating (A 1 ) and (A 2 ) Ten, for all j N tere olds u (j) A(u D ) and E(u (j+1) ) E(u (j) ) Tere olds w (j) (strongly) in H 1 and tere exists a subsequence (u (k) : k N) and a armonic map u A(u D ) suc tat u (k) u (weakly) in H 1

4 4 3 FE DISCRETIZATION AND NUMERICAL ANALYSIS OF (A 1 ) AND (A 2 ) In order to make difficulties in a finite element discretization of (A 1 ) and (A 2 ) more clear we will occasionally consider a two-dimensional situation in tis section Terefore, we assume tat n = 2 or n = 3 and tat is a bounded, polygonal or polyedral respectively, Lipscitz domain in R n Given a regular triangulation T of into triangles (n = 2) or tetraedra (n = 3), let N denote te set of nodes in T Te lowest order finite element space related to T is denoted by S 1 (T ) H 1 () Te nodal basis functions (ϕ z : z N) S 1 (T ) satisfy ϕ z (z) = 1 and ϕ z (z ) = for z N and z N \ {z} We define S 1 (T ):= {v S 1 (T ) : v = } Trougout tis section, m is a positive integer Te pointwise constraint v = 1 is satisfied solely by functions v S 1 (T ) m wic are constant in Terefore, assuming u D C( ; R m ), a reasonable finite element discretization of (P) replaces A(u D ) by te set A (T, u D ):= {v S 1 (T ) m : z N v (z) = u D (z), z N v (z) = 1} and seeks a local minimizer of E in A (T, u D ): { Find u A (T, u D ) wic is a local (P ) minimizer for E in A (T, u D ) Existence of solutions for te finite dimensional problem (P ) follow from compactness arguments Te computation of a solution owever is not obvious We propose a discrete version of Alouges algoritm and state sufficient conditions for its stability and convergence Algoritm (A ) Input: (T, u (), δ), were T is a regular triangulation of, u() A (T, u D ) is a starting value, and δ > is a termination parameter (a) Set j := (b) Solve te optimization problem { Minimize E(u (j) v ) subject to Denote te solution by w (j) (c) If Dw (j) L 2 () δ set (u, w ):= (u (j) (d) Define v S 1(T )m and v (z) u (j) (z) = for all z N u (j+1) := z N (e) Set j := j + 1 and go to (b) Output: (u, w ) A (T, u D ) S 1 (T )m, w(j) ) and stop u (j) (z) w(j) (z) (z) w(j) (z) ϕ z u (j) A discrete version of (21) is necessary for stability of step (d) in te algoritm It will turn out tat suc an estimate in general only olds on acute type triangulations

5 31 Validity and possible failure of an energy decreasing property of (A ) Te following definition gives a sufficient criterion for stability of step (d) in Algoritm (A ) Definition 31 A regular triangulation T of is said to satisfy an energy decreasing condition (ED) if for all v S 1 (T ) m satisfying v (z) 1 for all z N, v (z) = 1 for all z N, and w S 1 (T ) m defined by tere olds w := z N v (z) v (z) ϕ z E(w ) E(v ) Te next lemma implies tat acute type triangulations [18] allow for condition (ED) Lemma 32 Let T be a regular triangulation of and suppose tat ϕ z ϕ y dx for all z N \ and y N \ {z} Ten T satisfies condition (ED) Proof For z, y N set k zy := ϕ z ϕ y dx Let φ S 1 (T ) m and define φ z := φ (z) for all z N Since y N k zy = for all z N and since k zy = k yz for all z, y N we ave φ 2 L 2 () = k zy φ z φ y = k zy φ z (φ y φ z ) = 1 2 z,y N z,y N k zy φ z (φ y φ z ) z,y N z,y N k zy φ y (φ z φ y ) = 1 2 z,y N k zy φ z φ y 2 Suppose tat v and w are as in Definition 31 and let v z := v (z) and w z := w (z) for z N Let z, y N be suc tat z y If z N \ or y N \ we ave k zy and ence by Lipscitz-continuity of te mapping {s R m : s 1} R m, s s/ s, wit Lipscitz constant 1 tat 1 2 k zy w z w y 2 = 1 2 k zy w z w z w y w y k zy v z v y 2 If z, y N we ave w z = v z and w y = v y and ence w 2 L 2 () = 1 k zy w z w y 2 1 k zy v z v y 2 = v 2 L (), wic proves te lemma z,y N z,y N Remark 33 (i) Suppose n = 2 Given neigboring nodes z N \ and y N \ {z} let T 1, T 2 T be suc tat T 1 T 2 equals te interior edge connecting z and y Let α zy (1) and α zy (2) be te angles of T 1 and T 2, respectively, opposite to te edge connecting z and y Tere olds [11] ϕ z ϕ y dx = cot α zy (1) cotα(2) zy Sufficient for ϕ z ϕ y dx is tat α (1) zy + α (2) zy π (ii) Suppose n = 3 and let z N \ and y N \ {z} be suc tat z, y T for some 5

6 6 T T Given any T T suc tat z, y N T let α zyt be te angle between te two faces F (1) zyt, F (2) zyt T wic do not contain bot z and y Tere olds [18] F (1) (2) zyt F zyt ϕ z ϕ y dx = cosα zyt, 9 T T T, z,y N T were F (l) (l) zyt is te surface measure of F zyt for l = 1, 2 and T denotes te volume of T Sufficient for ϕ z ϕ y dx is tat α zyt π/2 for all T T suc tat z, y N T Te conditions of Remark 33 are sarp in te sense of te following example Example 34 Let < β < 1/2 and := (, 1) (, β) Let z 1 := (, ), z 2 := (1/2, ), z 3 := (1, ), z 4 := (, β), z 5 := (1/2, β), z 6 := (1, β), z 7 := (1/4, β/2), z 8 := (3/4, β/2) and T := {T 1, T 2,, T 8 } be defined troug T 1 := conv{z 1, z 2, z 7 }, T 2 := conv{z 2, z 8, z 7 }, T 3 := conv{z 2, z 3, z 8 }, T 4 := conv{z 3, z 6, z 8 }, T 5 := conv{z 8, z 6, z 5 }, T 6 := conv{z 7, z 8, z 5 }, T 7 := conv{z 7, z 5, z 4 }, T 8 := conv{z 1, z 7, z 4 }, cf Figure 1 Define s:= 1/2 β and set v j := (1, ) for j = 1, 2,, 6, v 7 := ( 1, ), and v 8 := (1, s) Let v, w S 1 (T ) 2 be suc tat v (z j ) = v j and w (z j ) = w j := v j / v j for j = 1, 2,, 8 Tere olds E(w ) > E(v ) Proof For j, l = 1, 2,, 8 set k jl := ϕ z j ϕ zl dx Since w j = v j for j = 1, 2,, 7 we ave (cf te proof of Lemma 32) δ:= v 2 L 2 () w 2 L 2 () = 1 2 = 8 ( k jl vj v l 2 w j w l 2) j,l=1 6 ( k j8 (1, ) v8 2 (1, ) w 8 2) ( k 78 ( 1, ) v8 2 ( 1, ) w 8 2) j=1 We use (1, ) v 8 2 = s 2, ( 1, ) v 8 2 = 4 + s 2 and abbreviate κ 2 1 := (1, ) w 8 2 = 2 2/ 1 + s 2, κ 2 2 := ( 1, ) w 8 2 = 2 + 2/ 1 + s 2 Using tat 8 j=1 k j8 = we verify 6 j=1 k j8 = k 78 k 88 and obtain δ = (s 2 κ 2 1)(k 78 + k 88 ) k 78 (4 + s 2 κ 2 2) = k 88 (s 2 κ 2 1) k 78 (4 + κ 2 1 κ 2 2) Elementary calculations sow tat k 88 = (12β 2 + 5)/(4β), k 78 = (1 4β 2 )/(4β)

7 Wit a function φ suc tat 1 + s 2 = s2 + φ(s 2 ) we deduce 4β 1 + s 2 δ = ( 12β )( 1 2 s4 + s 2 φ(s 2 ) 2φ(s 2 ) ) ( 1 4β 2)( 2s 2 + 4φ(s 2 ) ) Using tat β 2 = 1 4 s + s2 we verify 4β 1 + s 2 δ = ( 8 12s + 12s 2)( 1 2 s4 + s 2 φ(s 2 ) 2φ(s 2 ) ) 16 ( s s 2)( 1 2 s2 + φ(s 2 ) ) = 8s s 4 6s 5 + 6s 6 + φ(s 2 ) ( s 12s s 4) = 6s 3( 1 2s) 6s 5 (1 s) + 4sφ(s 2 ) ( 2 3s 2 + 3s 3) 2 ( s 3 + 8φ(s 2 ) ) Since < s < 1/2 and φ(s 2 ) <, te first tree terms on te rigt-and side are negative A Taylor expansion proves s 4 /8 φ(s 2 ) and implies tat te last term on te rigt-and side is non-positive Tis sows δ < and proves te lemma 7 β T 7 T 6 T 5 β/2 T 8 T 4 T 1 T2 T 3 1/4 1/2 3/4 1 FIGURE 1 Triangulation T in Example 34 tat does not satisfy condition (ED) for < β < 1/2 We include anoter sufficient criterion for validity of condition (ED) tat allows to construct triangulations of a large class of tree-dimensional domains Lemma 35 Suppose n = 3 and assume tat eac T T as tree mutually perpendicular edges Ten T satisfies condition (ED) Proof Let v, w S 1 (T ) m be as in Definition 31 and let T T Let b 1, b 2, b 3 R 3 \ {} be mutually perpendicular and suc tat b j = z j y j, j = 1, 2, 3, were z j, y j N T for j = 1, 2, 3 After an appropriate rotation we may assume b j = b j e j for j = 1, 2, 3, were e j is te j-t canonical basis vector in R 3 Since s s/ s for s R m wit s 1 is Lipscitz continuous wit Lipscitz constant 1 we deduce for j = 1, 2, 3 tat w T x j = 1 wic implies te lemma v (z j ) z j y j v (z j ) v (y j ) v (y j ) 1 z j y j v (z j ) v (y j ) = v T x j, Remark 36 It can be sown [17] tat if n = 3 and eac T T as 3 mutually perpendicular edges wic do not pass troug te same vertex ten T is of acute type, ie, satisfies te conditions of Remark 33 (ii)

8 8 Te following example defines a triangulation of te unit cube wic satisfies te conditions of Lemma 35 It allows to construct triangulations tat satisfy condition (ED) of unions of finitely many quadrilaterals Oter constructions and acute type refinement strategies of tetraedra can be found in [18, 17] Example 37 ([19, 5]) Set N := {z 1, z 2,, z 8 } for z 1 := (,, ), z 2 := (1,, ), z 3 := (,, 1), z 4 := (1,, 1), z 5 := (, 1, ), z 6 := (1, 1, ), z 7 := (, 1, 1), z 8 := (1, 1, 1) Define T := {T 1, T 2,, T 6 } wit T 1 := conv{z 1, z 2, z 3, z 6 }, T 2 := conv{z 2, z 4, z 3, z 6 }, T 3 := conv{z 3, z 4, z 8, z 6 }, T 4 := conv{z 3, z 8, z 7, z 6 }, T 5 := conv{z 7, z 5, z 3, z 6 }, T 6 := conv{z 3, z 5, z 1, z 6 } Ten T is a regular triangulation of (, 1) 3 and satisfies te assumptions of Lemma 35, cf Figure 2 z7 z 8 z 4 z 3 z 5 z 6 z 2 z 1 FIGURE 2 Triangulation T of te unit cube defined in Example 37 suc tat eac element in T as tree mutually perpendicular edges 32 Well-posedness and termination of Algoritm (A ) Te following lemma sows tat all steps in (A ) are well defined and tat te algoritm terminates witin a finite number of iterations, provided tat T satisfies condition (ED) Lemma 38 Suppose T satisfies condition (ED) Given δ > and u () A (T, u D ), Algoritm (A ) wit input (T, u (), δ) terminates witin a finite number M of iterations wit output (u, w ) A (T, u D ) S(T 1 ) m suc tat Dw L 2 () δ and E(u ) E(u () ) Proof We proceed by induction to sow u (j) A (T, u D ) and E(u (j+1) ) E(u (j) ) Suppose tat for some j we are given u (j) A (T, u D ) Te set L (j) := { v S(T 1 ) m : z N v (z) u (j) (z) = } is a subspace of S 1(T )m Hence, tere exists w L (j) suc tat (31) Dw : Dv dx = : Dv dx Du (j)

9 for all v L (j) Tis is equivalent to E(u (j) w ) E(u (j) v ) for all v L (j) Tus, w = w (j) is te unique solution in step (b) of Algoritm (A ) Since w (j) (z) u(j) (z) = and u(j) (z) = 1 tere olds u(j) (z) w(j) (z) 1 for all z N Hence, u (j+1) is well defined and u (j+1) A (T, u D ) Since L (j) and T satisfies condition (ED) tere olds E(u (j+1) ) E(u (j) w(j) ) E(u(j) ) Equation (31) wit v = w = w (j) proves E(u(j) w(j) ) = E(u(j) ) E(w(j) ) and a combination wit te previous assertion sows E(w (j) ) E(u(j) ) E(u(j+1) ) Since ( E(u (j) ) : j N) is monotonically decreasing and bounded from below we conclude tat it is a Caucy sequence and ence Dw (M) L 2 () δ for M sufficiently large 33 Convergence for Te following teorem sows tat for a sequence of triangulations wit maximal mes-size tending to te sequence of outputs of Algoritm (A ) provides a weakly convergent subsequence wose weak limit is a armonic map Te important questions weter tis weak limit is (globally) energy minimizing in (P) or weter weak convergence can be improved to strong convergence are left for future researc Teorem 39 Suppose u D H 1 ( ; R 3 ) Let (T k : k N) be a sequence of regular triangulations of satisfying condition (ED) wit maximal mes-sizes ( k : k N) satisfying k for k and let (δ k : k N) be a sequence of positive numbers suc tat δ k for k Suppose tat u () k A (T k, u D ) and tere exists C > suc tat Du () k L 2 () C for all k N For eac k N, let (u k, w k ) be te output of Algoritm (A ) applied to te input (T k, u () k, δ k) Ten tere exists a subsequence (u l : l N) and a armonic map u A(u D ) suc tat u l u (weakly) in H 1 and E(u ) lim inf l E(u l) Te following lemma is essential in te proof of te teorem For c R 3 and a matrix A R 3 3 wit columns a 1, a 2, a 3 R 3 we let c A R 3 3 be te matrix wose columns equal c a j for j = 1, 2, 3 Lemma 31 ([8]) A function u A(u D ) is a armonic map if and only if ( ) (32) u Du : Dφ dx = for all φ H 1 (; R 3 ) L (; R 3 ) 9

10 1 Proof Suppose tat u is a armonic map, ie, for all w H 1(; R3 ) L (; R 3 ) tere olds Du : Dw dx = Du 2 u w dx Let φ H 1 (; R3 ) L (; R 3 ) and set w:= u φ Using (a b) c = (a c) b for any a, b, c R 3 we verify Du : Dw = Du : ( Du φ + u Dφ ) = Du : ( u Dφ ) = ( u Du ) : Dφ almost everywere in and since u w = we find tat u satisfies (32) Suppose now tat u satisfies (32) and set φ:= u w Te identity (a b) (c d) = (a c)(b d) (b c)(a d) yields D(u w) : ( u Du) = ( Du w ) : ( u Du) + ( u Dw ) : ( u Du) = ( u T Du ) (w T Du ) Du 2 u w + u 2 Dw : Du ( u T Dw ) (u T Du ) almost everywere in Te identity u 2 = 1 implies u T Du = almost everywere in An integration over finises te proof of te lemma Proof of Teorem 39 By Lemma 38 and te boundedness of (u () k ) in H1 tere olds Du k L 2 () Du () k L 2 () C for all k N Hence, tere exists a subsequence (u l : l N) and u H 1 (; R 3 ) suc tat u l u (weakly) in H 1 Weak lower semicontinuity of E implies E(u ) lim inf l E(u l ) Since u l (z) = 1 for all z N l we ave, by a T elementwise application of Poincaré s inequality and u l 1 almost everywere in, u l 2 1 L 2 () C P l 2u T l Du l L 2 () 2C P C l Since u l u almost everywere in we deduce u = 1 almost everywere in Moreover, we ave u l u L D C 2 ( ) l u D / s L 2 ( ) (ere, u D / s denotes te surface gradient of u D along ) and compactness of te trace operator as a mapping from H 1 (; R 3 ) into L 2 ( ; R 3 ) (cf [26] for details) implies u = u D It remains to sow tat u is a armonic map For all Ψ l S(T 1 l ) 3 wit Ψ l (z) u l (z) = for all z N l tere olds by definition of w l D(u l w l ) : DΨ l dx = Given φ C (; R3 ) let Φ l := φ u l and coose Ψ l := I l (φ u l ), were I l denotes te nodal interpolation operator on T l We ten ave (33) Du l : D(φ u l ) dx = D(u l w l ) : D(Φ l Ψ l ) dx + Dw l : DΦ l dx Using Du l : D(φ u l ) = Du l : (Dφ u l + φ Du l ) = Du l : (Dφ u l ) = Dφ : (u l Du l )

11 11 and u l u (strongly) in L 2, Du l Du (weakly) in H 1 we deduce (34) Du l : D(φ u l ) dx = Dφ : (u l Du l ) dx Dφ : (u Du ) dx Since u l is T elementwise affine tere olds for eac T T D(Φ l Ψ l ) L 2 (T) = D ( φ u l I l (φ u l ) ) L 2 (T) C l D 2 (φ u l ) L 2 (T) C l ( D 2 φ L 2 (T) + Dφ L () Du l L 2 (T)), and ence Φ l Ψ l (strongly) in H 1 Notice tat u l w l is uniformly bounded in H 1 so tat (35) D(u l w l ) : D(Φ l Ψ l ) dx Since Φ l is bounded in H 1 and w l (strongly) in H 1 we ave (36) Dw l : DΦ l dx A combination of (33)-(36) yields Dφ : (u Du ) dx = wic, according to Lemma 31, sows tat u is a armonic map 4 NUMERICAL EXPERIMENTS I In tis section we report on some numerical experiments We first discuss te implementation of Algoritm (A ) 41 Uzawa iteration for te efficient solution of step (b) Step (b) of Algoritm (A ) requires te solution of a quadratic optimization problem wit linear constraints Tis can be solved directly, but may be inefficient We tus propose te use of an Uzawa iteration Te optimization problem may be rewritten as a saddle point problem and te related optimality conditions read: Find w S 1 (T )3 and λ R K suc tat, for all v S 1(T )3, (SP ) Dw : Dv dx + λ z u (j) (z) v (z) = Du (j) : Dv dx, z K w (z) u (j) (z) = for all z K Here, K:= N denotes te set of free nodes in N Te problem can be recast as: Find x R 3N and λ R N suc tat (SP ) A x + B T λ = b, Bx = In tis formulation, x R 3N contains te values of w in te free nodes and we set N := card(k) Te constraint u (z) w (z) =, z K, is realized by te matrix B R 3N N Te positive definite matrix A R 3N 3N is te restriction of A to S 1 (T ) 3, were A is te stiffness matrix defined troug te nodal basis in S 1 (T ) 3 Finally, b is given by te

12 12 restriction of Au to te free nodes, assuming tat u contains te nodal values of u (j) in N Te efficient iterative solution of (SP ) is realized by an Uzawa algoritm wit conjugate directions and an LU decomposition of A (cf, eg, [6]) 42 Numerical examples For te first numerical experiments we specify (P) in te following example Example 41 Set := ( 1/2, 1/2) 3 and u D (x):= x/ x, x Ten, u(x) = x/ x, x, is te unique solution of (P) [2] In order to satisfy te conditions tat guarantee convergence in Teorem 39 we construct triangulations of tat satisfy condition (ED) by scaling, translating, and assembling copies of te triangulation T from Example 37 Example 42 Given an integer k 1 set k := 1/k, C k := { k (l, m, n) : l, m, n k 1 } (1, 1, 1)/2, and define, wit T from Example 37, T k := { c + k T : c C k, T T } Ten, T k is a regular triangulation of = ( 1/2, 1/2) 3 wit maximal mes-size 3/k and satisfies condition (ED) We used four triangulations T k, specified troug k = 4, 8, 16, 32 in Example 41, wit 3N k = 375, 2187, 14739, degrees of freedom (ie, N k free nodes in T k) We set δ k := 1 4 /log 2 (k), and define initial functions u () k A (T, u D ) by { u () z/ z, for z k (z):= Nk, (, 1, ), for z N k In all experiments te Uzawa iteration was stopped wen te l 2 norm of te residual Bx in (SP ) was less tan 1 6 In most of te experiments tis stopping criterion was satisfied after at most 2 iterations Figure 3 displays te decay of te energy E(u (j) k ), j = 1, 2,, in te iteration of Algoritm (A ) wit input (T k, u () k, δ k) for k = 4, 8, 16, 32 Te plot sows tat te decrease in te energy is largest for te first few iterations Tis yields te conjecture tat te coice of te termination criteria δ k = 1 4 /log 2 k is inefficient in tis example if one is only interest in an asymptotic beavior for Figure 4 sows te projection of te vector fields u (j) 32 (,, ) obtained from Algoritm (A ) onto {(x, y, z) R 3 : x = } in ( 1/2, 1/2) 2 for j =, 1, 5, 315 We observe tat only a few iterations are needed to rotate vectors in suc a way tat only one degree one singularity is present Te subsequent iterations move tis singularity to te origin After 317 iterations Algoritm (A ) wit input (T 32, u () 32, δ 32 ) terminates and te nodal values of te output u 32 appear to be very close to te exact solution away from Te value of te numerical solution at, were te exact solution as a singularity, as no particular meaning and seems to depend on te triangulation and te initial value

13 (j) E(u ) 2 15 k=4 k=8 k=16 k= FIGURE 3 Decay of te energy in te iteration of Algoritm (A ) on T k wit k = 4, 8, 16, 32 in Example 41 and initial u () k A (T, u D ) j FIGURE 4 Projection of te vector fields u (j) 32 (,, ) onto {(x, y, z) R 3 : x = } in ( 1/2, 1/2) 2 for j =, 1, 5, 315 in Example 41 and initial u () k A (T, u D )

14 14 We assume tat our definition of u () k is suboptimal as it admits large gradients in a neigborood of In particular, tis coice does not satisfy Du () k L 2 () C for k However, even if for all z K, ξ(z) is a random unit vector in R 3 and te starting value ũ () k A (T, u D ) is defined by { z/ z, for z k (z):= Nk, ξ(z), for z N k ũ () ten we observe in Figure 5 tat te energy still decreases rapidly in te first iterations and becomes stationary almost as fast as for te previous coice We assume tat te number of iterations depends on te initial energy and can be reduced wit an optimal coice of u () k Indeed, te proof of Lemma 38 sows tat te sequence of corrections w(j) k for all l l j= Dw (j) k 2 L 2 () Du() k 2 L 2 (), satisfies and assuming tat Du () k L 2 () C (for a k-independent constant C > ) ten te number of iterations can be expected to grow less fast tan in te presented experiments Figure 6 sows te projection of te vector fields u (j) 32 (,, ) onto {(x, y, z) R 3 : x = } in ( 1/2, 1/2) 2 for j =, 1, 5, 165 produced by Algoritm (A ) wit initial data ũ () 32 We observe tat te algoritm immediately canges te igly unordered initial configuration into a more stable one; after ten iterations only one degree one singularity wit ig symmetry can be seen Te subsequent iterations move te singularity to te origin 5 LOCAL REFINEMENT CRITERIA Te main assertion of tis section is a modification of te assumptions of Teorem 39 It replaces te assumption tat te maximal mes-size tends to and tat te employed triangulations are of acute type by te weaker assumption tat certain computable quantities tend to Moreover, te assertion is independent of a particular sceme since te computable quantities are entirely determined by an approximation u Given a regular triangulation T of let T L () be te T -elementwise constant function satisfying T T = diam(t) for all T T F denotes te set of all faces in T and F is defined on F troug F F := diam(f) for all F F For a T -elementwise smoot (eg, T -elementwise constant) function Σ L (; R 3 3 ) we set [ ] Σ nf F := ( ) (Σ T2 ) F (Σ T1 ) F nf, were F F, T 1, T 2 T suc tat T 1 T 2 = F, and n F R 3 is te unit normal vector to F pointing from T 1 into T 2 Definition 51 Given any u S 1 (T ) 3 let w S(T 1 ) 3 satisfy w (z) u (z) = for all z N and Dw : Dv dx = Du : Dv dx

15 (j) E(u ) 2 15 k=4 k=8 k=16 k= FIGURE 5 Decay of te energy in te iteration of Algoritm (A ) on T k wit k = 4, 8, 16, 32 in Example 41 wit random initial data ũ () k A (T, u D ) j FIGURE 6 Projection of te vector fields u (j) 32 (,, ) onto {(x, y, z) R 3 : x = } in ( 1/2, 1/2) 2 for j =, 1, 5, 165 in Example 41 wit initial data ũ () 32 A (T, u D )

16 16 for all v S 1 (T )3 wit v (z) u (z) = for all z N For eac T T set η 1 (T, u ) 2 := u L 2 (T) + u u D 2 L 2 ( T ), η 2 (T, u ) 2 := 1/2 F [D(u w ) n F ] 2 + Dw L 2 ( T ) 2 L 2 (T) Note tat te following assertion does not assume tat u is obtained by Algoritm (A ), tat te maximal mes-sizes tend to, or tat te triangulations satisfy condition (ED) Proposition 52 Suppose tat (T k : k N) is a sequence of regular triangulations of, and let (u k : k N) S 1 (T k ) 3 be suc tat Du k L 2 () C 1 for some C 1 > and all k Suppose tat T T k η 1 (T, u k ) 2 + η 2 (T, u k ) 2 for k Ten tere exists a subsequence (u l : l N) and a armonic map u A(u D ) suc tat u l u (weakly) in H 1 and (51) E(u ) lim inf l E(u l) Proof Te boundedness of Du k 2 L 2 () + u k 2 L 2 () implies te existence of a weakly convergent subsequence (u l : l N) and a weak limit u H 1 (; R 3 ) Since T T η 1(T, u l ) 2 one verifies tat u A(u D ) Te weak lower semicontinuity of E proves (51) It remains to sow tat u is a armonic map Given any φ C (; R3 ) we set Φ l := φ u l and let Ψ l := I l Φ l be te nodal interpolant of Φ l on T l As in te proof of Teorem 39 we ave to sow tat Du l : D(φ u l ) dx = D(u l w l ) : D(Φ l Ψ l ) dx + Dw l : DΦ l dx A T l -elementwise integration by parts and standard interpolation estimates yield D(u l w l ) : D(Φ l Ψ l ) dx = [D(u w ) n Fl ] (Φ l Ψ l ) ds F F l, F F ( ) 1/2 DΦl C η 2 (T, u l ) 2 L 2 () Hölder s inequality implies ( ) 1/2 DΦl Dw l : DΦ l dx η 2 (T, u k ) 2 L 2 () T T k T T Te proof of Teorem 39 sows Du l : D(φ u l ) dx Dφ : (u Du ) dx A combination of te assertions wit Lemma 31 and DΦ l L 2 () C sows tat u is a armonic map

17 17 6 NUMERICAL EXPERIMENTS II 61 Adaptive Algoritm Proposition 52 motivates te following adaptive mes refinement algoritm It realizes uniform mes-refinement for Θ = and adaptive mesrefinement for Θ = 1/2 Te idea is to iterate steps (b) and (d) of Algoritm (A ) as long as te energy is significantly decreasing A termination criterion tat may be based on smallness of te local refinement indicators η j (T, u ) can easily be included Algoritm (A Θ ) Input: (T, u(), κ), were T is a regular triangulation of, u() A (T, u D ), and κ > (a) Set j := (a1) Solve te optimization problem { Minimize E(u (j) v ) subject to Denote te solution by w (j) (a2) Define (a3) If E(u (j+1) v S 1(T )3 and v (z) u (j) (z) = for all z N u (j+1) := z N u (j) (z) w(j) (z) (z) w(j) (z) ϕ z u (j) ) E(u (j) ) κ set j := j + 1 and go to (a1) (b) Set u := u (j) (c) For eac T T compute η(t) 2 := 2 j=1 η j(t, u ) 2 (d) Mark all T T wic satisfy η(t) Θ max S T η(s) for refinement and generate a new regular triangulation T suc tat all marked elements are refined (e) Set T := T, construct u () A (T, u D ) by interpolating nodal values of u, and go to (a) 62 Numerical example We ran Algoritm (A Θ ) wit Θ = and Θ = 1/2 in Example 41 and an initial triangulation of into 5 tetraedra We cose te termination criterion κ:= 1 4 for te iteration in step (a) of Algoritm (A Θ ) Te mes refinement was realized by a bisection strategy for Θ = 1/2 and by uniform (red-) refinement for Θ = Te left plot in Figure 7 displays te L 2 error u u L 2 () for uniform and adaptive mes refinement wit te iterates u of Algoritm (A Θ ) We used a logaritmic scaling on bot axes to identify a relation between te number of degrees of freedom and te L 2 error We observe tat te L 2 error is significantly smaller at comparable numbers of degrees of freedom wen te refinement indicators of Proposition 52 are used to refine te mes locally Moreover, te experimental convergence rate for uniform meses is only O() (owing to = N 1/3 for uniform meses) instead of te optimal convergence rate O( 2 ) Te adaptive refinement strategy leads to an improved experimental convergence rate Te rigt plot in Figure 7 displays te discrete energies E(u ) for uniform and adaptive

18 uniform refinement adaptive refinement 38 e 2 L ( ) u-u 1 1 E(u ) 375 uniform refinement adaptive refinement dof N dof N x 1 4 FIGURE 7 L 2 error for uniform and adaptive mes refinement in Example 41 (left) Discrete energies E(u ) for uniform and adaptive mes refinement (rigt) mes refinement and we observe tat te adaptive strategy reaces a stable value for a smaller number of degrees of freedom tan te uniform refinement strategy Figure 8 displays te adapted triangulation generated by four iterations of Algoritm ) Te dots in te plot indicate te location of a midpoint of a tetraedron and we observe a refinement towards te origin, were te exact solution as a point singularity (A 1/2 63 Instability of a degree two singularity Te final numerical example discusses a situation wic leads to more tan one degree one singularity Example 61 ([1, 15]) Let π s : S 2 C denote te stereograpic projection of te unit spere S 2 R 3 into te complex numbers C and given a positive integer p let ω(z):= z p Set := ( 1/2, 1/2) 3 and u D (x):= πs 1 ω π s (x/ x ) for x We employed Algoritm (A Θ ) wit Θ = 1/2 in Example 61 for p = 2 and an initial triangulation of into 5 tetraedra We defined an initial function u () by nodal interpolation of te initial data Figure 9 displays projections of intermediate solutions restricted to {(x, y, ) : 1/2 x, y 1/2} on te adapted meses after, 4, 8, and 12 iterations of te algoritm We observe tat te initial degree two singularity splits into two degree one singularities and te mes is refined mostly between te two singularities in wic te discrete vector field as a large gradient We ran te adaptive algoritm (A 1/2 ) in Example 61 wit p = 4 and p = 8 Te exact solution subject to te corresponding boundary data is expected to ave 4 respectively 8 well separated degree one singularities Te projection of te midpoints of te tetraedra to te plane {(x, y, z) R 3 : z = } in te left plot of Figure 1 sows tat Algoritm (A 1/2 ) automatically refines te mes around four separated points wic is in good agreement wit te expected beavior of te exact solution Te results for p = 8 displayed in te rigt plot of Figure 1 sow a local refinement towards eigt points close to te boundary were te separated degree one singularities are expected

19 y x z FIGURE 8 Midpoints of tetraedra (indicated by dots) in te adaptively generated triangulation T after four iterations of Algoritm (A 1/2 ) in Example FIGURE 9 Intermediate solutions u (,, ) in ( 1/2, 1/2) 2 after, 4, 8, and 12 iterations of Algoritm (A 1/2 ) in Example 61

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