PROJECTION-FREE APPROXIMATION OF GEOMETRICALLY CONSTRAINED PARTIAL DIFFERENTIAL EQUATIONS

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1 PROJECTION-FREE APPROXIMATION OF GEOMETRICALLY CONSTRAINED PARTIAL DIFFERENTIAL EQUATIONS Abstract. We devise algoritms for te numerical approximation of partial differential equations involving a nonlinear, pointwise olonomic constraint. Te elliptic, parabolic, and yperbolic model equations are replaced by sequences of linear problems wit a linear constraint. Stability and convergence olds unconditionally wit respect to step sizes and triangulations. In te stationary situation a multilevel strategy is proposed tat iteratively decreases te step size. Numerical experiments illustrate te accuracy of te approac. 1. Introduction Partial differential equations wit pointwise constraints ave attracted considerable attention among pure and applied matematicians in te last decades. Matematically callenging problems like partial regularity, caracterization of singularities, and occurrence of finite-time blow-up as well as modern applications in micromagnetics, geometry, continuum mecanics, and general relativity ave motivated to develop and analyze numerical scemes for te approximate solution of tis class of nonlinear partial differential equations. Te nonlinear caracter of te problem and te lack of regularity of solutions cause tat blackbox optimization routines perform poorly for tis class of equations and necessitates te development of customized approximation scemes. Numerical metods tat provide accurate numerical solutions and converge under restrictions on step sizes or conditions on underlying triangulations or require te solution of nonlinear systems of equations ave been devised and analyzed in [LL89, Alo97, LW00, Alo08, Bar05, BBFP07, BP07, BFP08, BP08, Bar09b, Bar10, San12]. We aim at developing numerical scemes tat are unconditionally stable and convergent, provide accurate approximations wit a small number of degrees of freedom, and only require te solution of linear systems of equations. Harmonic maps (into te spere) are stationary points of te Diriclet energy I(u) = 1 2 Ω u 2 dx Date: August 20, Matematics Subject Classification. 65N12 (65N15 65N30). Key words and prases. Finite elements, partial differential equations, pointwise constraints, armonic maps, armonic map eat flow, wave maps. 1

2 2 among vector fields u : Ω R m subject to te unit-lengt constraint u(x) = 1 for almost every x Ω and Diriclet boundary conditions u ΓD = u D. Te Euler- Lagrange equations caracterize critical points and seek weak solutions of u = u 2 u, u ΓD = u D, u 2 = 1, ν u ΓN = 0. Here, Γ N = Ω \ Γ D and ν u = u ν is te outer normal derivative on Ω. Te factor λ = u 2 on te rigt-and side of te differential equation is te Lagrange-multiplier associated to te pointwise constraint. Gradient flows provide an attractive tool to find solutions of te Euler-Lagrange equations since tese decrease te Diriclet energy along trajectories. Te simplest case corresponds to te L 2 gradient flow and reads t u u = u 2 u, u(t, ) ΓD = u D, u(0) = u 0, u 2 = 1, ν u(t, ) ΓN = 0. Tis evolutionary partial differential equation called armonic map eat flow (into te spere) is of interest in its own rigt, in particular since it is closely related to te Landau-Lifsitz-Gilbert equation in micromagnetics. A yperbolic variant specifies critical points of an action functional subject to te pointwise constraint and leads to te nonlinear wave equation 2 t u u = ( u 2 t u 2) u, u(t, ) ΓD = u D, u(0) = u 0, u 2 = 1, ν u(t, ) ΓN = 0, t u(0) = v 0. Solutions of tis nonlinear initial boundary value problem are called wave maps (into te spere). Te unit-lengt constraint in te equations enforces te vector fields to attain teir pointwise values in te unit spere. Various applications motivate to consider oter submanifolds, e.g., given as te zero level set of a function g : R m R, so tat te pointwise constraint becomes g(u(x)) = 0 for almost every x Ω or g(u(t, x)) = 0 for almost every (t, x) [0, T ] Ω. We abbreviate tese conditions by g(u) = 0. Te most general approac to te numerical solution of nonlinearly pointwise constrained partial differential equations is based on imposing te constraint at te nodes of an underlying triangulation in combination wit predictor-corrector approaces tat linearize te constraint to compute an update wic is ten projected onto te target manifold under consideration. To guarantee well posedness and stability of suc metods restrictive conditions on te step size or properties of te underlying triangulation tat permit to use monotonicity arguments need to be imposed. Alternatively, in igly symmetric situations as is provided by te case of te unit spere equivalent reformulations of te nonlinear equations tat are constraint preserving may be employed. Appropriate discretizations reflect tis property but typically require te solution of nonlinear systems of equations. In tis paper we sow tat te projection step in predictor-corrector approaces can often be omitted. Tis leads to a violation of te constraint at te nodes of an underlying triangulation tat is controlled by te step size independently of te number of iterations. Since it is in general impossible to satisfy te constraint almost everywere in a numerical approximation sceme te additional error introduced by omitting te projection step does not seem to be relevant. Tis idea

3 APPROXIMATION OF GEOMETRIC PDE 3 as previously been employed by te autor in [Bar13] for te development of approximation scemes for large bending isometries. We explain te main idea for a semi-discrete in time approximation of te armonic map eat flow wit Γ D =. Te unit-lengt constraint implies tat we ave t u u = 0 and if u k 1 is an approximation of u(t k 1 ) and v k of t u(t k ) we may impose te ortogonality relation v k u k 1 = 0. If u k 1 is given we compute v k as te unique vector field in H 1 (Ω; R m ) wit v k u k 1 = 0 almost everywere in Ω and (v k, w) + ( [u k 1 + τv k ], w) = ( u k 2 u k 1, w) = 0 for all w H 1 (Ω; R m ) wit w u k 1 = 0 almost everywere in Ω. Here, τ > 0 is a time-step size and (, ) denotes te L 2 inner product. Te existence and uniqueness of v k is an immediate consequence of te Lax-Milgram lemma. Tus, we may define te new approximation u k = u k 1 + τv k. Even if u k 1 is a unit-lengt vector field te new approximation u k will in general not satisfy te pointwise constraint. In fact, owing to te pointwise ortogonality of u k 1 and v k we ave for l 1 tat and an inductive argument implies u l 2 = u l 1 + τv l 2 = u l τ 2 v l 2 l u l 2 = 1 + τ 2 v k 2 provided tat u 0 = 1 almost everywere in Ω. Tis yields tat u l 2 1 l τ 2 L1 v k 2 (Ω) L 2 (Ω). Upon coosing w = v k in te time-discrete evolution equation and employing te binomial formula 2(a + b)a = a 2 + ((a b) 2 b 2 ) we find v k 2 + τ 2 vk ( u k 2 u k 1 2) = 0. 2τ Multiplication by τ and summation over k = 1, 2,..., l lead to l I(u k ) + τ v k 2 + τ 2 l v k 2 = I(u 0 ). 2 A combination of te estimates tus implies for all l 1 tat u l 2 1 L1 (Ω) τi(u0 ). Tis proves tat te approximations satisfy te constraint in te limit τ 0 and tat te error in te approximation of te constraint is bounded independently of te number of time steps or te value of te time orizon. Te left plot of Figure 1 illustrates te numerical sceme. Te corrections v k are computed in te tangent spaces of te level surfaces of te function g(p) = p 2 1. Te rigt plot of Figure 1 illustrates te numerical sceme augmented

4 4 by a projection step. Altoug te projection step is well defined its stability is critical in fully discrete scemes. u 2 u 2 τv 2 τv 2 u 0 τv1 u 1 u 1 τv 1 u 0 Figure 1. Omitting te projection step in te semi-implicit L 2 flow leads to approximations tat violate te unit-lengt constraint but te corresponding error in L 1 (Ω) is bounded independently of te number of iterations and controlled by te step size (left). A projection step leads to an accurate treatment of te constraint but requires restrictive conditions in fully discrete situations to guarantee stability (rigt). We will apply te stategy tat omits te projections to te fully discrete approximation of armonic maps, te armonic map eat flow, and wave maps into a large class of target manifolds caracterized as te zero level set of a function g C 2 (R m ). Wit te elp of numerical experiments we will demonstrate tat te metod leads to accurate approximations. 2. Preliminaries 2.1. Notation. We use standard notation for Lebesgue and Sobolev spaces and abbreviate te norm in L 2 (Ω; R l ) by. We let x = min{m Z : m x} denote te smallest integer above a given number x R. Trougout tis article c stands for a generic positive constant tat is independent of discretization parameters Finite elemement spaces. We assume tat te bounded Lipscitz domain Ω R d, d = 2, 3, as a polyedral boundary and let T be a regular triangulation of Ω into triangles or tetraedra for d = 2, 3, respectively. If a nonempty, closed boundary part Γ D Ω is given we assume tat it is resolved exactly by te edges or faces of te triangulation. Te set of vertices or nodes of te triangulation is denoted by N and we let (ϕ z : z N ) be te nodal basis of te space S 1 (T ) = {w C(Ω) : w T affine for all T T } of continuous, elementwise affine functions. For Γ D Ω we let HD 1 (Ω) be te set of Sobolev functions in H 1 (Ω) tat vanis on Γ D and set S 1 D(T ) = S 1 (T ) H 1 D(Ω). Te nodal interpolation operator is denoted by I : C(Ω) S 1 (T ) and is applied componentwise to vector fields. Te diameter of an element T T is given by te number T and we use te convention tat te index is up to a constant an upper bound for te maximal mes-size in T, i.e., T c for all T T. In tis way we consider a sequence of triangulations (T ) >0, were > 0 indicates a sequence of positive real numbers tat accumulate at 0. Te minimal mes size is defined

5 APPROXIMATION OF GEOMETRIC PDE 5 as min = min T T T. Letting z = diam(ω z ) be te diameter of te node patc ω z = supp ϕ z for every z N we ave for all 1 r < te norm equivalence (2.1) c 1 w r L r (Ω) z N d z w (z) r c w r L r (Ω) for all w S 1 (T ). We say tat T is weakly acute if ( ϕ z, ϕ y ) 0 for all distinct z, y N. If d = 3 ten sufficient for tis is te restrictive condition tat all angles between faces of tetraedra are bounded by π/ Difference quotients. For te discretization of time-dependent problems we will employ backward difference quotients to approximate time derivatives. For a time-step size τ > 0 and a sequence (a k ) k=0,...,k of real number, functions, or vector fields tat are associated to te time steps t k = kτ, we denote d t a k = 1 τ (ak a k 1 ). We note tat for every θ [0, 1] we ave te binomial identity (a k 1 + θτd t a k ) d t a k = d t 2 ak 2 + τ 2 (2θ 1) d ta k 2. We define te continuous interpolant â τ of a sequence (a k ) k=0,...,k troug â τ (t) = t t k 1 a k t t k τ τ for t [t k 1, t k ] and te piecewise constant interpolants for t (t k 1, t k ). a + τ (t) = a k, a τ (t) = a k 1 a k Discrete constraint. Te following lemma sows tat an approximate treatment of te equality constraint at te nodes of triangulations is sufficient to guarantee te validity of te constraint for cluster points of approximate solutions. We assume trougout tat g C 2 (R m ) and tat tere exist c 1, c 2, c 3 > 0 wit g(p) g(q) c 1 ( 1 + g (p) + g (q) ) p q, g (p) c 2 (1 + p ), D 2 g(p) c 3 for all p, q R m, were we denote g = Dg. Lemma 2.1 (Constraint approximation). Assume tat c d/2 1 min [log( 1 min )] 1/2. If (u ) >0 is a bounded sequence in H 1 (Ω; R m ) suc tat u S 1 (T ) m for all > 0, u u in L 2 (Ω; R m ) for some u H 1 (Ω; R m ) as 0, and I g(u ) L1 (Ω) 0 as 0 ten we ave g(u) = 0 almost everywere in Ω. Proof. Two applications of te triangle inequality sow tat g(u ) L 1 (Ω) g(u) g(u ) L 1 (Ω) + g(u ) I g(u ) L 1 (Ω) + I g(u ) L 1 (Ω). Owing to te assumptions of te lemma we ave tat te tird term on te rigtand side tends to zero as 0. Te assumptions on g imply tat g(u) g(u ) L1 (Ω) c u u

6 6 so tat also te first term on te rigt-and side vanises as 0. We use an elementwise interpolation estimate in L 2 (Ω; R m ) to verify g(u ) I g(u ) 2 L 1 (Ω) c 4 T D 2 [g(u )] 2 L 2 (T ) T T c D 2 g(u ) 2 L (Ω) 4 T u 4 L 4 (T ) T T c D 2 g(u ) 2 L (Ω) 4 T u 2 L 2 (T ) u 2 L (T ). T T Wit te unform boundedness of D 2 g and te inverse estimate u L (T ) u L (T ) we deduce tat c 1 T g(u ) I g(u ) 2 L 1 (Ω) c2 u 2 L (Ω) u 2. Finally, te estimate u L (Ω) c 1 d/2 min log( 1 min ) u H 1 (Ω) wic follows from inverse estimates and te Sobolev inequality v L s (Ω) c(d r) 1 v W 1,r (Ω) for 1 r < d and s = dr/(d r) yields tat g(u ) I g(u ) 2 L 1 (Ω) c2 2 d min log( 1 min ) u 4 H 1 (Ω) Hence, as 0 we find g(u ) 0 in L 1 (Ω). For an appropriate subsequence > 0 we ave u u and g(u ) 0 almost everywere in Ω and ence g(u) = 0. Remarks 2.1. (i) Te condition c d/2 1 min [log( 1 min )] 1/2 is a mild restriction on te strengt of te grading of a triangulation. For quasiuniform triangulations we ave min c and te condition is satisfied. (ii) Te boundedness of te sequence (u ) >0 in H 1 (Ω; R m ) implies te existence of a weakly convergent subsequence wic converges strongly in L 2 (Ω; R m ) Projection estimates. Te renormalization of a finite element function increases its Diriclet energy in general. Te following lemma sows for te case of te unit spere tat tis increase is uniformly bounded. Lemma 2.2 (Projection). Tere exists c Π > 0 tat only depends on te geometry of te triangulation T suc tat for all φ S 1 (T ) m wit φ (z) 1 for all z N we ave Π φ c Π φ wit Π φ S 1 (T ) m defined by for all z N, i.e., Π φ = I [φ / φ ]. Π φ (z) = φ (z) φ (z) Proof. We prove te estimate for every element T T wic implies te global statement by summation. For T = conv{z 0, z 1,..., z d } we let Φ T : T T denote te affine linear mapping to te standard simplex T = conv{0, e 1,..., e d } wit Φ T (z j ) = e j for j = 1, 2,..., d and Φ T (z 0 ) = 0. Setting φ = φ Φ 1 T and Π φ = (Π φ ) Φ 1 T we ave by Lipscitz continuity of te mapping p p/ max{1, p } for p Rm wit constant 1 tat Π φ φ (z j ) = x j φ (z j ) φ (z 0 ) φ (z j ) φ (z 0 ) φ = φ (z 0 ) x j

7 APPROXIMATION OF GEOMETRIC PDE 7 for j = 1, 2,..., d, i.e., Π φ L2 ( T ) φ L2 ( T ). Te equivalence d/2 1 T ŵ L2 ( T ) w L 2 (T ) for all w S 1 (T ), cf., e.g., [Ran08], implies tat Π φ L2 (T ) c T φ L2 (T ) wit a constant c T > 0 tat depends on te geometry of T but not on its diameter. Remark 2.1. If T is weakly acute ten te estimate olds wit c Π = 1, cf. [Bar05]. 3. Harmonic maps Given a closed subset Γ D Ω of positive surface measure and u D = ũ D ΓD for some ũ D C(Ω; R m ) H 1 (Ω; R m ) wit g(ũ D ) = 0 in Ω we aim at approximating critical points for I(u) = 1 u 2 dx subject to g(u) = 0. 2 Ω Te direct metod of te calculus of variations implies te existence of solutions and tese are caracterized by ( u, w) = 0 for all w H 1 D (Ω; Rm ) wit w g (u) = 0 in Ω. Te following algoritm iteratively computes approximate solutions for tis problem. Algoritm 1 (Harmonic maps). Let u 0 S1 (T ) m wit u 0 (z) = u D(z) for all z N Γ D and g ( u 0 (z)) = 0 for all z N. Coose τ, ε stop > 0 and set k = 1. (1) Compute v k S1 D (T ) m suc tat v k(z) g ( u k 1 (z) ) = 0 for all z N and ( v k, w ) + ( [u k 1 + τv k ], w ) = 0 for all w SD 1 (T ) m wit w (z) g ( u k 1 (z) ) = 0 for all z N. (2) Set u k = uk 1 + τv k and stop if vk ε stop. Oterwise increase k k + 1 and continue wit (1). Te algoritm terminates witin a finite number of iterations and provides an approximate armonic map in te sense of te following proposition. Proposition 3.1 (Termination). Algoritm 1 terminates witin a finite number of iterations and provides a function u = uk S1 (T ) m for some K N suc tat u (z) = u D(z) for all z N Γ D and ( u, w ) = R (w ) for all w SD 1 (T ) m wit I [w g (u )] = 0 and a bounded linear functional R : HD 1 (Ω; Rm ) R satisfying R H 1 D (Ω) ε stop. Moreover, we ave and I(u ) I(u0 ). Proof. Given u k 1 I g(u ) L1 (Ω) cτi(u 0 ) S 1 (T ) m te set F [u k 1 ] = {w SD(T 1 ) m : w (z) g ( u k 1 (z) ) = 0 for all z N } is a closed subspace of S 1 D (T ) m so tat te Lax-Milgram lemma implies te existence of a unique solution v k F [u k 1 ] in every iteration. Coosing w = v k =

8 8 d t u k, incorporating te binomial formula 2(a+b)a = a2 +((a+b) 2 b 2 ), and noting u k = uk 1 + τv k lead to v k ( u k 2τ 2 u k 1 2) + τ 2 vk 2 = 0. Multiplication by τ and summation over k = 1, 2,..., l sow 1 2 ul 2 + τ ( 1 + τ ) l v k 2 2 = 1 2 u0 2. Tis implies v k 0 as k and ence te convergence of te iteration and te estimate u l u0 for all l 0. At termination wit some K N we ave for u = uk = uk 1 + τv K tat ( u, w ) = ( v K, w ) for all w SD 1 (T ) m wit w (z) g ( u (z)) = 0 for all z N. Since v K ε stop tis implies te asserted identity. For every node z N and l 1 we ave by a Taylor expansion wit some ξ z R m tat g ( u l (z) ) = g ( u l 1 (z) + τv(z) l ) = g ( u l 1 (z) ) + τg ( u l 1 (z) ) v(z) l + τ 2 2 D2 g(ξ z )[v(z), l v(z)] l = g ( u l 1 (z) ) + τ 2 2 D2 g(ξ z )[v(z), l v(z)]. l Recalling tat D 2 g is uniformly bounded it follows tat g ( u K (z) ) D 2 τ 2 g L (R m ) 2 K v(z) l 2. Multiplication by d z and summation over z N togeter wit te norm equivalences (2.1) and Poincaré s inequality imply te asserted estimate. Remarks 3.1. (i) If M = g 1 ({0}) is te boundary of a convex set, i.e., M = C for a convex set C R m, and if T is weakly acute ten one may define te update u k via u k ( (z) = Π C u k 1 (z) + τv(z) k ) for all z N wit te ortogonal projection Π C onto C. Since u k 1 (z) + τv k(z) int(c) we ten ave u k (z) M = C and te Lipscitz continuity of Π C wit constant 1 implies u k [u k 1 l=1 + τv] k u k 1. Tis allows to devise an algoritm wose iterates satisfy te constraint at te nodes exactly and converges wit τ = 1, cf. [Alo97, Bar05, Bar10]. Weak acuteness is a restrictive condition if d = 3 and if te triangulation is not weakly acute te step size restriction τ = O() as to be imposed to guarantee stability wen te projection step is included, cf. [Bar10]. (ii) Te proposition proves global convergence of Algoritm 1. Newton iterations are in general only locally convergent but can be combined wit Algoritm 1 as in [Bar09a]. (iii) Convergence of approximations as (, τ, ε stop ) 0 for te case g(p) = p 2 1 can be sown wit te elp of Lemma 2.1 and weak compactness arguments,

9 APPROXIMATION OF GEOMETRIC PDE 9 cf. [Alo97, Bar05]. Te case of target manifolds different from te unit spere is difficult in general owing to te lack of related compactness results, cf. [Bar10] for certain convergence results if d = 2. In te case of te unit spere, i.e., g(p) = p 2 1, we ave u k (z) 1 for all z N and k 0. Te renormalization is tus well-defined and tis motivates te following multilevel strategy tat starts wit a large step size and stopping criterion, carries out te iteration of Algoritm 1, renormalizes te output, decreases te step size, and repeats tese steps until a prescribed step size and stopping criterion is attained. Algoritm 2 (Multilevel strategy). Let u 0 S1 (T ) m wit u 0 (z) = u D(z) for all z N Γ D and u 0 (z) 2 = 1 for all z N and coose τ, ε stop > 0, and L 0. Set ε stop = 2 L ε stop, τ = 2 L τ, l = 0, k = 1, and u l,0 = u0. (1) Compute v l,k S1 D (T ) m suc tat v l,k (z) ul,k 1(z) = 0 for all z N and ( v l,k, w ) + ( [u l,k 1 + τv l,k ], w ) = 0 for all w SD 1 (T ) m wit w (z) u l,k 1 (z) = 0 for all z N. (2) Set u l,k = ul,k 1 + τv l,k. If vl,k > ε stop increase k k + 1 and continue wit (1). (3) Stop if l = L. Oterwise, increase l l+1, decrease τ τ/2, ε stop ε stop /2, define u l,0 S1 (T ) m by u l,0 ul 1,k (z) = (z) u l 1,k (z) for all z N, set k = 1, and continue wit (1). Wit te arguments of te proof of Proposition 3.1 we obtain te following result. Proposition 3.2 (Multilevel convergence). Algoritm 2 is well defined, terminates witin a finite number of iterations, and provides a function u S1 (T ) m suc tat u (z) = u D(z) for all z N Γ D and ( u, w ) = R (w ) for all w SD 1 (T ) m wit I [w u ] = 0 and a bounded linear functional R : HD 1 (Ω; Rm ) R satisfying R H 1 D (Ω) ε stop. Moreover, wit c Π > 0 suc tat [ I φ ] φ c Π φ for all φ S 1 (T ) m wit φ (z) 1 for all z N we ave I u 2 1 L 1 (Ω) τc L ΠI(u 0 ). Proof. Te proof follows from a repeated application of Proposition 3.1 and noting tat u l,0 c Π u l 1,k for l = 1, 2,...L. Remarks 3.2. (i) If te underlying triangulation is weakly acute ten we ave c Π = 1 and I(u ) I(u0 ). (ii) Te upper bound for te error in te approximation of te constraint can be replaced by τi(u L,0 ) wic is expected to be a sarper bound for te error I u 2 1 L 1 (Ω) since te Diriclet energy is decreased on every level.

10 10 4. Harmonic map eat flow Te iterative sceme devised and analyzed in te previous section computes approximations of armonic maps wit te elp of te corresponding H 1 gradient flow. Te related L 2 gradient flow is te armonic map eat flow into M = g 1 ({0}) defined troug ( t u, w) + ( u, w) = 0, g(u) = 0, u(0) = u 0, for almost every t [0, T ] and all w H 1 D (Ω; Rm ) wit w g ( u(t, ) ) = 0 togeter wit te boundary conditions u(t, ) ΓD = u D, ν u(t, ) ΓN = 0. Here, te case Γ D = is not excluded. Tis initial boundary value problem can be approximately solved by canging te inner product used in Algoritm 1. We incorporate a parameter θ [0, 1] tat allows to analyze a class of midpoint scemes. We assume tat te initial data u 0 H 1 (Ω; R m ) is continuous and satisfies u 0 ΓD = u D and g(u 0 ) = 0 in Ω. Algoritm 3 (Harmonic map eat flow). Let u 0 S1 (T ) m be defined troug u 0 (z) = u 0(z) for all z N and set k = 1. (1) Compute v k S1 D (T ) m suc tat v k(z) g ( u k 1 (z) ) = 0 for all z N and (v k, w ) + ( [u k 1 + θτv k ], w ) = 0 for all w SD 1 (T ) m wit w (z) g ( u k 1 (z) ) = 0 for all z N. (2) Set u k = uk 1 + τv k and stop if k K = T/τ. Oterwise increase k k + 1 and continue wit (1). Te properties of Algoritm 3 are similar to tose of Algoritm 1. Proposition 4.1 (Stability). Algoritm 3 computes a sequence of functions (u k ) k=0,...,k S 1 (T ) m suc tat u k (z) = u D(z) for all z N Γ D and k = 0, 1,..., K, and l I(u l ) + τ v k 2 + (2θ 1) τ 2 l v k 2 2 = I(u 0 ) for all l = 1, 2,..., K. Wit te interpolants û,τ, u,τ, u+,τ : [0, T ] Ω Rm of te sequence (u k ) k=0,...,k we ave ( t û,τ, w ) + ( [(1 θ)u,τ + θu+,τ ], w ) = 0 for almost every t [0, T ] and all w S 1 D (T ) m wit I [ w g ( u,τ (t))] = 0 in Ω and if θ 1/2 ten I g(u + (t, )) L 1 (Ω) cτi(u 0 ). Proof. Coosing w = v k = d tu k in te equation of Algoritm 3 te proof follows te lines of te proof of Proposition 3.1. Remarks 4.1. (i) A projection step can be included if te target manifold is te boundary of a convex set. Stability of te iteration ten requires tat T is weakly acute or tat τ = O( 2 ). (ii) Unconditional convergence to solutions of subsequences of numerical approximations as (, τ) 0 in te case g(p) = p 2 1 and θ 1/2 follows wit te

11 APPROXIMATION OF GEOMETRIC PDE 11 tecniques from [BBFP07, Alo08]. If θ < 1/2 ten te condition τ c 2 min needs to be imposed. (iii) If m = 3 and g(p) = p 2 1 ten te armonic map eat flow into te unit spere can equivalently be defined troug te equation t u + u (u u) = 0 togeter wit initial and omogeneous Neumann boundary conditions. Te Landau- Lifsitz-Gilbert equation seeks a mapping u : [0, T ] Ω R 3 wit u(t, x) = 1 in [0, T ] Ω and t u + αu (u u) u u = 0 wit a small damping parameter α > 0. Te discretization of tis equation is analogous to te discretization of te armonic map eat flow, cf. [AJ06, Alo08]. In particular, projection steps can be omitted. 5. Wave maps Te partial differential equation tat defines wave maps reads ( 2 t u, w) + ( u, w) = 0, g(u) = 0, u(0) = u 0, t u(0) = v 0, for all w H 1 D (Ω; Rm ) wit w g (u) = 0 togeter wit te boundary conditions u(t, ) ΓD = u D, ν u(t, ) ΓN = 0, were Γ D = is admissible. We assume tat te data functions u 0 H 1 (Ω; R m ) and v 0 L 2 (Ω; R m ) are continuous and compatible and employ te following algoritm to approximate solutions. Algoritm 4 (Wave maps). Let u 0, v0 S1 (T ) m be defined troug u 0 (z) = u 0(z) and v 0(z) = v 0(z) for all z N and set k = 1. (1) Compute v k S1 D (T ) m suc tat v k(z) g ( u k 1 (z) ) = 0 for all z N and (d t v k, w ) + ( [u k 1 + θτv k ], w ) = 0 for all w SD 1 (T ) m wit w (z) g ( u k 1 (z) ) = 0 for all z N. (2) Set u k = uk 1 + τv k and stop if k K = T/τ. Oterwise increase k k + 1 and continue wit (1). Proposition 5.1 (Stability). Algoritm 4 computes sequences of functions (u k ) k=0,...,k, (v k ) k=0,...,k S 1 (T ) m suc tat u k (z) = u D(z) for all z N Γ D and k = 0, 1,..., K and 1 2 vl ul 2 + τ 2 2 l d t v k 2 +(2θ 1) τ 2 2 l v k 2 = 1 2 v u0 2. for all l = 1, 2,..., K. Wit te interpolants v,τ, û,τ, u,τ, u+,τ : [0, T ] Ω Rm of te sequences we ave ( t v,τ, w ) + ( [(1 θ)u,τ + θu+,τ ], w ) = 0 for almost every t [0, T ] and all w S 1 D (T ) m wit w I g ( u,τ (t)) = 0 in Ω and if θ 1/2 ten I g(u +,τ (t, )) L 1 (Ω) cτ(t + 1)( v u 0 2 ).

12 12 Proof. Coosing w = v k = d tu k in te equation of Algoritm 4 leads to d t 2 vk 2 + τ 2 d tv k 2 + d t 2 uk 2 + (2θ 1) τ 2 2 vk 2 = 0. Summation over k = 1, 2,..., l and multiplication by τ imply te first asserted identity. If θ 1/2 ten we ave v k c for all k = 0, 1,..., K and wit te arguments of te proof of Proposition 3.1 we deduce te asserted bound for I g(u +,τ (t, )) L 1 (Ω). Remarks 5.1. (i) Te linearized treatment of te constraint proibits te use of te test function w = (u k 1 + u k )/2 wic would be desirable in te case θ = 1/2 to verify weter te sceme is dissipation-free. (ii) Unconditional convergence to solutions of subsequences of approximations as (, τ) 0 in te case g(p) = p 2 1 can be sown for θ 1/2 as in [BFP08, Bar09b]. If θ < 1/2 ten te condition τ c 2 min needs to be imposed. 6. Numerical experiments We report in tis section te practical performance of te projection-free approximation scemes for two- and tree-dimensional settings. Te implementations were realized in Matlab wit a direct solution of linear systems of equations. Te employed triangulations were obtained by uniform red-refinements of coarse triangulations of Ω = ( 1/2, 1/2) d wit mes size 1. We will refer to te triangulation tat is obtained by l successive uniform refinements troug te mes-size = 2 l Harmonic maps. We tested Algoritms 1 and 2 in an example tat leads to te singular solution u(x) = x/ x, x Ω, if d = 3 and a smoot solution if d = 2. Example 6.1. Let m = 3, and g(p) = p 2 1 for p R 3, Ω = ( 1/2, 1/2) d for d = 2, 3, Γ D = Ω, Γ N =, and u D (x) = x/ x for x Γ D. In Tables 1 and 2 we displayed te error in te approximation of te constraint wit respect to te L 1 and L norm, te initial energy, and te number of iterations of Algoritm 1 for different mes-sizes wit d = 2 and d = 3, respectively. We employed different mes-sizes = 2 l for l = 3, 4, 5, 6 and l = 2, 3, 4, 5 for d = 2 and d = 3, respectively, and te parameters τ =, ε stop =. Te algoritm was initialized wit vector fields defined troug { u 0 u D (z) for z N Γ D, (z) = ξ z / ξ z for z N \ Γ D, were ξ z [ 1/2, 1/2] 3 denotes for every z N a random vector. Te initial energy I(u 0 ) is tus strongly mes-dependent and grows like 2 wic is suboptimal in view of te bound for te approximation error for te constraint of Proposition 3.1. Neverteless, we observe an experimental convergence of te approximation error in te constraint as 0. Te analysis of Algoritm 1 sows tat te number of iterations depends on te value of te initial energy and for te suboptimal coice of u 0 in tis experiment and te stopping criterion ε stop = we see tat te number of iterations increases like 2. Table 3 illustrates te performance of te multilevel iteration defined by Algoritm 2 wit different numbers of levels L and for fixed d = 2, = 2 5, τ = ε stop =. We

13 APPROXIMATION OF GEOMETRIC PDE 13 I u 2 1 L 1 I u 2 1 L I(u 0 ) K iter Table 1. Error in te constraint approximation in different norms, initial energy, and iteration numbers for Algoritm 1 in Example 6.1 wit d = 2, = 2 l, l = 3, 4, 5, 6, and τ = ε stop =. I u 2 1 L 1 I u 2 1 L I(u 0 ) K iter Table 2. Error in te constraint approximation in different norms, initial energy, and iteration numbers for Algoritm 1 in Example 6.1 wit d = 3, = 2 l, l = 3, 4, 5, 6, and τ = ε stop =. L I u 2 1 L 1 I u 2 1 L K iter Table 3. Error in te constraint approximation and (total) iteration numbers for te multilevel iteration of Algoritm 2 in Example 6.1 wit d = 2, = 2 5, τ = ε stop =, and number of levels L = 0, 1,..., 5. L I u 2 1 L 1 I u 2 1 L K iter Table 4. Error in te constraint approximation and (total) iteration numbers for te multilevel iteration of Algoritm 2 in Example 6.1 wit d = 3, = 2 4, τ = ε stop =, and number of levels L = 0, 1,..., 4.

14 14 see tat te increasing number of levels decreases te total number of iterations compared to te single level iteration corresponding to L = 0. Moreover, te approximation of te constraint is significantly improved wic is in agreement wit Remark 3.2. Since te triangulation is weakly acute Algoritm 1 wit step size τ = 1 and a projection step may be used in tis example. In tis case te sceme terminated after 35 iterations. Table 4 sows te corresponding results for te tree-dimensional situation wit = 2 4. Again we observe a significant decrease of te total number of iterations compared to te iteration wit a fixed step size. Our triangulations of te unit cube in R 3 are not weakly acute so tat te projection sceme wit step size τ = 1 cannot be applied in tis situation Harmonic map eat flow. It is well understood tat te armonic map eat flow can develop singularities from smoot initial data. Te following example from [CDY92] leads to suc a situation and provides a worst-case scenario for numerical approximation scemes. Example 6.2. Let d = 2, T = 1, Ω = ( 1/2, 1/2) 2, Γ D = Ω, Γ N =, m = 3, g(p) = p 2 1, and u 0 (x) = 1 ( x1 sin φ(2 x ), x 2 sin φ(2 x ), x cos φ(2 x ) ) x for x = (x 1, x 2 ) Ω and φ(s) = (3π/2) min{s 2, 1}. We approximated te nonlinear evolution problem specified by Example 6.2 wit Algoritm 3 and te discretization parameters θ = 1, = 2 l, l = 6, 7, 8, 9, and τ =. Figure 2 visualizes te development of te approximation error I u k 2 L 1 (Ω) in dependence of te number of time steps k = 0, 1,..., K and te mes-size. Initially tis error increases rapidly but ten attains a constant value wen te evolution becomes almost stationary. Te experimental error decays only nearly linearly wit te step size τ = witin te considered time interval and does not sow te full linear convergence proved in Proposition 4.1. Tis is expected to be related to te global caracter in time of te estimate. Te approximation error appears large compared to te values of te employed step sizes wic reflects te influence of te large initial energy I(u 0 ) Wave maps. Te partial regularity properties and occurrence of singularities in wave maps is less well understood tan for te armonic map eat flow. Altoug finite-time blow-up is known to occur [KST08] no explicit examples seem to be available. Te initial data defined in te following example leads to maximal, unbounded gradients in numerical experiments. Example 6.3. Let d = 2, T = 1, Ω = ( 1/2, 1/2) 2, Γ D =, Γ N = Ω, m = 3, g(p) = p 2 1, v 0 = 0, and { (2a( x )x1, 2a( x )x u 0 (x) = 2, a( x ) 2 x 2) / ( a( x ) 2 + x 2) for x 1/2, (0, 0, 1) for x 1/2, for x = (x 1, x 2 ) Ω and a(s) = (1 2s) 4. We ran Algoritm 4 wit θ = 1, = 2 l for l = 5, 6, 7, 8, τ = /4, and te discrete initial data functions u 0 = I u 0 and v 0 = I v 0 = 0. Figure 3 sows te temporal development of te constraint violation, i.e., te sequence I u k 2 1 L 1 (Ω)

15 APPROXIMATION OF GEOMETRIC PDE = 2 6 = 2 7 = 2 8 = t Figure 2. Development of te constraint approximation error I u +,τ (t) 2 1 L 1 for different mes-sizes = 2 l, l = 6, 7, 8, 9, and τ = in te approximation of te armonic map eat flow specified in Example 6.2 wit Algoritm 3. Te error grows rapidly in an initial time interval and ten remains nearly constant. It decays nearly linearly wit te step size τ =. for k = 0, 1,..., K wit K = T/τ. We note tat for te initial data defined in Example 6.3 we ave tat te quantity 1( v u 0 2) is uniformly bounded and tis is reflected in te decay of te approximation error of te constraint, i.e., we observe experimentally a linear decay of te approximation error I u k 2 L 1 (Ω) as = τ 0 at fixed times t k. Te error increases in time proportionally to t wic is in agreement wit te error bound derived in Proposition 5.1. References [AJ06] [Alo97] [Alo08] [Bar05] [Bar09a] [Bar09b] [Bar10] François Alouges and Pascal Jaisson, Convergence of a finite element discretization for te Landau-Lifsitz equations in micromagnetism, Mat. Models Metods Appl. Sci. 16 (2006), no. 2, François Alouges, A new algoritm for computing liquid crystal stable configurations: te armonic mapping case, SIAM J. Numer. Anal. 34 (1997), no. 5, , A new finite element sceme for Landau-Lifcitz equations, Discrete Contin. Dyn. Syst. Ser. S 1 (2008), no. 2, Sören Bartels, Stability and convergence of finite-element approximation scemes for armonic maps, SIAM J. Numer. Anal. 43 (2005), no. 1, (electronic)., Combination of global and local approximation scemes for armonic maps into speres, J. Comput. Mat. 27 (2009), no. 2-3, , Semi-implicit approximation of wave maps into smoot or convex surfaces, SIAM J. Numer. Anal. 47 (2009), no. 5, , Numerical analysis of a finite element sceme for te approximation of armonic maps into surfaces, Mat. Comp. 79 (2010), no. 271,

16 = 2 5 = 2 6 = 2 7 = t Figure 3. Development of te constraint approximation error I u +,τ (t) 2 1 L 1 for different mes-sizes = 2 l, l = 5, 6, 7, 8, and τ = in te approximation of a wave map evolution specified in Example 6.3 wit Algoritm 4. Te error grows linearly in time and decays linearly wit te step size τ = /4. [Bar13], Approximation of large bending isometries wit discrete Kircoff triangles, SIAM J. Numer. Anal. 51 (2013), no. 1, [BBFP07] Jon W. Barrett, Sören Bartels, Xiaobing Feng, and Andreas Prol, A convergent and constraint-preserving finite element metod for te p-armonic flow into speres, SIAM J. Numer. Anal. 45 (2007), no. 3, [BFP08] Sören Bartels, Xiaobing Feng, and Andreas Prol, Finite element approximations of wave maps into speres, SIAM J. Numer. Anal. 46 (2007/08), no. 1, [BP07] Sören Bartels and Andreas Prol, Constraint preserving implicit finite element discretization of armonic map flow into speres, Mat. Comp. 76 (2007), no. 260, (electronic). [BP08], Convergence of an implicit, constraint preserving finite element discretization of p-armonic eat flow into speres, Numer. Mat. 109 (2008), no. 4, [CDY92] Kung-Cing Cang, Wei Yue Ding, and Rugang Ye, Finite-time blow-up of te eat flow of armonic maps from surfaces, J. Differential Geom. 36 (1992), no. 2, [KST08] Joacim Krieger, Wilelm Sclag, and Daniel Tataru, Renormalization and blow up for carge one equivariant critical wave maps, Invent. Mat. 171 (2008), no. 3, [LL89] San Yi Lin and Mitcell Luskin, Relaxation metods for liquid crystal problems, SIAM J. Numer. Anal. 26 (1989), no. 6, [LW00] Cun Liu and Noel J. Walkington, Approximation of liquid crystal flows, SIAM J. Numer. Anal. 37 (2000), no. 3, (electronic). [Ran08] Rolf Rannacer, Numerisce Matematik 2 (Numerik partieller Differentialgleicungen), Tec. report, University of Heidelberg, Germany, 2008, Lecture Notes (available online). [San12] Oliver Sander, Geodesic finite elements on simplicial grids, Internat. J. Numer. Metods Engrg. 92 (2012), no. 12, Department of Applied Matematics, Matematical Institute, University of Freiburg, Hermann-Herder-Str 9, Freiburg i. Br., Germany address: bartels@matematik.uni-freiburg.de