Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements

Transcription

1 Optimal A Priori iscretization Error Bouns for Geoesic Finite Elements Philipp Grohs, Hanne Harering an Oliver Saner Bericht Nr. 365 Mai 13 Key wors: geoesic finite elements, iscretization error, a priori bouns AMS Subject Classifications: 65N15, 65J99 Institut für Geometrie un Praktische Mathematik RWTH Aachen Templergraben 55, 556 Aachen (Germany)

2 Optimal A Priori iscretization Error Bouns for Geoesic Finite Elements Philipp Grohs, Hanne Harering an Oliver Saner May 14, 13 Abstract We prove optimal bouns for the iscretization error of geoesic finite elements for variational partial ifferential equations for functions that map into a nonlinear space. For this we first generalize the well-known Céa lemma to nonlinear function spaces. In a secon step we prove optimal interpolation error estimates for pointwise interpolation by geoesic finite elements of arbitrary orer. These two results are both of inepenent interest. Together they yiel optimal a priori error estimates for a large class of manifol-value variational problems. We measure the iscretization error both intrinsically using an H 1 -type Finsler norm, an with the H 1 -norm using embeings of the coomain in a linear space. To measure the regularity of the solution we propose a nonstanar smoothness escriptor for manifol-value functions, which bouns aitional terms not capture by Sobolev norms. As an application we obtain optimal a priori error estimates for iscretizations of smooth harmonic maps using geoesic finite elements, yieling the first high orer scheme for this problem. 1 Introuction This article investigates the numerical iscretization of partial ifferential equations (PEs) for functions whose coomain is a nonlinear Riemannian manifol M. Such problems arise, for example, in Cosserattype material moels [38, 47], liqui crystal physics [, 8], an in image processing [5, 53]. Further we mention variational splines in manifols [3], multi-boy ynamics [31], an the investigation of harmonic maps into manifols [14]. In signal processing of manifol-value signals (see, e.g., [4]) any generalization of a linear variational metho leas to a variational problem with values in a manifol. The numerical approximation of solutions to such PEs is ifficult, because the relevant function spaces o not possess a linear structure. Therefore, stanar iscretization methos like finite elements cannot be use. Instea, various a hoc methos have been propose in the literature to iscretize iniviual PEs with particular coomains M. For example, to compute harmonic maps into the unit sphere S, Bartels an Prohl [6, 7] embee S into R 3, an use first-orer Lagrangian finite elements, constraining only the vertex values to be in S. In the literature on geometrically exact shells, the irection of the shell surface normal is frequently expresse as a set of angles, an the angles are iscretize separately using finite elements [56]. For Cosserat continua (with values in R 3 SO(3)), an alternative approach, use by Münch [36] an Müller [35], interpolates rotation vectors in so(3) instea of in the group of rotations SO(3). Finally, Simo et al. [48, 49] i not interpolate rotations at all. Rather, they kept the orientation at each quarature point as a history variable, an upate it with linear interpolants of the corrections coming from a Newton metho. All these approaches have their shortcomings. Bartels an Prohl rely on an isometric embeing with corresponing projection, which is not always cheaply available in spaces other than S (as an example we mention the space SP(3) of symmetric positive efinite 3 3 matrices, frequently use in applications [18]). Also, it is unclear whether their metho can achieve higher than first-orer convergence. The approach use by Münch an Müller fails when values close to the cut locus of the ientity on SO(3) are involve. Also, the epenence on a fixe tangent space of the coomain breaks objectivity. For the approach by Simo an coworkers [48], Crisfiel an Jelenić [11] showe that it introuces a spurious epenence of the solution on the initial iterate an the parameters of the path-following mechanism. 1

3 With the notable exception of Bartels an Prohl, who prove weak convergence of their iscrete solutions to weakly harmonic maps (see also Remark 7.3 below), no analytical investigations of any of the above iscretization methos appear in the literature. Hence it is generally unknown whether these methos converge, an whether the nominal rate of of the approximation spaces is actually achieve. For the numerical approximation of explicitly given functions with values in a manifol, several theoretical results have been achieve in the recent years [13, 1,, 5, 4, 55, 57]. These methos are base on subivision schemes, an it is unclear how they can be use for solving PEs. Recently, geoesic finite elements (GFE) have been introuce for partial ifferential equations with nonlinear coomains [4, 43, 44]. Base on the Karcher mean (or Riemannian center of mass), they form a natural generalization of Lagrangian finite elements of arbitrary orer to the case where the coomain M is a nonlinear Riemannian manifol. Geoesic finite elements o not rely on an embeing of M into a linear space, an form a conforming iscretization in the sense that geoesic finite element functions are H 1 -functions [44, Thm. 5.1]. Also, they are equivariant uner isometries of M. In mechanics, this leas to the esirable property that iscretizations of objective problems are again objective. In [4, 43, 44], numerical stuies of the iscretization error were performe. These stuies involve geoesic finite elements of orer up to three for functions mapping into the unit sphere S an the special orthogonal group SO(3). In all cases optimal convergence orers in the L - an H 1 -norms were observe. However, no analytical investigation of the iscretization error was given at all. We make up for this with the present article, proviing a complete, intrinsic convergence theory for geoesic finite elements for problems of variational type. By variational type we mean the following setting. For a omain R an M a Riemannian manifol, we look at minimization problems u : M, u = arg min J(w), (1) w H with J : H R a nonlinear functional. The omain H of J is a set of functions M of H 1 smoothness, which we iscuss in etail in later sections. By construction, GFE functions are H 1 functions, an the set V h of GFE functions for a given gri is a subset of H. We can therefore formulate a iscrete problem by restricting J to V h. The iscrete solution is u h = arg min w h V h J(w h ), () i.e., we minimize the original energy functional over a finite-imensional subset of the original set H. As in the linear case, assessing the error of this numerical proceure is one in two steps. First, uner an ellipticity assumption on the energy J, we show that u h is a quasioptimal solution in the approximation space V h, that is, the error between u h an u is comparable to the approximation power of the space V h (inspire by the linear theory we call such a result a Céa lemma). As it turns out, such a result can be prove easily in general metric spaces, using only certain convexity properties of the energy along geoesics, see Theorem 3.1. However, for the crucial H 1 -type istance this convexity is ifficult to verify in practice. We therefore also give a more elaborate result (Theorem 3.3), which allows to boun the H 1 istance using variations of the energy along geoesic homotopies. The results are inepenent of the construction of geoesic finite elements, an also cover other iscretization methos. Then, in a secon step, the approximation power of the GFE spaces is assesse. In Theorem 5.4 we fin that, provie that the solution u has a certain smoothness, the best approximation error of u in V h ecays like a power of the mesh size h. 1 We obtain the same orers as in the corresponing linear cases. All our arguments are completely intrinsic an the epenence of the approximation quality on the geometry of M turns out to only consist of iterate covariant erivatives of the logarithm mapping of M. Combining these two results yiels optimal convergence orers for the iscretization error of geoesic finite element iscretizations of general nonlinear elliptic variational problems (1) in Theorems 6.1 an 6.. Compare to known results in the linear setting, the only important aitional restriction of our results is that we require the solution u to have boune first erivatives. As an application we give optimal a priori 1 Similar results are shown in previous work by Grohs [4] for univariate nonlinear interpolation functions an the L norm, albeit with ifferent methos.

4 error estimates for GFE iscretizations of harmonic maps in Theorem 7.1 uner certain assumptions of the sectional curvature of M. We woul like to emphasize that the two aforementione results, viz. the nonlinear Céa lemmas an the approximation error estimate are highly interesting in their own right. For instance, the Céa lemmas apply to general approximation spaces other than GFE spaces, for example, the interpolation metho use in [35, 36]. The approximation error estimates are also useful in the general context of approximating manifol-value functions (see, e.g., [4, 4]). A elicate issue is the proper choice of error measures in a nonlinear function space. In classical a priori bouns in linear spaces, a Sobolev-type half-norm u of the solution u bouns the error u u h. Since there is no subtraction efine on the set H, we nee to replace u u h by a suitable istance metric in the function space H. We present two such metrics in Section., which reuce to u u h H 1 if M is a linear space. To generalize the term u, the covariant Sobolev half-norm is an obvious choice. However, in our expression for the interpolation error, terms appear that cannot be controlle by a Sobolev half-norm alone. In Section.4 we therefore introuce a slightly stronger concept, which we call the smoothness escriptor. We show that it provies information that is comparable to the actual Sobolev (half-)norms, but it oes iffer from them even in linear spaces. The question of whether our bouns also hol for covariant Sobolev norms is open. We have structure the article as follows. In Chapter we iscuss the nonlinear spaces mae up by functions M of Sobolev smoothness. We propose two istance notions, an introuce the smoothness escriptor. In Chapter 3 we prove ifferent forms of a nonlinear Céa lemma. Only then geoesic finite elements are introuce in Chapter 4. The secon important part of the proof, the interpolation error boun, is shown in Chapter 5. This allows us to state a priori bouns for the iscretization error for the iscrete problem () in Chapter 6. Finally, in Chapter 7 we apply our results to harmonic maps an some of their generalizations. Uner some regularity an curvature assumptions we obtain optimal error bouns for iscrete harmonic maps of all approximation orers. Nonlinear Function Spaces escribing regularity of functions with a nonlinear coomain is a much less unifie fiel than the corresponing linear theory. We introuce the notions that will be use in this article..1 Sobolev Spaces Let R be open an boune with Lipschitz bounary. On we use canonical coorinates x 1,..., x. We use the notation k for the (weak) partial erivative of a -variate function with respect to the multiinex k = (k 1,..., k ) N, i.e., k k = ( x ) k,... ( x 1 ) k1 where we have written k := k k. For a function v : R an an integrability parameter p [1, ) we efine the usual Sobolev half norms an norms v p W := k,p k v(x) p x, v p := W k,p k =k k v p W. (3) j,p We enote by W k,p (, R N ) the set of functions R N for which this quantity is finite componentwise. This set of functions forms a linear space. As an extension the space W k, (, R N ) is efine as the set of all functions R N for which v W k, := sup k v(x) x x k k is finite. For a simpler notation we will sometimes write H k (, R N ) for W k, (, R N ). j= 3

5 Let now (M, g) be an n-imensional Riemannian manifol with scalar prouct, g an inuce istance ist : M M R +. The following efinition of a Sobolev space for functions with values in M is stanar (see, e.g., [45]). efinition.1. Let i : M R N be an isometric embeing (which always exists by [37]), k N an p N { }. efine W k,p (, M) := { v W k,p (, R N ) : v(x) i(m), a.e. }. Again we will write H k (, M) for W k, (, M). We shall also use the notation C(, M) to enote continuous functions from to M. For nonlinear M these spaces obviously o not form vector spaces. However, uner certain smoothness conitions the manifol structure of M is inherite. The following result is prove in [39]. Lemma.1. If k > /p, the spaces W k,p (, M) are Banach manifols. Unfortunately, this lemma exclues the important case of W 1, (, M) with. However, even when W k,p (, M) is not a manifol we can still consier vector fiels that are attache to a general continuous M-value function. efinition.. Let u C(, M). We say that W : T M is a vector fiel along u if W (x) T u(x) M for all x. The set of all vector fiels along u is enote by u 1 T M. For each continuous u : M, the set u 1 T M forms a linear space which we now equip with two norms. The first is of L p -type. efinition.3. Let u C(, M). For a vector fiel W u 1 T M, an p [1, ] we set W p L := W (x) p p g(u(x)) x, with the obvious moifications for p =. The secon one is a W 1, -type norm, involving erivatives with respect to x. With x we enote α the covariant partial erivative along u with respect to x α. In coorinates on an M it reas x α W l (x) := W l x α (x) + Γl ij(u(x)) ui x α W j (x), where we sum over repeate inices an enote with Γ l ij the Christoffel symbols associate to the metric of M. efinition.4. Let u W 1, (, M) an assume that the coorinate functions associate to the vector fiel W u 1 T M are in H 1 (, R). We set W H 1 := W L + x W (x) g x := W L + α=1 x α W (x) For this norm we can show the following version of the Poincaré inequality. g(u(x)) x. (4) Lemma. (Poincaré Inequality). Let u W 1, (, M), an assume that W u 1 T M with W =. Then we have W L C 1() x α W (x) x, α=1 with C 1 () the Poincaré constant of the omain. g(u(x)) Proof. By the Poincaré inequality for f : x W (x) g(u(x)) R we get W L = W (x) g(u(x)) x = f L C 1 4 f x α α=1 L.

6 Using the Cauchy inequality for g we may then calculate an the assertion follows. f x α (x) = W (x), x W α g(u(x)) W (x) g(u(x)) x α W (x), g(u(x)) We will frequently work with functions whose erivatives are boune by a fixe constant K >. We therefore introuce the notation { } H K = H K (, M) := v W 1, (, M) : max α=1..., x α v(x) K. (5) g(v(x)) Note that functions in H K are necessarily continuous.. istance Measures in Nonlinear Function Spaces To quantify the error between a function u W k,p (, M) an an approximation v of u in the same space, we nee a istance measure on the nonlinear function space W k,p (, M). There are several ways to construct such a istance. The most natural one uses the embeing i use in efinition.1 to efine the space W k,p (, M). efinition.5. For all u, v W k,p (, M) efine ist emb,w k,p(u, v) := i(u) i(v) W k,p. (6) Since i is an isometry, the efinition yiels a istance metric. Also, it naturally egenerates to the stanar Sobolev istance if M is a linear space. This istance is convenient to evaluate, an efine even for functions u, v of little smoothness. However, aesthetically it is somewhat unpleasing, because it epens on the embeing i. A purely intrinsic istance can be efine using minimizing paths. efinition.6. Let H be a set of functions M, an u, v H. Suppose there is at least one continuously ifferentiable path γ from u to v in H. We enote by γ : [, 1] M the push-forwar along γ, i.e., the vector fiel efine by of t γ(x, t) := t γ(x, t) T γ(x,t)m. For each γ(t) H let there be a norm G on the space of vector fiels along γ(t), an efine ist G (u, v) := 1 inf γ path from u to v For each norm G we obtain a corresponing istance. efinition.7. For each u, v C(, M) efine an ist L (u, v) := ist L (u, v) := 1 inf γ path from u to v inf γ path from u to v Finally, for each u, v W 1, (, M) efine ist W 1,(u, v) := ( γ(t) L t = 1 γ(t) G t. ist(u(x), v(x)) x γ(t) L t = sup ist(u(x), v(x)). x 1 inf γ path from u to v ) 1/ γ(t) H 1 t. (7) 5

7 The minimizing curves with respect to ist L are calle geoesic homotopies. They have the following useful property. Remark.1. Let γ : [, 1] C(, M) be a geoesic homotopy. Then for each x, the curve γ(x, ) is a geoesic on M. Two functions that can be connecte by a geoesic homotopy are calle geoesically homotopic. efining istance using minimizing paths is a very elegant way of efining a istance, but it can be ifficult to work with. Insie our proofs we will therefore frequently use a thir error measure. It has a lot less mathematical structure than the two istance notions introuce above. However, we show below that it bouns both the embee an the path-inuce istance from above. For the efinition we nee the exponential map exp(, ) of M, as well as its inverse log(, ). For both maps, the first argument enotes the base point p M. That is, exp(p, ) : T p M M an log(p, ) : M U T p M. efinition.8. Let u, v W 1, (, M). efine the quantity 1, (u, v) := log(u(x), v(x)) g(u(x)) x + α=1 log(u(x), v(x)) xα g(u(x)) x. (8) In the linear case, this efinition coincies with the usual H 1 error. It is, however, not a metric, since it is neither symmetric nor oes it fulfill the triangle inequality. The following lemma states that 1, (u, v) provies an upper boun for i(u) i(v) H 1 for u, v H K as efine in (5). In the following we will write A B to say that a quantity A is boune by a quantity B times a constant. If also the converse estimate hols we will sometimes write A B. Lemma.3. For u H K an M isometrically embee into Eucliean space we have the estimate i(u) i(v) H 1 1, (u, v), with the implicit constant only epening on K, the embeing i, an the geometry of M. Proof. For simplicity we abuse notation an write i(u) = u, i(v) = v. Clearly we have u(x) v(x) ist(u(x), v(x)) = log(u(x), v(x)) g(u(x)) for almost all x, which takes care of the first term in the efinition of H 1. For the term associate with the erivative we put v(x) = exp(u(x), log(u(x), v(x))) an compute, using the notation 1 exp(p, w) = p exp(p, w), exp(p, w) = w exp(p, w) that x α v(x) = 1 exp ( u(x), log(u(x), v(x)) ) x α u(x) + exp ( u(x), log(u(x), v(x)) ) log(u(x), v(x)). xα Then, since 1 exp(p, )w = w for all p M, w T p M, we have x α u(x) = 1 exp(u(x), ) x α u(x). Hence, we can write the ifference x u(x) α x v(x) as a sum of the terms α [ I := 1 exp(u(x), ) 1 exp ( u(x), log(u(x), v(x)) )] x α u(x) an II := exp ( u(x), log(u(x), v(x)) ) log(u(x), v(x)). xα The quantity II can be boune in moulus by x log(u(x), v(x)), up to a constant. By the Lipschitz α continuity of 1 exp in its secon argument an the fact that x u(x) K by assumption, we can boun α I up to a constant by log(u(x), v(x)). This proves the statement. 6

8 Using a uniformity property of geoesic homotopies (which we prove in the following section), we can show that 1, also bouns the istance ist W 1, introuce in efinition.7. Lemma.4. For each u, v H K we have where C is the constant efine in (9). ist W 1,(u, v) C 1, (u, v), Proof. Let Γ a geoesic homotopy (L -geoesic) from u to v. Then ist W 1,(u, v) 1 Γ(t) H 1 t sup Γ(t) H 1 C inf Γ(t) H 1 C Γ() H 1 = C 1, (u, v). t [,1] t [,1].3 H 1 -Uniformity of Geoesic Homotopies The curves that inuce the ist W 1,-istance are ifficult to work with. The following result shows that geoesic homotopies are in some sense similar to these curves, provie the erivatives are boune by a constant K. This will allow us to work with geoesic homotopies, an still obtain bouns in the ist W 1,-istance. Lemma.5. Assume that u, v H K (, M) an that Γ is a geoesic homotopy from u to v. Then Γ(, t) H 1 C inf Γ(, t) H 1 with sup t [,1] t [,1] C = + / C 3 Rm g K ist L (u, v), (9) where Rm g is the maximum norm of the Riemann curvature tensor Rm [1], an C 3 only epens on the geometry of M. Proof. Since t Γ(x, t) is a geoesic we have that inepenent of t. Hence where we have efine We note that as well as the fact that Γ(x, t) = ist(u(x), v(x)), for almost all x, Γ(, t) H 1 = ist L (u, v) + U (t), (1) U (t) := satisfies the Jacobi ifferential equation α=1 x Γ(x, α t) x α t Γ(x, t) = t J α (x, t) := g(γ(x,t)) Γ(x, t), xα Γ(x, t) xα x. t J α (x, t) = Rm ( J α (x, t), Γ(x, t) ) Γ(x, t). 7

9 Using this we can write for every α = 1,..., t x Γ(x, α t), x Γ(x, α t) = g(γ(x,t)) t x Γ(x, α t), = t J α (x, t), ( = Rm x Γ(x, α t) x α Γ(x, t) For simplicity we shall omit the subscript g(γ(x, t)) from now on. Since we can write J α as J α (x, t) = g(γ(x,t)) g(γ(x,t)) J α (x, t), Γ(x, ) t) Γ(x, t), x Γ(x, α t) g(γ(x,t)) Rm gj α (x, t) g(γ(x,t)) x Γ(x, α t) Γ(x, t) g(γ(x,t)) x α exp ( u(x), t log(u(x), v(x)) ), we see that there exists a uniform constant C 3, only epening on the geometry of M such that ( ) J α (x, t) C 3 max x α u(x), x α v(x) C 3 K. g(γ(x,t)). We can use the previous consierations to boun the time erivative of U. Together with the Cauchy Schwarz inequality we have ( t U (t) C 3 Rm g K ist L (u, v) x Γ(x, 1/ α t) x) ist L (u, v) We ivie by U(t) to get U(t) = t The results above imply that α=1 (+1)/ C 3 Rm g K ist L (u, v) ist L (u, v)u(t). t U (t) U(t) ( 1)/ C 3 Rm g K ist L (u, v) ist L (u, v). U(t ) U(t 1 ) ( 1)/ C 3 Rm g K ist L (u, v) ist L (u, v) (11) for any t 1, t [, 1]. Now we can use (1) together with (11) to see that for t 1, t [, 1] we have Γ(, t 1 ) H 1 Γ(, t ) H 1 which finally proves the esire estimate..4 The Smoothness escriptor ist L (u, v) + U(t 1) ist L (u, v) + U(t ) [ 1 + U(t ] ) U(t 1 ) ist L (u, v) + U(t ) [ ] 1 + ( 1)/ C 3 Rm g K ist L (u, v) ist L (u, v) ist L (u, v) + U(t ) + / C 3 Rm g K ist L (u, v), We have given one efinition of Sobolev regularity of functions u : M in Section.1. A natural alternative is the covariant Sobolev norm u H k cov := ( β u(x) 1/ k g(u(x))) x, u = H k u cov H i cov. im β=k 8

10 Here the symbol β u means covariant partial ifferentiation along u with respect to the multi-inex β in the sense that β u := x β... k x β x u, β {1,..., } k, k N. (1) β1 Aitionally we efine β u := 1 (a constant function R) if im β =. For a shorter notation we introuce the symbol [] := {1,..., }. Note that (1) iffers from the usual multi-inex notation, which cannot be use because covariant partial erivatives o not commute. Clearly, for linear M, these efinitions coincie with the usual Sobolev half norms an norms (3). However, they cannot control all terms appearing in the nonlinear Bramble Hilbert-Lemma in Section 5 (etails are given in Remark 5.1). Therefore, we efine the following alternative. efinition.9 (Smoothness escriptor). For a function u : M, k 1 an p [1, ], efine the homogeneous k-th orer smoothness escriptor Θ p,k, (u) := β j [] m j, j=1,...,k kj=1 m j =k ( k β j u(x) j=1 p g(u(x)) x) 1/p, with the usual moifications for p =. Further, we efine the L p part ( 1/p Θ p,, (u) := min ist(u(x), q) x) p, q M an the corresponing inhomogeneous smoothness escriptor Θ p,k, (u) := k i= Θ p,i, (u). We will be mostly ealing with the case p =, for which we will omit the parameter p in the notation, i.e., Θ k, := Θ,k, an Θ k, := Θ,k,. Note that we use a superpose ot to enote homogeneous quantities. Remark.. A function u with Θ k, (u) < must be uniformly continuous if k > /. Furthermore, in that case we have iam(u) := sup ist(u(x), u(y)) Θ k, (u). x,y Both these assertions are irect consequences of the Sobolev embeing theorem. To better present the smoothness escriptors Θ we iscuss their relationships to other measures of regularity. For simplicity we restrict our analysis to the case p =. First, it follows irectly from the efinition that the smoothness escriptor Θ is a stronger notion than the covariant Sobolev norm. Lemma.6. u H k cov (,M) Θ k, (u). Proof. The proof follows immeiately by noting that all terms which occur in the efinition of u H k cov (,M) also occur in the efinition of Θ k, (u). In the other irection we show that the Sobolev norm with respect to an embeing also bouns Θ from above, if k is sufficiently large. Lemma.7. Let i be an isometric embeing of M into a Eucliean space. For k > we have Θ k,(u) i u k H k. Note that the smoothness escriptor is boune by the k-th power of the corresponing norm. 9

11 Proof. Ientify u with i u for simplicity. We nee to estimate terms of the form ( k β j u(x) x g(u(x)) j=1 ) 1 (13) with β j [] mj, j = 1,..., k, an k j=1 m j k. It will be no loss of generality to assume the most ifficult case k j=1 m j = k. First we euce from the efinition of the covariant erivative that any term of the form (13) can be estimate by a finite linear combination of terms of the form ( k k j u(x) x g(u(x)) j=1 ) 1, (14) with k j=1 k j = k. Now, for any values p j, j 1,..., k with k j=1 1 p j 1, by Höler s inequality we can boun (14) by ( k k j u(x) x g(u(x)) j=1 ) 1 k ( j=1 k j u(x) pj g(u(x)) ) 1 p j x k u W k j,p j. j=1 We make the specific choice 1 = 1 p j k k j + (k 1)(k /). k With this choice an k = k we have that k j=1 1 = k k(k 1) (k /)(k 1) + = 1 p j. We shall now use the Sobolev embeing theorem which states that whenever u W l,p u W k,, 1 k l < 1 p. Setting l = k j an p = p j for each j = 1,..., k we arrive at the esire statement. A result similar to Lemma.7 can also be establishe for p. In summary, our smoothness escriptor is an appropriate covariant way to measure smoothness of an M-value function. We finally show that the smoothness escriptor has a particular homogeneity property, also enjoye by conventional Sobolev seminorms in linear spaces. efinition.1. Let T 1, T be two omains in R, an F : T 1 T a C -iffeomorphism. For l N we say that F scales with h of orer l if we have sup x T k F 1 (x) h k for all k N, k =,..., l, (15a) et ( F(x)) h for all x T 1 (where F is the Jacobian of F), (15b) sup x T 1 x α F(x) h 1 for all α = 1,...,. Such an F will be use to move finite element functions to the reference element an back, without losing approximation orers, see Section 5 below. (15c) 1

12 Lemma.8. Let T 1, T be two omains in R, an F : T 1 T a map that scales with h of orer l. Then for any u : T 1 M, k l an p [1, ] we have Θ p,k,t (u F 1 ) h /p h k Θ p,k,t1 (u). Note that we boun the homogeneous smoothness escriptor by the inhomogeneous one. Proof. It follows irectly from the chain rule an the prouct rule that for any m N an β [] m the expression β ( u F 1) can be written as a linear combination of terms of the form τ u(f 1 (x)) l k i ( F 1 ) j i (x) with τ [] n, l k i = m, n m, l m, an ( F 1) j i enoting the ji -th coorinate of F 1. Using the scaling assumption (15a) we can therefore estimate the quantity β ( u F 1 ) g(u(f 1 (x))) by terms of the form τ u(f 1 (x)) l k i ( F 1 ) j i g(u(f (x) h m τ u(f 1 (x)). 1 (x))) g(u(f 1 (x))) Therefore, every integran k β j ( u F 1 ) (x) p g(u(f 1 (x))) j=1 in the efinition of the homogeneous smoothness escriptor Θ p,k,t (u F 1 ) can be estimate pointwise by terms of the form k h pk τj u ( F 1 (x) ) p (16) g(u(f 1 (x))) with j=1 τ j [] nj, n j N, k n j k. Now, integrating (16) over T, an using the substitution y = F 1 (x), introuces an aitional factor h /p. Together with the scaling assumption (15b) we obtain the esire estimate. Remark.3. The attentive reaer will have notice that only properties (15a) an (15b) have been use for the proof of Lemma.8. We will require the thir assumption (15c) later, when we use scaling to erive local element-wise interpolation error estimates in Theorem 5.3 below. 3 Ellipticity an Céa s Lemma Recall that we are trying to approximate the solution u of the variational problem j=1 u = arg min J(w) (17) w H by a minimizer v on a set V H v = arg min J(w), (18) w V 11

13 where H is a suitable set of functions, possibly fulfilling irichlet conitions. The classical linear Céa lemma assumes that H is a Hilbert space, an gives an estimate for the error between v an u in terms of the optimal approximation error inf w V u w H of the approximation space V [8]. In this section we show analogous results when H consists of manifol-value functions. We procee in two steps. Céa-type lemmas can be formulate an prove elegantly in general metric spaces. We show this in Section 3.1, an also give a reformulation for the case that H has a smooth structure with a Finsler norm. These results require certain convexity or ellipticity properties of the energy along istance-realizing curves. They are of inepenent interest, but they also illustrate some of the ieas of the subsequent section. There we allow variations over certain nonminimizing curves. The resulting Céa lemma is the basis of the iscretization error bouns for geoesic finite elements in Chapter Céa s Lemma Base on Variations Along Curves We start in an abstract setting. Suppose H is a metric space with istance function ist(, ), an u an v are solutions of the minimization problems (17) an (18), respectively. We will refer to v as a quasioptimal solution if ist(u, v) C inf ist(u, w) w V for a constant C >. The main assumption leaing to quasioptimality is a notion of strong convexity along curves. The following efinition is taken from [3]. efinition 3.1. A functional J : H R is calle λ-convex along the curve γ : [, 1] H if there is a λ > such that for all t [, 1]. J(γ(t)) (1 t) J(γ()) + t J(γ(1)) 1 λt(1 t) ist(γ(), γ(1)) With this assumption, a metric version of the Céa Lemma follows almost immeiately. Theorem 3.1. Assume that H is a metric space, an let J : H R. Suppose that u H is a minimizer of J an let V be a subset of H for which the minimization problem v := arg min J(w) w V has a solution. Assume that there exists a curve γ with γ() = u an γ(1) = v, along which the energy J is λ-convex. Further, assume that J is quaratically boune aroun u in the sense that there is a constant Λ > such that J(w) J(u) Λ ist(u, w) (19) for all w V. Then Proof. Inserting t = 1 ist(u, v) Λ λ inf ist(u, w). w V into the efinition of λ-convexity yiels that Since v is a minimizer on V, we can write ist(u, v) 4 (J(v) J(u)). λ ist(u, v) 4 λ inf (J(w) J(u)). w V By (19), the right-han-sie can be boune as esire. 1

14 A slightly more involve argument allows to get ri of the factor. We now consier the case that H has a ifferentiable structure, which implies that we can have curves γ : [, 1] H with well-efine tangent vectors γ. We also assume that there is a norm efine on these tangent vectors. The following alternative conition on J is frequently convenient. efinition 3.. We say that J is elliptic along a ifferentiable curve γ : [, 1] H if it is twice continuously ifferentiable along γ, an if there exist positive constants λ, Λ such that for all t [, 1]. λ γ(t) t J(γ(t)) Λ γ(t) () This concept is relate to convexity in the following way. Assume that for each pair w 1, w H there is a ifferentiable path from w 1 to w, parametrize by arc length, that realizes the istance ist(w 1, w ). We call such paths (constant-spee) geoesics. Lemma 3.1. Let J be elliptic in the sense of efinition 3. along a given constant spee geoesic γ : [, 1] H. Then J is λ-convex along that curve. If aitionally γ() is a minimizer of J, then γ is quaratically boune in the sense of (19) along γ, with constant Λ. In particular, we see that the requirements of Theorem 3.1 are strictly weaker, because they are implie by ellipticity, but require no smoothness. Proof. Set f := J(γ) : [, 1] R. By the ellipticity assumption f is twice continuously ifferentiable. We first show that the lower boun on f implies that J is λ-convex along γ. Pick t 1 an apply Taylor s formula to f at t. This gives an f() f(t) + f (t)( t) + 1 λ γ(s 1) ( t) f(1) f(t) + f (t)(1 t) + 1 λ γ(s ) (1 t), where s 1 t an t s 1. Since γ is a constant spee geoesic we have γ(s 1 ) = γ(s ) = ist(γ(), γ(1)). Multiply the first inequality by t, the secon one by 1 t, an a them to obtain the assertion. Next we show that f (t) Λ γ(t) implies J(γ(1)) J(γ()) Λ ist(γ(), γ(1)). Using that f () = by assumption we can irectly compute J(γ(1)) J(γ()) = 1 f (t) t 1 f () t = 1 = 1 t (1 t)f (t) t f (s) s t 1 (1 t)λ γ(t) t = Λ ist(γ(), γ(1)). As an immeiate consequence of Theorem 3.1 together with Lemma 3.1 we get the following result. Theorem 3.. Let H be a Banach manifol with norm, an J : H R a functional. Assume that u H is a minimizer of J, an that J is elliptic along constant spee geoesics, with constants λ, Λ. For a V H set v := arg min J(w), w V assuming that this is well-efine. Then we have that ist(u, v) Λ λ inf ist(u, w). w V 13

15 Using this theorem for the space W 1, (, M) together with the norm H 1 efine in efinition.4 yiels the following corollary, provie that W 1, (, M) is a Banach manifol. Unfortunately the latter only hols for = 1. Corollary 3.1. Assume that W 1, (, M) is a Banach manifol, an let J : W 1, (, M) R. Assume that u W 1, (, M) is a minimizer of J, an that J is elliptic along constant spee geoesics in W 1, (, M), starting in u. Let V W 1, (, M) an Then we have that v = arg min J(w) w V ist W 1,(u, v) (the iscrete solution). Λ λ inf ist W 1,(u, w). w V This corollary is the natural extension of the stanar Céa lemma to nonlinear function spaces. 3. Céa s Lemma Using Geoesic Homotopies When trying to apply the results of the previous section we encounter two problems. First, for the energies J an omains of our interest we consier variational problem formulations in W 1, (, M) an in general this space oes not possess the structure of a Banach manifol [16, 6, 7]. Hence the results base on Banach manifols cannot be use. Seconly, even if the space W 1, (, M) turns out to be a Banach manifol, it is ifficult to work with constant spee geoesics in these spaces. In particular it is not easy to verify ellipticity properties along these curves for important energies, such as the harmonic energy. To overcome these issues we generalize the approach somewhat. Instea of geoesics in W 1, we now consier geoesic homotopies. However, we still obtain bouns in terms of a W 1, -like measure, namely the quantity 1, introuce in (8). While this quantity is of little interest in itself, the result will allow to boun the iscretization error of geoesic finite elements in terms of the embee istance (6) an the geoesic istance (7). The proof is base on the H 1 -uniformity of geoesic homotopies shown in Section.3. The price we pay is that we aitionally have to assume the existence of a constant K > such that u H K (the first weak erivatives are boune by K), an V H K. Theorem 3.3. Let H W 1, (, M), J : H R an K >. Assume that u H H K (, M) is a stationary point of J w.r.t. variations along geoesic homotopies in H H K (, M) starting in u. For a secon constant L > enote H u L := { v : ist L (u, v) L }, an assume that J : H K HL u H R is elliptic along geoesic homotopies that start in u. Let V H K HL u H an v := arg min J(w). w V Then we have that Λ 1, (u, v) C λ inf 1,(u, w), w V with C the uniformity constant (9), only epening on, the prouct KL an the curvature of M. Proof. For w V efine f w (t) := J(Γ(t)) with Γ : [, 1] H K HL u H a geoesic homotopy from u to w. We have J(w) J(u) = 1 f w(t) t 1 By the ellipticity assumption () we have λ 1 f w() t = 1 t f w(s) s t = 1 1 (1 t) Γ(t) H t J(w) J(u) Λ (1 t) Γ(t) 1 H t (1 t)f w(t) t.

16 Now we use Lemma.5 which shows that λ Γ() H J(w) J(u) 1 ΛC Γ() H 1, C where the constant C epens only on, K, L an the curvature of M. Noting further that immeiately yiels that 1, (u, v) C λ for all w V. Furthermore we have 1, (u, w) = Γ() H 1 [ ] C [ ] J(v) J(u) J(w) J(u) λ J(w) J(u) C Λ 1,(u, w). Together, we obtain that 1, (u, v) C Λ λ inf w V 1,(u, w). Replacing Γ() H 1 by Γ(1) H 1 in the arguments above shows the following corollary. Corollary 3.. With the notation of the previous theorem, we also have the estimate Λ 1, (v, u) C λ inf 1,(w, u). w V The restriction that u H K oes not appear in the linear theory. The question whether Theorem 3.3 can be shown without it is open. Remark 3.1. Requiring that the approximation space V consists only of functions with erivatives boune by K may lea to a restriction when consiering families of approximation spaces V h associate with a mesh with h >. If V h are chosen as GFE spaces as introuce in Section 4 we show in Theorem 6. that the conition V h H K can be ispense with, provie that u is sufficiently regular. 4 Geoesic Finite Elements In Chapter 3 very little has been require from the approximation spaces V. For the theory base on istance-realizing curves in Section 3.1 only the existence of a minimizer of J in V was aske. In Section 3. we aitionally neee that the approximating functions that make up V have their erivatives boune by a constant K. In this section we present geoesic finite elements (GFE) as one particular example of a suitable space V. They have originally been introuce in [43, 44], but for completeness we give a brief review. The efinition consists of two parts. First, nonlinear interpolation functions are constructe that interpolate values given on a reference element. Then, for a given gri, these interpolation functions are piece together to form global finite element functions. 4.1 Geoesic Interpolation Let T ref be an open boune subset of R, which we will call reference element. Particular instances are the reference simplex { x R x α, α = 1,...,, α=1 xα 1 }, an the reference cube [, 1]. On T ref we assume the existence of a set of Lagrangian interpolation polynomials, i.e., a set of Lagrange noes a i T ref, i = 1,..., m, an corresponing polynomial functions λ i : T ref R of orer p such that λ i (a j ) = δ ij for all 1 i, j m, 15

17 an m λ i 1. (1) We now generalize Lagrange interpolation to values in a manifol. Let v i M, i = 1,..., m be given values at the Lagrange noes a i T ref. We want to construct a function Υ v : T ref M such that Υ v (a i ) = v i for all i = 1,..., m. The following efinition was given an motivate in [44]. efinition 4.1. Let {λ i, i = 1,..., m} be a set of p-th orer scalar Lagrangian shape functions, an let v i M, i = 1,..., m be values at the corresponing Lagrange noes. We call p-th orer geoesic interpolation on M. Υ : M m T ref M m Υ(v 1,..., v m ; x) = arg min q M λ i (x) ist(v i, q) () For fixe coefficients v 1,..., v m we set Υ v ( ) := Υ(v 1,..., v m ; ) an obtain the esire function. It is easy to verify that this efinition reuces to p-th orer Lagrangian interpolation if M is a linear space an ist(, ) the stanar istance. For the nonlinear case an p = 1, well-poseness of the efinition uner certain restrictions on the v i is a classic result by Karcher [34]. For p, where the λ i can become negative, well-poseness has been prove in [44]. The interpolation function Υ is infinitely ifferentiable both as a function of the v i an of the local coorinates x. This an several other features is iscusse in [43, 44]. Since the values of Υ v are efine as solutions of a minimization problem, we can also characterize them by the corresponing first-orer optimality conition (see, for instance, [34]). We will make use of this representation in the interpolation error boun in Chapter 5. Lemma 4.1. For any q M enote by log(q, ) : M U T q M the inverse of the exponential map of M at q. Then q := Υ(v 1,..., v m ; x) is (locally uniquely) characterize by the first-orer conition m λ i (x) log(q, v i ) = T q M. (3) Interpolation error bouns for geoesic finite elements are base on the fact that the shape functions λ i are exact on polynomials of egree no greater than p, meaning that m λ i (x)q(a i ) = q(x) (4) for all polynomials q : T ref R of egree less than or equal to p. Using this we can prove the following technical property. Lemma 4.. For all multi-inices l with l p an all functions f : T ref R we have m λ i (x) (a i x) l f(x) =. Proof. We start by fixing some arbitrary x T ref. Then we can write m m λ i (x) (a i x ) l f(x ) = λ i (x)p x (a i ) where p x (y) := (y x ) l f(x ) 16

18 is a polynomial of egree l. By (4) we get m λ i (x) (a i x ) l f(x ) = p x (x). Since by efinition p x (x ) =, this implies m λ i (x ) (a i x ) l f(x ) =, which, by the arbitrariness of x, implies the statement. 4. Geoesic Finite Element Functions Let now be a omain in R. Suppose we have a conforming gri G for with elements not necessarily restricte to simplices. Let n i, i = 1,..., n be a set of Lagrange noes such that for each element T of G there are m noes a T,i containe in T, an such that the p-th orer interpolation problem on T is well pose. efinition 4. (Geoesic Finite Elements). Let G be a conforming gri on, an let M be a Riemannian manifol. We call v h : M a geoesic finite element function for M if it is continuous, an for each element T G the restriction v h T is a geoesic interpolation in the sense that v h T (x) = Υ ( v T,1,..., v T,m ; F T (x) ), where the F T : T T ref are element mappings (typically affine or multilinear), an the v T,i are values in M corresponing to the Lagrange noes a T,i. The space of all such functions v h will be enote by V M p,g. This efinition reuces to stanar (vector-value) Lagrangian finite elements if M is a linear space with the usual Eucliean istance. The following property is crucial for our analysis, because we always assume that the approximation space V is a subset of the solution space. The proof is given in [43] an [44]. Theorem 4.1. V M p,g () H1 (, M) for all p 1. While this hols for all gris G an all polynomial orers p, we note that geoesic finite element spaces are generally not neste. This means that in general Vp,G M V p+1,g M an V p,g M V p,g M if G is a uniform refinement of G. See [44, Chap. 4] for a brief iscussion. Remark 4.1. In numerical algorithms one uses the algebraic representation of Vp,G M, that is, a function v h Vp,G M is ientifie with a set of noal coefficients v M n. However, note that Vp,G M is not globally homeomorphic to M n ; in fact, it is not even globally a manifol. This is so because for certain sets of coefficients there is more than one interpolation function (see [4] for a simple example). On the other han, it is shown in [44] that for many v M n there is only a single interpolating function v h, an then there is a iffeomorphism mapping a neighborhoo of v in M n to a neighborhoo of v h in Vp,G M. In this sense the space Vp,G M contains many small manifol patches. Its global structure, however, remains unclear. To prove quasi-optimality in Theorem 3.3 we ha to make the assumption that the iscrete space V contains only functions with first erivatives boune by a global constant K. While it is obvious that each GFE function has boune first erivatives, a global boun for all functions of a space Vp,G M exists only if M has finite iameter. This global boun epens on the gri size h. The specific nature of this epenence will allow us in Theorem 6. to circumvent the restriction V H K an obtain iscretization error bouns without constraints on the ansatz space. For later use there we therefore state the following simple result, which hols for all orers p, an for M with boune or unboune iameter. Lemma 4.3. Let G be such that F T scales with h of orer p for each element T of G. Then for each function v h Vp,G M we have Θ,1, (v h ) Ch 1, where the constant C epens on the values of v h at the Lagrange noes. 17

19 In orer to assess the approximation properties of the spaces Vp,G M, we finally construct the pointwise interpolation operator mapping continuous functions with values in M to elements in Vp,G M. As in the classical linear case we first efine the interpolant on a reference element. We start by fixing the reference element T ref with Lagrangian interpolation noes a i an corresponing local basis functions λ i, i = 1,..., m. Given a function u : T ref M its local Lagrangian interpolant on T ref is efine by I Tref u(x) := Υ (u(a 1 ),..., u(a m ); x). Likewise, for a general element T with associate mapping F T : T T ref, the local Lagrangian interpolant is given by ( ) I T u(x) = I Tref u F 1 T (FT (x)), x T. With these notions at han we can efine the geoesic Lagrange interpolant of a continuous function u : M. efinition 4.3. For each continuous function u : M efine the geoesic Lagrange interpolant I G u V M p,g by I G u(x) = I T u(x), x T, T G. Note that unlike in the linear case, this interpolant is not always unique. 5 Interpolation Error Estimates The goal of this section is to erive estimates of optimal orer for the interpolation error between a function u : M an its interpolant I G u. To motivate our proof, we briefly review how interpolation error estimates can be obtaine in the linear case M = R. There, we start with an error boun on the reference element. Theorem 5.1. Let u : T ref R satisfy u H k (T ref ) with k > /. Then we have u I Tref u H1 (T ref ) u H k (T ref ). In orer to turn Theorem 5.1 into an estimate for a small element T, say, T = ht ref with F T (x) := h 1 x an a function u : T R we use the fact that the Sobolev seminorm satisfies the subhomogeneity property u F 1 T H k (T ref ) h k h / u Hk (T ). (5) We obtain the factor h k from the k-fol application of the chain rule to u F 1 T (x) = u(hx), an the factor h / from the integral transformation formula. Then, enoting v := u F 1 T : T ref R an using x F α T = h 1 for all α = 1,...,, we get u I T u H 1 (T ) = h α=1 We can now invoke Theorem 5.1 an get the estimate T ( x α v ) x α I T ref v F T (x) x = h h v I Tref v H 1 (T ref ). u I T u H 1 (T ) h h u F 1 T H k (T ref ) which, together with (5) yiels the classical estimate u I T u H 1 (T ) h k 1 u H k (T ). To obtain a similar result for nonlinear coomains M we first nee a generalization of Theorem 5.1. We prove such a result in Section 5.1, where the norm on the left becomes the quantity 1,, an the norm on the right becomes the smoothness escriptor Θ. Then, in Section 5. we assemble these local estimates to establish optimal approximation rates for the geoesic finite element spaces Vp,G M. This works because the smoothness escriptor Θ also has the subhomogeneity property (5) (Lemma.8). 18

20 5.1 Nonlinear Elementwise Estimates In this section we prove a nonlinear generalization of the linear element-wise approximation result of Theorem 5.1. We cope with the implicit efinition () of the interpolant by a clever use of the equilibrium conition (3). Let log(p, ) : M T p M be the inverse of the exponential map at p. enote by 1, the covariant erivative of a bivariate function with respect to the first an secon argument, respectively. In particular, for l N we will require the erivatives an more precisely their norms an l log(p, q) = l 1 log(p, q) = l log(p, q) : (T q M) l T p M l 1 log(p, q) : T p M (T q M) l T p M; l log(p, q) (v 1,..., v l ) g(p) sup l v 1,...,v l T qm v i g(q) l 1 log(p, q) (w, v 1,..., v l ) g(p) sup l v 1,...,v l T qm w g(p) v. i g(q) w T pm Now we can state an prove a nonlinear elementwise approximation result. Theorem 5.. Let u W k, (T ref, M), an I Tref u its p-th orer geoesic interpolation. For k > / an p k 1, we have T ref log(itref u(x), u(x)) g(i Tref u(x)) x C 1,u(T ref ) Θ k,tref (u), (6) an for any α = 1,..., x α log(i T ref u(x), u(x)) where an T ref C 1,u (T ref ) := sup g(i Tref u(x)) 1 l k C,u (T ref ) := sup 1 l k x ( C 1,u(T ref ) + C,u(T ref ) ) Θk,Tref (u), (7) sup p I Tref u(t ref ) q u(t ref ) sup p I Tref u(t ref ) q u(t ref ) l log (p, q) l 1 log (p, q). The implicit constants are inepenent of u an M an only epen on the basis functions λ i. Note that the left han sies of (6) an (7) make up the quantity 1, (I Tref u, u). Proof. We split the proof into eight steps. Step 1 We first prove (6). Using the balance law (3) we obtain for any x T ref that m λ i (x) log (I Tref u(x), u(a i )) =. Aing a zero we rewrite this as log(i Tref u(x), u(x)) = log(i Tref u(x), u(x)) m λ i (x) log (I Tref u(x), u(a i )), an call the right han sie ε(x) T ITref u(x)m. To obtain (6), we nee to control the L -norm of the function ε. 19

21 Step Next we efine the helper function G(x, y) := log (I u(x), u(y)), an perform a Taylor expansion of G in its secon argument aroun y = x (note that for fixe x, the function G takes its values in a vector space). In what follows we shall use the notation k y G(x, y) for the partial erivatives of G with respect to its secon argument an the multi-inex k. The Taylor expansion then reas where G(x, y) = l <k (y x) l l y G(x, x) + R k (x, y)(y x) k, (8) l! k =k R k (x, y) = k 1 (1 t) k 1 k y G(x, x + t(y x)) t. k! We can express the terms log (I Tref u(x), u(a i )) occurring in the efinition of ε in the form (8) an get m ε(x) = G(x, x) λ i (x) (a i x) l l y G(x, x) + R k (x, a i )(a i x) k. l! l <k k =k Using that the weight functions λ i form a partition of unity on T ref (1) we get ε(x) = m < l <k λ i (x) (a i x) l l! l y G(x, x) + where the zeroth orer erivative cancels with G(x, x). m λ i (x) R k (x, a i )(a i x) k, (9) Step 3 By the assumption p k 1 we can apply Lemma 4. with f(x) = l y G(x, x) to each sum m λ i(x) (ai x) l l! l y G(x, x) in (9) an see that the first aen in (9) is zero. Hence we can write ε(x) as the sum m ε(x) = ε i (x) with ε i (x) := λ i (x) R k (x, a i )(a i x) k. We now treat each term ε i separately. For simplicity, we may assume, after a suitable translation (epening on the inex i) that a i = an thus we arrive at the pointwise estimate ε i (x) g(itref u(x)) k 1 t k 1 k y G(x, tx)x k t k! k =k where we have use that the λ i are boune on T ref. k =k k =k g(itref u(x)) k 1 k! t k 1 k =k k y G(x, tx) g(itref u(x)) x k t, (3) Step 4 In orer to untangle the erivatives of log an u in the expression k y G(x, y), we use the chain rule which yiels k ( ) l β y G(x, y) log (I Tref u(x), u(y)) 1 β u(y),..., l u(y). (31) g(itref u(x)) We repeat that we use the notation 1 l k, β j [] m j l j=1 mj=k l log(p, q) : (T q M) l T p M to enote the l-th orer covariant erivative of the function q log(p, q), which is an l-multilinear form.

22 Remark 5.1. We recor here that this is the point where the smoothness escriptor Θ (efine in Section.4) becomes necessary. Inee, (31) alreay inicates that control over proucts of covariant erivatives of lower orer is require whenever l log. Note also that in the linear case we have l log = for all l > 1 an therefore the usual Sobolev seminorm H k is sufficient to obtain the esire control over terms of the form (31). Keeping this in min it is easy to see that in the linear case our proof yiels exactly the expecte bouns for the interpolation error in the Sobolev seminorm. By (3) we get ε i (x) g(i Tref u(x)) x T ref ( 1 T ref ( 1 t k 1 1 l k, β j [] m j l j=1 mj=k t k 1 1 l k, β j [] m j l j=1 mj=k k y G(x, tx) x t) k x g(itref u(x)) T ref ( ) l β log (I Tref u(x), u(tx)) 1 β u(tx),..., k u(tx) x t) g(itref k x u(x)) T ref ( 1 t k 1 β H 1,..., β l (x, tx)x t) β x, where we have put H β 1,..., β l (x, y) := l log (I Tref u(x), u(y)) l β j u(y). g(u(y)) Step 5 In the appenix we have collecte a few estimates for remainer terms of Taylor series. We can β use Lemma A.1 for the functions H 1,..., β l, which gives us the boun ε i (x) β g(i Tref u(x)) x sup H 1,..., βl (x, y) g(u(y)) y T ref x T ref 1 l k, β j [] m j l j=1 mj=k sup 1 l k sup p I Tref u(t ref ) q u(t ref ) T ref = C 1,u(T ref ) Θ k,tref (u). This conclues the first part of the proof. l log (p, q) j=1 1 l k, β j [] m j l j=1 mj=k T ref j=1 l β j u(x) x g(u(x)) Step 6 We now turn to the estimate for the first erivatives. To that en we nee to boun the L -norm of x ε(x). Since the functions λ α i have uniformly boune first erivatives on T ref, by the prouct rule, we can further reuce the problem to bouning the L -norm of for α = 1,...,. Using the chain rule we get x α k =k x α k =k k 1 t k 1 k y G(x, tx)x k t k! k 1 t k 1 k y G(x, tx)x k t = I(x, x) + II(x), k! 1