A Higher Order Finite Element Method for Partial Differential Equations on Surfaces

Transcription

1 A Higher Order Finite Element Method for Partial Differential Equations on Surfaces Jörg Grande and Arnold Reusken Preprint No. 403 July 2014 Key words: Laplace Beltrami equation, surface finite element method, high order, gradient recovery, error analysis AMS Subject Classifications: 58J32, 65N15, 65N30, 76D45, 76T99 Institut für Geometrie und Praktische Mathematik RWTH Aachen Templergraben 55, D Aachen, Germany) Institut für Geometrie und Praktische Mathematik, RWTH-Aachen University, D Aachen, Germany; Institut für Geometrie und Praktische Mathematik, RWTH-Aachen University, D Aachen, Germany;

2 A HIGHER ORDER FINITE ELEMENT METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS ON SURFACES JÖRG GRANDE AND ARNOLD REUSKEN Abstract. A new higher order finite element method for elliptic partial differential equations on a stationary smooth surface is introduced and analyzed. We assume is characterized as the zero level of a level set function φ and only a finite element approximation φ h of degree k 1) of φ is known. For the discretization of the partial differential equation, finite elements of degree m 1) on a piecewise linear approximation of are used. The discretization is lifted to h, which denotes the zero level of φ h, using a quasi-orthogonal coordinate system that is constructed by applying a gradient recovery technique to φ h. A complete discretization error analysis is presented in which the error is split into a geometric error, a quadrature error, and a finite element approximation error. The main result is a H 1 )- error bound of the form ch m + h k+1 ). Results of numerical experiments illustrate the higher order convergence of this method. Key words. Laplace Beltrami equation, surface finite element method, high order, gradient recovery, error analysis AMS subject classifications. 58J32, 65N15, 65N30, 76D45, 76T99 1. Introduction. In the past decade the study of numerical methods for PDEs on surfaces has been a rapidly growing research area. The development of finite element methods for solving elliptic equations on surfaces can be traced back to the paper [8], which considers a piecewise polygonal surface and uses a finite element space on a triangulation of this discrete surface. This approach has been further analyzed and extended in several directions, see, e.g., [9, 10] and the references therein. Another approach has been introduced in [4] and builds on the ideas of [2]. The method in that paper applies to cases in which the surface is given implicitly by some level set function and the key idea is to solve the partial differential equation on a narrow band around the surface. Unfitted finite element spaces on this narrow band are used for discretization. Another surface finite element method based on an outer bulk) mesh has been introduced in [12] and further studied in [11, 6]. The main idea of this method is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface by considering traces of the bulk finite element space on the surface, instead of extending the PDE off the surface, as in [2, 4]. Most of the methods mentioned above have been studied both for stationary and evolving surfaces. In all the papers we know of, except for [5], the discretization that is studied is based on piecewise linear finite elements. The paper [5] is the only one in which higher order finite element methods for partial differential equations on stationary) surfaces are studied. We outline the key results of that paper. For a smooth bounded and connected surface R 3 we consider the Laplace-Beltrami problem: for given f L 2 ) with f ds = 0 determine u H1 ) := { u H 1 ) u ds = 0 } such Institut für Geometrie und Praktische Mathematik, RWTH-Aachen University, D Aachen, Germany; grande@igpm.rwth-aachen.de Institut für Geometrie und Praktische Mathematik, RWTH-Aachen University, D Aachen, Germany; reusken@igpm.rwth-aachen.de 1

3 that u v ds = fv ds for all v H 1 ). 1.1) It is assumed that is represented as the zero level of a smooth signed distance function d. The exact surface is approximated by a quasi-uniform shape-regular polyhedral surface ˆ h having triangular faces, and with vertices on. Based on the distance function d a parametric mapping, consisting of piecewise polynomial mappings of degree k, is defined on ˆ h, which results in a corresponding discrete surface ˆ k h. Using the same mapping a standard higher order finite element space on ˆ h is lifted to ˆ k h. This lifted space on ˆ k h is used for the discretization of 1.1). An extensive error analysis of this method is presented in [5], resulting in optimal error bounds. For example, for the H 1 ) error where the discrete solution is lifted to ) a bound of the form ch m + h k+1 ) is proved. Here k is the degree of the polynomials used in the parametrization of hat k h and m the degree of the polynomials in the finite element space on ˆ h. We emphasize that in this method explicit knowledge of the exact signed distance function to is an essential requirement. In many applications the exact signed distance function to the surface is not known. One often encounters situations in which is the zero level of a smooth level set function φ not necessarily a signed distance function) and one only has a finite element approximation of φ available. This paper deals with the question: how) can one develop a higher order finite element method in such a setting? We will present a constructive affirmative answer to this question. We restrict ourselves to the model problem 1.1) with a stationary surface. We assume to be sufficiently smooth. Our approach is fundamentally different from the one in [5], in the sense that we do not need the exact distance function d. Instead, we only!) need a finite element approximation φ k h of a level set function φ, which has as its zero level. The discrete level set function φ k h comes from a standard finite element space on a quasi-uniform triangulation of a bulk domain that contains. In the error analysis we assume that φ k h satisfies an error bound of the form φ k h φ L U) + h φ k h φ H 1 U) ch k+1, 1.2) where U is a small) neighborhood of in R 3. The zero level of φ k h is denoted by k h. Note that for k > 1, k h cannot be easily constructed. From 1.2) it follows that dist, k h ) chk+1 holds. The method that we introduce is new and is built upon the following key ingredients: For k = 1 the function ˆφ h := φ 1 h is piecewise linear, hence its zero level is piecewise planar, consisting of quadrilaterals and triangles, and can easily be determined. The quadrilaterals are subdivided into triangles. The resulting triangulation is denoted by ˆ h. This triangulation is in general very shape-ir regular. Nevertheless, the trace of an outer finite element space or a standard finite element space directly on ˆ h turns out to have optimal approximation properties [12, 13]. Such a finite element space on ˆ h is denoted by Ŝh. We take k > 1. For the parametrization of k h we use a quasi-normal field, as introduced in [14]. Given φ k h we apply a gradient recovery method which results in a Lipschitz continuous vector field n h that is close to the normal field n that corresponds to φ. Using this quasi-normal field, there is a unique 2

4 decomposition x = p h x) + d h x)n h p h x)) for all x in a neighborhood of k h, with p hx) k h and d h R an approximate signed distance function. It can be shown that p h : ˆh k h is a bijection. This p h is used for the parametrization of k h. For given x ˆ h its image p h x) k h can be determined with high accuracy) using the known field n h and only few evaluations of φ k h. Using the parametrization p h the finite element space Ŝh on ˆ h is lifted to k h and used for a Galerkin type discretization of 1.1), i.e. we take 1.1) with replaced by k h, H1 ) replaced by the lifted finite element space, f suitably extended, and instead of we use the tangential gradient along k h. Only evaluations of p h and Dp h can be computed. Hence, quadrature is needed. The finite element space is pulled back to ˆ h, integrals over k h are transformed to integrals over ˆ h and quadrature is applied on triangles in ˆ h. We then only) need evaluations of p h, Dp h, and of exact normals on ˆ h and on k h. The latter are easily determined using φk h. The method is described more precisely in section 5. The implementation is discussed in section 14. Apart from the new discretization method outlined above, the main contribution of this paper is an error analysis of this method. A key point related to this is the following. On each triangle T of the base triangulation ˆ h the parametrization p h is only Lipschitz. This low regularity is due to the construction of the quasi-normal field n h. Hence, the bilinear form pulled back to ˆ h consists of a sum of integrals of the form T G ˆh û h ˆh ˆv h dŝ h with a function G that has very low smoothness not even continuous). Due to this the analysis of the quadrature error is not straightforward. This lack of smoothness is also an important reason why the analysis in this paper is even) more technical than the one in [5]. The structure of the error analysis is outlined in section 6. As a main result, cf. Theorem 13.1, we prove an H 1 ) error bound where the discrete solution is lifted to ) of the form ch m + h k+1 ). Here m is the degree of the polynomials used in the finite element space Ŝh. 2. Preliminaries. Let φ be a smooth function with a smooth, bounded and connected zero level set Ω R 3, and let Ω 1 = {x Ω φx) 0} be the enclosed compact) region. Furthermore U is a small) open subset of R 3 with U Ω. This neighborhood is sufficiently small such that on U we have a local coordinate system x = px) + dx)npx)), 2.1) with n the normal vector field on pointing out of Ω 1 ), p : U and d the signed distance function to negative in Ω 1 ). For every x U the normal field has the unique value nx) = npx)). We assume that φx) c 0 > 0 for all x U holds. Let {T h } h>0 be a family of regular quasi-uniform tetrahedral triangulations on Ω. Furthermore V k h denotes a standard FE space on T h consisting of continuous piecewise polynomial functions of degree k. Remark 1. Some of the assumptions introduced above are used to simplify the presentation and not essential for the applicability of the method or the validity of the error analysis. For example, R 3 could easily be replaced by R n, with n 2. Also the extension to a surface with a finite number of connected components is straightforward. Finally, with minor modifications the analysis also applies if {T h } h>0 is a shape-regular not necessarily quasi-uniform) family of triangulations. The assumption that is a smooth surface is an essential one. 3

5 Let φ k h V k h be an approximation of φ that satisfies φ k h φ L U) + h φ k h φ H 1 U) ch k ) In the remainder we take a fixed value k 1. To simplify notation, we write φ h = φ k h. The linear finite element approximation φ 1 h plays a special role and is denoted by ˆφ h. The zero level sets of φ h and ˆφ h are denoted by h and ˆ h, respectively. The outward pointing normal fields on h and ˆ h are denoted by n h and ˆn h, respectively. From 2.2) we obtain, cf. [5]: dist, ˆ h ) ch 2, dist, h ) ch k+1, n ˆn h L h ) ch, n n h L h ) ch k. 2.3) The two eigenvalues of the Weingarten map H := D 2 d R 3 3 corresponding to the eigenvectors orthogonal to nx) are denoted by κ i x), i = 1, 2. These are related to the principal curvatures of by the formula κ i x) = κ i px))/ 1 + dx)κ i px) ). In [5] it is shown that for the surface measures ds h on h and ds on we have the relation µ h x) ds h x) = dspx)), with µ h x) = nx) T n h x) 2 1 dx)κ i x)), x h. 2.4) i=1 Using this formula and the results in 2.3) one obtains: µ h 1 L h ) c d L h ) + c 1 n T n h L h ) = c d L h ) + c n n h 2 L h ) chk ) Here and in the remainder, c is used to denote different constants, which are all independent of h. We need an Oh) neighborhood of h, denoted by Ω h, consisting of all tetrahedra with distance to h smaller than ch, with a given c > 0. We assume that h is sufficiently small such hold, cf. 2.3). h Ω h U and ˆ h Ω h 3. Quasi-normal field. In this section we define the notion of a quasi-normal field, as introduced in [14]. Such a quasi-normal field is constructed using a simple) gradient recovery technique. Only this field, and not the gradient recovery technique, is then used in the finite element method further on. A gradient recovery operator is a mapping G h : V k h V k h )3, which has to satisfy certain reasonable approximation and stability conditions. Assumption 3.1. Let I h be the nodal interpolation in the finite element space V k h. We assume that for φ sufficiently smooth the gradient recovery method G h : V k h V k h )3 satisfies: G h I h φ) φ L U) ch k, 3.1) G h v h L U) c v h H 1 U e ) for all v h V k h. 3.2) 4

6 Here U e denotes the neighborhood U enlarged with a suitable patch of surrounding elements. Remark 2. In the literature gradient recovery techniques are known and often used in error estimators, cf. [1]. In such a setting one usually requires a power k + 1, instead of k, in 3.1). In [14] the polynomial-preserving recovery PPR) technique is considered. For the PPR technique, 3.2) and 3.1) with k + 1 are shown to hold in two dimensions in [15]. To indicate that the conditions 3.1) and 3.2) are mild ones, as an example we describe a very simple gradient recovery technique satisfying Assumption 3.1. It is used in the experiments in section 14. The set of finite element nodes is denoted by N h. To each finite element node ξ N h we assign the set T ξ of all tetrahedra containing ξ. For ξ U e this T ξ is chosen such that T T ξ T U e. Let n ξ := T ξ. The gradient recovery is defined by simple local averaging, namely Gv h )ξ) := 1 n ξ T T ξ v h Tξ ξ) for all ξ. Let φ h = I h φ be the nodal interpolation of a smooth function φ. From standard interpolation theory we get max G 1 hφ h )ξ) φξ) = max ξ N h U e ξ N h U e n ξ Hence, using I h G h φ h ) = G h φ h we get T T ξ φ h Tξ ξ) φξ)) c max φ T T h U e h φ L T ) ch k φ H k+1 U ). e G h φ h φ L U) I h G h φ h φ) L U) + I h φ) φ L U) c max G hφ h )ξ) φξ) + ch k φ H ξ N h U e k+1 U e ) ch k φ H k+1 U ), e and thus the condition in 3.1) is satisfied. With similar arguments, using stability properties of I h, one can verify that for this simple recovery operator condition 3.2) is satisfied, too. Properties of different gradient recovery techniques with respect to the construction of a quasi-normal field will be analyzed in a forthcoming paper. For the analysis in this paper it suffices to assume that we use a gradient recovery technique that has the properties given in Assumption 3.1. Given the gradient recovery operator G h we apply it to φ h = φ k h and define the quasi-normal field: n h x) = G hφ h )x) G h φ h )x), x Ω h. 3.3) Note that this field is only Lipschitz continuous; a main point in the analysis is that n h can be approximated by a smooth vector field cf. Lemma 7.1). The result 3.5) in the following lemma explains why we call n h a quasi-normal field. A proof of this lemma is given in [14] where the power k + 1 instead of k is assumed in 3.1)). Since the lemma is of fundamental importance for the analysis in this paper, we include a proof. By Bx; r) we denote the ball with center x and radius r. Lemma 3.1. Let Assumption 3.1 be satisfied. Let r x > 0 depending on x) be small enough such that Bx, r x ) U for all x h. There exist constants c and h 0 > 0 such that for all h h 0 and all x h the following holds: n h x) n h y) c x y, for all y Bx; r x ), 3.4) n h x), x y ch k x y + c x y 2, for all y h Bx; r x ). 3.5) 5

7 Proof. Take x h and y Bx; r x ) U. From the definition of n h we get n h x) n h y) 2 G hφ h )x) G h φ h )y). 3.6) G h φ h )x) We write G h φ h = G h φ h I h φ) + G h I h φ) φ ) + φ, and using 3.1), 3.2), 2.2), φx) c 0 > 0 and an interpolation bound we get G h φ h )x) φx) c φ h I h φ H 1 U e ) c G h I h φ) φ L U) c 0 ch k 1 2 c 0, 3.7) provided h is sufficiently small. The vector function G h v h Vh 3 is Lipschitz continuous and G h φ h )x) G h φ h )y) 1 0 G h φ h )x + tx y)) dt x y 3.8) holds. We write z := x + tx y) Bx; r x ) and note that G h φ h )z) G h φ h I h φ) ) L U) + I h φ) L U). Using an inverse inequality and the boundedness of I h in H, 1 3.1) and 3.2) we get G h φ h )z) ch 1 G h φ h I h φ) L U) + c ch 1 G h φ h I h φ) L U) + ch 1 G h I h φ) φ L U) + ch 1 φ I h φ) L U) + c ch 1 φ h I h φ H 1 U e ) + ch k 1 + c ch k 1 + c c. 3.9) Using this result in 3.8), in combination with 3.7) and 3.6) proves 3.4). Now assume y h. The definition of n h and the lower bound in 3.7) yield n h x), x y 2 c L G h φ h )x), x y. 3.10) Since x, y h we have 0 = φ h x) φ h y) = 1 0 φ hx + ty x)), x y dt, hence, G h φ h )x), x y = G h φ h )x) G h φ h )x + ty x)), x y dt G h φ h )x + ty x)) φ h x + ty x)), x y dt. 3.11) From the Lipschitz continuity estimate 3.8)-3.9), with y replaced by x + ty x) Bx, r x ) we get G h φ h )x) G h φ h )x + ty x)), x y c x y ) For the second term on the right-hand side in 3.11) we get, using 3.1) and 3.2), G h φ h )x + ty x)) φ h x + ty x)), x y G h φ h φ h L U) x y ) G h φ h I h φ) L U) + G h I h φ) φ L U) + φ φ h L U) x y c φ h I h φ H 1 U e ) + h k) x y ch k x y. 6

8 Using this and 3.12) in 3.11) in combination with 3.10) proves 3.5). The quasi-normal field can be used to define a local coordinate system similar to 2.1). Given n h we define F : h R R 3, F z, t) := z + tn h z). In Lemma 3.1 and Theorem 3.2 in [14] it is proved that from 3.4) and 3.5) it follows that this mapping is a bijection between h [ ɛ, ɛ] ɛ > 0, sufficiently small) and a sufficiently small) neighborhood of h in R 3. This neighborhood is again denoted by U. Hence there is a unique decomposition x = p h x) + d h x)n h p h x)), x U, 3.13) with the skew projection p h : U h and d h an approximate signed distance function to h, d h x) = x p h x). This decomposition resembles the one in 2.1). In the latter, however, one needs the exact level set function φ to compute npx))), whereas 3.13) is based on the quasi-normal field, which can be determined from the finite element approximation φ h. Furthermore, d h x) = 0 iff x h holds, and we have the useful formula d h x) = x p h x), n h p h x)). 4. Parametrization of h. We use ˆ h the zero level of the piecewise linear function ˆφ h ) and the quasi-normal field n h for a computable parametrization of h the zero level of the higher order finite element function φ h ). From the assumptions above, it follows that p h ˆh : ˆ h h is a bijection. Note that this bijection is only) Lipschitz. The Lipschitz manifold ˆ h consists of triangles and convex quadrilaterals. Each quadrilateral is subdivided into two triangles. The resulting triangular triangulation of ˆ h is denoted by F h, i.e., ˆ h = {T T F h }. 4.1) The family {F h } h>0 may be quite shape-irregular, but this does not cause problems, cf. remark 3 below. The mapping p h is used for the parametrization of h. We need a transformation formula between integrals over T ˆ h and over p h T ), which is derived in the following lemma. Lemma 4.1. For T F h, let H R 3 be the plane containing T, and let x U x + u be a parametrization R 2 H with an orthogonal matrix U R 3 2. Then, for any measurable function g : p h T ) R the transformation formula gy) dσy) = gp h x)) ˆµ h x)dσx) 4.2) p h T ) T holds, with ˆµ h x) = detu T Dp h x) T Dp h x)u). Proof. Let F : R 2 R 3 be an injective Lipschitz-mapping, and let T R 2 be Lebesgue-measurable. We recall the transformation rule gy) dσy) = gf x)) µx)dσx), µx) = detdf x) T DF x)). F T ) T We apply this formula to the parametrization x = F x) = U x + u of H. The surface measure on H is dσx) = detu T U) d x = d x, 4.3) 7

9 because U is orthogonal. We also apply this formula to the parametrization y = F x) := p h x) = p h U x + u). The surface measure on this set is dσy) = detu T Dp h x) T Dp h x)u) d x = detu T Dp h x) T Dp h x)u) dσx), by the result in 4.3). 5. Finite element discretization. We introduce the finite element discretization of the Laplace-Beltrami equation 1.1). Our method has some similarity with the one presented in [5], but an essential difference is that we only) need the finite element approximations φ h and ˆφ h of φ. From φ h the quasi-normal field n h can be determined. Let Ŝh be a finite element space of piecewise polynomials of degree m 1 on the triangulation F h of ˆ h, cf. 4.1): Ŝ h = { ˆv h Cˆ h ) ˆv h T P m for all T F h }. 5.1) Remark 3. We briefly discuss two possible choices for the space Ŝh. A first possibility is to use a trace space as introduced and analyzed in [12]. Such a space is constructed by taking the trace of a standard outer finite element space, e.g. the space V m h used for the approximation of the level set function, cf. section 2. Its optimal) approximation properties depend on the shape-regularity of {T h } h>0 not on the shape-regularity of the family {F h } h>0. A second possibility is to define standard polynomial spaces directly on the triangulation F h. Although this triangulation is in general very shape irregular, it has a maximal angle property in three dimensions: in [13] it is shown, that if in the construction of F h the quadrilaterals are subdivided in two triangles in a suitable way, the maximal inner angles in the resulting triangulation are uniformly bounded away from π. Hence, standard finite element spaces on such a triangulation have optimal approximation quality, cf. [13] for more information. We lift the space Ŝh to h by using the bijection p h : ˆ h h : S h := { v h = ˆv h p 1 h ˆv h Ŝh }. 5.2) For the discretization we need a sufficiently accurate) extension of the data f on to h. This extension is denoted by f h and is such that h f h ds = 0 holds. Remark 4. One possible choice for the extension f h is f h = f e 1 h h f e ds h, where f e denotes the constant extension along the exact normals on. This, however, is not feasible, since in our setting it is not reasonable to assume that the normals to are known. Another possibility arises if we assume that f is a smooth) function that is defined in a neighborhood U of. As extension we may then take: f h x) := fx) c f for x h with c f := 1 f ds h. 5.3) h h In the remainder we restrict to the latter choice of the extension. For f we assume the smoothness property f H 1 U). The discrete problem is as follows: Determine u h S h with h u h ds h = 0 such that au h, v h ) = lv h ) for all v h S h, au h, v h ) := h u h h v h ds h, h lv h ) := f h v h ds h. h 8 5.4)

10 For the implementation of this method we pull the discretization back to ˆ h and apply quadrature on the triangulation F h of ˆ h. We first treat the pull back procedure. For this we derive a relation between the tangential gradient on h and the tangential gradient on ˆ h. For this we need several projectors, defined as follows, with ˆn h x) the exact normal on ˆ h : ˆQx) = I 1 ˆαx) ˆn hx) n h y) T, ˆαx) = ˆn h x) T n h y), y = p h x), x ˆ h, 5.5) ˆPx) = I ˆn h x)ˆn h x) T, x ˆ h, 5.6) Py) := I n h y) n h y) T, y h, 5.7) Note that ˆQx), x ˆ h, is an oblique projector which maps into the tangential space n h y). The following commutation relations hold: ˆQx) Py) = Py), Py) ˆQx) = ˆQx), ˆPx) ˆQx) = ˆPx), ˆQx) ˆPx) = ˆQx). 5.8) Lemma 5.1. For ˆv h Ŝh, let v h = ˆv h p h ˆh ) 1 S h. For the tangential gradients the relations ˆh ˆv h x) = ˆPx)Dp h x) T h v h y) = W x) h v h y) W x) := I ˆQx) + ˆPx)Dp h x) T, hold for almost all x ˆ h. Proof. As p h ˆh : ˆ h h is a bijection, we have: ˆv h x) = v h p h x)). with y = p h x), 5.9) This relation and the ones below hold for almost all x ˆ h. We apply the tangential gradient on ˆ h to both sides of the equation to obtain ˆh ˆv h x) = ˆPx) v h p h x)) = ˆPx)Dp h x) T h v h y). This proves the first relation in 5.9). From ˆPx)Dp h x) T h v h y) = ˆPx)Dp h x) T Py) h v h y), ˆPx)Dp h x) T Py) = W x) Py), we obtain the second relation in 5.9). From Lemma 9.1 below it follows that for h sufficiently small the matrix W is invertible. We assume that this condition on h is satisfied, i.e. W is invertible. We introduce the symmetric positive definite matrix function T h x) := W x)w x) T. 5.10) Using the transformation formulas in 4.2) and 5.9) we obtain the following pulled back equivalent formulation of the discrete problem 5.4): Determine û h Ŝh with ˆ h û h ˆµ h dŝ h = 0 such that û G ˆh h ˆv ˆh h dŝ h = ˆ h f h p h )ˆv h ˆµ h dŝ h ˆ h for all ˆv h Ŝh, 5.11) Gˆx) := T h ˆx) 1 ˆµ h ˆx), ˆx ˆ h. 9

11 Clearly, for the implementation of this discretization we need quadrature. We introduce quadrature along the same lines as in [3]. Let T be the unit triangle in R 2, T F h and M T : T T an affine mapping MT x = B T x + b T = x, x T, x T. We consider a quadrature rule on T of the form Q T φ) = L l=1 ω φ ξ l l ) with strictly positive weights ω l and quadrature nodes ξ l T. This induces a quadrature rule on T : Q T ˆφ) := L ˆω l ˆφˆξl ), ˆω l = T ω l, ˆξl = M T ξ l ). 5.12) l=1 Note that, although not explicit in the notation, ˆω l, ˆξ l depend on T. We apply quadrature to the discrete problem 5.11) as follows. First we consider the approximation of the bilinear form au h, v h ). Using the correspondence v h p h = ˆv h, we can represent au h, v h ) as follows, cf. 5.11): au h, v h ) = û G ˆh h ˆv ˆh h dŝ h = û G ˆh h ˆv ˆh h dŝ h 5.13) ˆ h = 3 i,j=1 T G ij i û h j ˆv h dŝ h, with i the ith component of the vector ˆh = ˆP. Quadrature results in an approximate bilinear form, given by a h u h, v h ) = Q T G ˆh û h ˆh ˆv h ). 5.14) For the right hand-side functional lv h ) = h f h v h ds h = ˆh f p h c f )ˆv h ˆµ h dŝ h we have the approximation l h v h ) = f q h ˆµ ) hˆv h, for all vh S h, ˆv h = v h p h, Q T f q h := f p h c q f, T cq f := 1 Q T f p h ˆµ h ), A := Q T ˆµ h ). A 5.15) The constant shift c q f is taken such that the consistency condition l h1) = 0 is satisfied. The final discrete problem, i.e., after quadrature, is as follows: Determine u q h S h with Q T û q h ˆµ h) = 0 such that a h u q h, v h) = l h v h ) for all v h S h. 5.16) Remark 5. In lemma 9.3 below we show a h v h, v h ) γ h v h 2 L 2 h ) for all v h S h under some conditions on Q T. Recall the Poincaré-inequality v h L 2 h ) c h v h L 2 h ) for all v h S h with h v h ds h = 0. This implies that the variational problem 5.16) has in S h a solution that is unique apart from a shift with the constant function on h. This shift is uniquely determined by the condition Q T û q h ˆµ h) = 0, which is a computable approximation of the standard condition h u q h ds h = 0. Hence, the final discrete problem has a unique solution. 10

12 6. Outline of the analysis. In the sections 7-12 we present an error analysis of the discrete problem 5.16). The analysis is rather technical and contains ingredients that are not standard in the literature. Therefore we outline the structure and main ideas of the analysis. Central in the analysis is the Strang Lemma 9.4, in which the discretization error is bounded by three different error components, namely an approximation error, a geometric error and a quadrature error. In the sections bounds for these three components are derived. Preliminaries for the analysis of these error components are derived in the sections 7 and 8. In these sections, properties of the quasi-normal field n h and the skew projection p h, which is used for the parametrization of h, are derived. In the discrete problem 5.16), besides the skew projection p h its Jacobian Dp h plays a prominent role. Key estimates for this Jacobian are derived in Lemma 8.2. An important result in this lemma is that by using suitable projections the error bound of order Oh k ) in 8.8) can be improved to Oh k+1 ) in 8.9), 8.10). In section 12 the error due to quadrature is analyzed. As far as we know, such quadrature errors have not been considered in other papers that treat error analyses of finite element methods for surface partial differential equations. The quadrature issue, however, is essential for the analysis of our method. The reason for this is that the discrete problem before quadrature 5.11) contains an integrand that is not smooth on the triangles T F h. This non-smoothness is caused by the use of the quasi-normal field, which is only Lipschitz. Due to the nonsmooth integrand, standard analyses of quadrature errors as in e.g. [3], do not yield satisfactory bounds. The analysis of the quadrature error in section 12 is based on the following idea. Consider an integral T Gg dŝ h, with a function g that is smooth on T and a function G that is not necessarily smooth on T. Assume that G s is a smooth approximation of G. For the quadrature error we use the splitting E T Gg) := Gg dŝ h Q T Gg) T = G G s )g dŝ h + E T G s g) + Q T G s G)g ). T The error terms T G Gs )g dŝ h and Q T G s G)g) can be controlled by suitable bounds for G G s as in Corollary 12.3). Since G s g is smooth the term E T G s g) can be bounded using standard quadrature error analysis. A smooth approximation of the matrix) function G is derived and analyzed in section One further nonstandard ingredient is the following. As expected, in the analysis of the quadrature error we use the affine transformation between a triangle T F h and the unit triangle T in R 2. We also need the usual relation between norms on T and on T as given in 12.1). In this estimate a Sobolev norm on T is bounded by the corresponding norm on T. In our setting, due to the fact that the triangulation F h is not shape-regular inner angles are not uniformly bounded away from zero), an estimate in the other direction, i.e., bounding û H n p T ) by ũ H n p T ), does not hold. Fortunately, only the estimate 12.1) and not the one in the other direction is needed in our analysis. In the analysis different skew) projections play a key role. For these projections we use boldface notation. For the readers convenience we summarize these projections 11

13 and the normal fields that are used: n : U R 3 exact normal on ), n h : Ω h R 3 quasi-normal field), ˆn h : ˆ h R 3 exact normal on ˆ h ), n h : h R 3 exact normal on h ), Px) = I nx)nx) T, x U, 6.1) ˆPx) = I ˆn h x)ˆn h x) T, x ˆ h, 6.2) Py) := I n h y) n h y) T, y h. 6.3) ˆQx) = I 1 ˆαx) ˆn hx) n h y) T, ˆαx) = ˆn h x) T n h y), y = p h x), x ˆ h, 6.4) Qx) = I 1 αx) n hx) n h x) T, αx) := n h x) T n h x), x h. 6.5) We use the following notation in many proofs below: For any x ˆ h sometimes x U), we let y := p h x) h, z := py) = p p h x), and ζ := px). 7. Properties of n h and p h. In this section we derive some properties of the quasi-normal field n h and the skew projection p h onto h that we need in the analysis further on. We start with a lemma in which it is shown that Dn h is close to a smooth matrix) function. Lemma 7.1. The following holds for all sufficiently small h: n h φ φ L Ω h ) ch k, 7.1) n h n L h ) ch k, 7.2) Dn h D φ ) L φ Ω h ) ch k ) Proof. For the gradient recovery operator applied to the finite element approximation φ h of the level set function φ we write G h = G h φ h. Using 3.1),3.2),2.2) and standard interpolation error results we get G h φ L U) G h φ h I h φ) L U) + G h I h φ) φ L U) c φ h I h φ H 1 U e ) + ch k ch k. 7.4) From this and n h = G h 1 G h the result in 7.1) follows. Using this result we get, for x Ω h : n h x) nx) ch k + nx) φx) φx) = ch k + φpx)) φpx)) φx) φx). For x h we have x px) ch k+1 and thus φx) φpx)) ch k+1. Hence we get the result 7.2). For the derivatives we have Dn h = D G h G h ) = 1 1 I G h G h 2 G hg T ) h DGh D φ ) 1 1 ) = I φ φ φ 2 φ φt D 2 φ )

14 Using an inverse inequality and the result in 7.4) we obtain G h φ H 1 U) G h I h φ) H 1 U) + I h φ) φ H 1 U) ch 1 G h I h φ) L U) + ch k ch 1 ) G h φ L U) + φ I h φ) L U) + ch k ch k 1. From this it follows that DG h D 2 φ L U) ch k 1 holds. Using this and the result in 7.4) in combination with the formulas 7.5) proves the result in 7.3). The next lemma quantifies how well p h approximates p and d h approximates d. Lemma 7.2. For h sufficiently small the following holds: p h p L Ω h ) ch k+1, p p h p L Ω h ) ch k+1, 7.6) d h d L Ω h ) ch k ) Proof. Take x Ω h and q h, d R such that q = x d npx)) = x d nq). Note that px) = pq) holds. Using diamω h ) ch we get d = q x q pq) + px) x dist h, ) + dx) ch. 7.8) In Lemma 4.1 in [14] is it shown that d h is uniformly Lipschitz on U, i.e. there is a constant c such that d h z 1 ) d h z 2 ) c z 1 z 2 for all z 1, z 2 U. Since q h we have d h q) = 0. Using this we get d h x) = d h x) d h q) c x q = c d ch. From p h x) = x d h x)n h p h x)) we obtain p h x) q = d nq) d h x)n h p h x)), and thus p h x) q 2 = d nq), p h x) q d h x) n h p h x)), p h x) q. For the first term we have, using 7.2) and 3.5), nq), p h x) q nq) n h q), p h x) q + n h q), p h x) q By 3.5), for the second term: Hence, ch k p h x) q + c p h x) q 2. n h p h x)), p h x) q ch k p h x) q + c p h x) q 2. p h x) q 2 ch k+1 p h x) q + ch p h x) q 2, and thus, for h sufficiently small, p h x) q ch k+1 holds. Using q px) = q pq) dist h, ) ch k+1 we thus get p h x) px) ch k+1, which proves the first estimate in 7.6). The second result in 7.6) follows from a triangle inequality: pp h x)) px) pp h x)) p h x) + p h x) px) dist h, )+ch k+1 ch k+1. 13

15 For the result in 7.7) we use the representation d h x) dx) = x p h x), n h p h x)) x px), npx)) = px) p h x), npx)) + x p h x), n h p h x)) np h x)) + x p h x), np h x)) npx)). The result in 7.7) follows from this if we use px) p h x) ch k+1, x p h x) = d h x) ch, n h p h x)) np h x)) ch k, np h x)) npx)) ch k+1, which completes the proof 8. Properties of the Jacobian Dp h. The Jacobian Dp h plays a key role in the discretization 5.11) cf. definition of W and ˆµ h ). In this section we derive properties of this Jacobian that we need in our analysis. First we consider the Jacobian of the exact projection p onto given in 2.1). Differentiating the relation 2.1) and using nx) = npx)) we get, for x U, I + dx)hpx)) ) Dpx) = Px), Px) = I nx)nx) T, Hy) = Dny). 8.1) This formula has equivalent representations due to Px) = Ppx)) and Hpx)) = Px)Hpx)) = Hpx))Px). We derive a formula for Dp h, cf. Lemma 8.1 below. It turns out that we need a skew projection as a substitute for the projection P in 8.1). This skew projection is as in 6.5): Qx) = I 1 αx) n hx) n h x) T, αx) := n h x) T n h x), x h. The following relations hold, with P as in 6.3): Q P = P, PQ = Q. 8.2) Lemma 8.1. For a. e. x U, the following relations hold with y = p h x) h, I + dh x)qy)dn h y) ) Dp h x) = Qy), 8.3) Qy)Dp h x) = Dp h x) = Dp h x)qy). 8.4) Proof. Let d h be the exact signed distance function to h. Differentiating d h p h x)) = 0, which holds for a.e. x U, yields Applying this to the differential of p h = id d h n h p h, n h y) T Dp h x) = ) Dp h = I d h Dn h y)dp h n h y) d T h, 8.6) yields 0 = n h y) T d h n h y) T Dn h y)dp h n h y) T n h y) d T h. This can be solved for d T h, d T h = 1 α nh y) T d h n h y) T Dn h y)dp h ). 14

16 Inserting this into 8.6) and rearranging completes the proof of 8.3). The equation Qy)Dp h = Dp h follows immediately from 8.5) and the definition of Q. The equation Dp h x) = Dp h x)qy) follows from 8.3). Below, we frequently use that I + M, M R n n, is invertible if ρm) < 1 and that I + M) 1 = I I + M) 1 M, M R n n, ρm) < ) Lemma 8.2. For sufficiently small h, the following holds, with projections P, ˆP, P defined in 6.1),6.2), 6.3): Dp Dp h L ˆ h ) chk, 8.8) P Dp Dp h ) ˆP L ˆ h ) chk+1, 8.9) P p h ) Dp Dp h ) ˆP L ˆ h ) chk ) Proof. Let x ˆ h be arbitrary and ζ = px). Let ñ = φ/ φ which is defined on U. As ñ n on, we have px) = x dx)ñζ). Differentiating this relation we obtain the following representation for the Jacobian Dp: Dpx) = I + dx)pζ)dñζ) ) 1 Pζ) = I + B1 ) 1 Pζ), B 1 = dx)pζ)dñζ). From 8.3) we get, with y = p h x) h : Dp h x) = I+d h x)qy)dn h y) ) 1 Qy) = I+B2 ) 1 Qy), B 2 = d h x)qy)dn h y). Using 7.6) we get ζ y ch k+1. Define R i := I + B i ) 1 B i, hence, by 8.7), I + B i ) 1 = I R i. From d h x) ch 2, dx) ch 2 and the definition of B i we get R i ch 2. From the definitions we obtain Dpx) Dp h x) = Pζ) Qy) + R 2 Qy) R 1 Pζ), 8.11) Pζ) Qy) = 1 αy) 1) n h y) n h y) T Using 7.2) and 2.3) we get + n h y) nζ) ) n h y) T + nζ) n h y) nζ) ) T. 8.12) n h y) ny) ch k, n h y) ny) ch k, and combining this with αy) 1 = 1 2 n hy) n h y) 2, the smoothness of n and ζ y ch k+1, we obtain 1 αy) 1 ch 2k, 8.13) n h y) nζ) n h y) ny) + ny) nζ) ch k, 8.14) n h y) nζ) n h y) ny) + ny) nζ) ch k. 8.15) From these results and 8.12) we get Qy) Pζ) ch k. 8.16) 15

17 Using the smoothness of ñ and the result in 7.3) we obtain: Dñζ) Dn h y) Dñζ) Dñy) + Dñy) Dn h y) ch k 1. Combining this with dx) ch 2 and dx) d h x) ch k+1, cf. 7.7), yields R 1 R 2 = I + B 1 ) 1 I + B 2 ) 1 c B 1 B ) c dx) Pζ) Qy) + c dx) Dñζ) Dn h y) + c dx) d h x) ch k+1. Combining this and 8.16) with the result in 8.11) proves the result in 8.8). Let P denote either Px) or Py). Using 8.11)-8.12) we get P Dpx) Dp h x) ) ˆPx) = M ˆPx) + R ˆPx), M := P nh y) nζ) ) n h y) T nζ) n h y) nζ) ) T ) ˆPx), R := P R 2 Qy) R 1 Pζ) ) + 1 αy) 1) Pnh y) n h y) T. 8.18) Combining 8.16) and R i ch 2 with 8.13), 8.17) we get R ch k+1. We finally consider the term M. From we get ˆn h x) n h y) ˆn h x) nx) + nx) ny) + ny) n h y) ch n h y) T ˆPx) = ˆPx) nh y) ˆn h x) ) ch. 8.19) If P = Px) then Pnζ) = Pnx) = 0 holds. If P = Py) we obtain from 8.15): Pnζ) = Py) nζ) n h y) ) nζ) n h y) ch k. Combining these results we get M n h y) nζ) n h y) T ˆPx) + Pnζ) nh y) nζ) ch k+1, which completes the proof. The estimates in lemma 8.2 play a key role in the error analysis of our method. In Section 14 we give results of a numerical experiment which show that the bounds in the estimates are sharp. In particular, for obtaining the h k+1 bounds the projections in the terms on the left hand-side are essential. 9. Strang Lemma. In this section we derive a Strang lemma. In the analysis we will need the constant extension of a function w on along the normals n to a function w e on U given by w e x) := w px) for all x U. 9.1) We also use the lift of a function defined on h or on ˆ h ) to a function defined on along the normals n. More precisely, for a function w defined on h or on ˆ h ), its lift w l to is given by w l px) = wx) for all x h or x ˆ h ). 9.2) 16

18 The lifted finite element space is denoted by S l h := { vl h v h S h }. We need the following matrix function on : A px)) = 1 µ h x) Px)[I dx)hx)] Px)[I dx)hx)]px), x h, 9.3) with µ h as in 2.4) and the projectors as in 6.1), 6.3). From [5, formula 2.14)] we have the integral identity h u h h v h ds h = A u l h vh l ds. h Using this we obtain that if u h solves 5.4), then the lifted function u l h Sl h satisfies A u l 1 h v h ds = fhv l h ds for all v h Sh. l 9.4) µ l h We also need the following estimates, which follow from the results 2.11), 2.12) in [5]: γ v e L 2 γ) c v L 2 ), v H 1 ), γ {ˆ h, h }, v l L2 ) c γ v L2 γ), v H 1 γ), γ {ˆ h, h }. 9.5) We first derive ellipticity of the bilinear form a h, ) in 5.14). For this we need that the matrix G, cf. 5.11), is positive definite. To derive this result, in the next lemma we first consider the matrix W. Lemma 9.1. For h sufficiently small the following holds: W I L ˆ h ) ch. 9.6) Proof. We recall the definition W x) = I ˆQx) + ˆPx)Dp h x) T. We drop the x- dependence in the notation. From 8.3) we get, due to d h x) ch 2, for h sufficiently small, Dp h x) = I + d h x)qy)dn h y) ) 1 Qy) = Qy) + Oh 2 ), with y = p h x). Hence, W = I ˆQ + ˆPQy) T + Oh 2 ) holds. Using 7.6) we get x y = x p h x) x px) + px) p h x) dist, ˆ h ) + ch k+1 ch 2. Using this and the results in 2.3), we obtain Hence, ˆn h x) n h y) ˆn h x) nx) + nx) ny) + ny) n h y) ch. ˆP n h y) = ˆP n h y) ˆn h x)) n h y) ˆn h x) ch. Using this, the definitions of the projections and ˆQ = ˆQ ˆP, cf. 5.8), yields W I I ˆQ) ˆP + ˆPQy) T I) + ch 2 = 1 ˆα ˆn h n h y) T ˆP + 1 αy) ˆP n h y)n h y) T + ch 2 c ˆP n h y) + ch 2 ch, 17

19 which completes the proof. Corollary 9.2. For h sufficiently small the matrix Gx) = T h x) 1 ˆµ h x), x ˆ h, is uniformly symmetric positive definite, i.e. there exists a constant λ min G) > 0 such that z T Gx)z λ min G) z 2 for almost all x ˆ h and all z R 3. Proof. From lemma 9.1 and 5.10), 5.11), we obtain for all sufficiently small h and arbitrary z R 3 that z T Gx)z = ˆµ h x) W x) 1 z 2 ˆµ hx) 2 z 2, x ˆ h. We recall the definition ˆµ h x) = detu T Dp h x) T Dp h x)u). The matrix U depends on the triangle T F h and satisfies ˆPx)U = U, cf. Lemma 4.1. For h sufficiently small we have Dp h x) = Qy) + Oh 2 ), y = p h x). With ζ = px), we have Pζ) = Px). From 2.3) and nx)nx) T ˆn h x)ˆn h x) T = nx) nx) ˆn h x) ) T + nx) ˆnh x) )ˆn h x) T it follows that Using this and the result in 8.16), we get P ˆP L ˆ h ) ch. 9.7) Qy) ˆPx) Qy) Pζ) + Pζ) Px) + Px) ˆPx) ch. This yields Dp h x) = ˆPx) + Oh) and consequently U T Dp h x) T Dp h x)u = U T ˆPx) ˆPx)U + Oh) = U T U + Oh) = I + Oh). Thus, for h sufficiently small, we have that ˆµ h x) = 1 + Oh), x ˆ h, 9.8) is uniformly in x) bounded from below by a strictly positive constant. Using the result of the previous corollary we can derive ellipticity of the bilinear form a h, ): Lemma 9.3. Assume that the quadrature rule Q T is exact for all polynomials of degree 2m 2. There exists a constants γ > 0 and h 0 > 0 such that for all h h 0 a h v h, v h ) γ h v h 2 L 2 h ) for all v h S h. 9.9) Proof. Let h be sufficiently small such that the matrix G is uniformly positive definite, cf. Corollary 9.2. From this positive definiteness and the fact that the quadrature weights are strictly positive we get Q T G ˆh ˆv h ˆh ˆv h ) c 0 Q T ˆh ˆv h 2), with c 0 = λ min G) > 0, independent of h. Since the quadrature rule Q T on T is exact for all polynomials of degree 2m 2 and the mapping M T between T and T 18

20 is affine this exactness property also holds for Q T on T. The functions i ˆv h) 2 are polynomials of degree 2m 2 on T, and thus we have a h v h, v h ) c 0 Q T with γ > 0 due to 5.9) and 9.6). ˆh ˆv h 2) = c 0 ˆh ˆv h 2 L 2 ˆ h ) γ h v h 2 L 2 h ) Based on this ellipticity property we apply standard arguments to derive the following variant of the Strang Lemma. Theorem 9.4. Assume h is sufficiently small such that a h, ) has the ellipticity property 9.3). Define the data extension error Ẽf := f 1 f µ l h l L 2 ). For the h solution u q h of 5.16) the following error bound holds: u u q h )l ) L2 ) [ c min u vh) l L v h S 2 ) + I A )P L ) h v h L 2 h ) h + sup w h S h /R av h, w h ) a h v h, w h ) ] lw h ) l h w h ) + c sup + h w h L 2 h ) w h S h /R h w h Ẽf. L 2 h ) 9.10) Proof. Take an arbitrary v h S h. We start with a triangle inequality and 9.5): u u q h )l ) L2 ) u v l h) L2 ) + v l h u q h )l ) L 2 ) u v l h) L 2 ) + c h v h u q h ) L 2 h ). We derive a bound for h e h L 2 h ), e h := u q h v h. Let c 1 be a constant such that ẽ h := e h +c 1 satisfies h ẽ h ds h = 0. For arbitrary constants c, there holds h c 0. In particular, by 5.14), we get the consistency property a h c, ẽ h ) = 0. Using this, 9.9), and the definition of the discrete problems 5.4), 5.16), we obtain h e h 2 L 2 h ) = h ẽ h 2 L 2 h ) γ 1 a h e h, ẽ h ) = γ 1 l h ẽ h ) lẽ h ) + au h, ẽ h ) a h v h, ẽ h ) ) = γ 1 au h v h, e h ) + av h, ẽ h ) a h v h, ẽ h ) + l h ẽ h ) lẽ h ) ). 9.11) We will derive the bound au h v h, e h ) c u v l h) L 2 ) + I A )P L ) h v h L 2 h ) + Ẽf ) h e h L2 h ), 9.12) and combination of this with the relation 9.11) and the triangle inequality above proves the result 9.10). For the derivation of 9.12) we note, cf. 9.4), that for all w h Sh l we have A u l 1 h w h ds = µ l fhw l 1 h ds = h µ l fh l f ) w h ds + fw h ds h 1 = fh l f ) w h ds + u w h ds. µ l h 19

21 Let c be a constant that is chosen below and ē h := e h c. From the previous equation, au h v h, e h ) = au h v h, ē h ) = h u h v h ) h ē h ds h h = A u l h vh) l ē l h ds 9.13) = u vh) l e l 1 h ds + fh l f ) ē l h ds + I A )P vh l e l h ds. holds. Now c is chosen as c := 1 h µ h e h ds h such that we have ē l h ds = e l h ds c ds = µ h e h ds h c = 0. h µ l h Hence, the Poincare inequality ē l h L 2 ) c e l h L 2 ) holds. Using this, the Cauchy-Schwarz inequality and e l h L 2 ) c h e h L2 h ), cf. 9.5), in 9.13), we get the estimate 9.12). In the total error there are three different components, namely a geometric error approximation of by h ), an approximation error results from using the finite element space) and a quadrature error. In section 10 we study the first term on the right hand-side in 9.10), which quantifies the approximation error. The second term and the fifth term are related to the geometric error and are analyzed in section 11. The third and fourth term on the right hand-side in 9.10) arise from the quadrature and are treated in section Approximation error. For the analysis of the approximation error we assume the following approximation quality of the finite element space Ŝh on ˆ h, cf. 5.1): there are m 1 and an interpolation operator I h : H m+1 ) Ŝh such that for s = 0,..., m: w e I h w 2 H s T ) ch2m+1 s) w 2 H m+1 ) for all w H m+1 ). 10.1) Such an approximation property holds for the two possible choices for Ŝh mentioned in Remark 3. The estimate 10.1) for s = 1 implies ˆh w e I h w) L 2 ˆ h ) chm w H m+1 ) for all w H m+1 ). 10.2) In the analysis we use the spaces Ŝh on ˆ h ), cf. 5.1), S h on h ), cf. 5.2), and the lifted space Sh l on ), cf. 9.2). The analysis requires smoothness of the solution of 1.1), Assumption The solution u of 1.1) satisfies u H m+1 ) H ). 2 In the analysis below we use the following test function u h, S h to prove an upper bound for the minimum over v h S h in 9.10): u h, := û h, p 1 h S h, û h, := I h u Ŝh. 10.3) Theorem Let m 1 be such that 10.2) and assumption 10.1 are fulfilled. For h sufficiently small the following holds, with u h, as in 10.3): min v h S h u v l h) L2 ) u u l h, ) L2 ) ch m u H m+1 ) + ch k+1 u H 2 ) 20

22 Proof. The test functions in 10.3) satisfy u h, p h x) = û h, x), u l h, py) = u h, y), y := p h x) h, x ˆ h, cf. 5.2), 9.2). Using û := u e H 1 ˆ h ), we define u H 1 h ), and ũ H 1 ) by u p h x) = û x), ũ py) = u y), y := p h x) h, x ˆ h. ) Note that ũ = u l on holds. From 5.9) it follows that h ˆv p 1 h L 2 h ) ˆv c ˆh L2 ˆ h ) for all ˆv H1 ˆ h ) holds. Using this and 9.5) we get ũ u l h, ) L 2 ) = u u h, ) l L 2 ) c h u u h, ) L 2 h ) c ˆh u e û h, ) L 2 ˆ h ). Hence, with the triangle inequality and 10.2) we obtain u u l h, ) L 2 ) c ˆh u e û h, ) L2 ˆ h ) + ũ u) L 2 ) ch m u H m+1 ) + ũ u) L 2 ). 10.4) We derive a bound for the term ũ u) L 2 ). Let x ˆ h be arbitrary, y = p h x) h, z = py), and ζ = px). From 7.6) it follows that y ζ ch k+1, z ζ ch k+1, x z ch ) Our starting point is the identity ũz) = uζ), which holds a. e. on ˆ h by definition of ũ. Taking the tangential gradient on ˆ h yields ˆPx)Dp h x) T Dpy) T ũz) = ˆPx)Dpx) T uζ). 10.6) From the smoothness of u and z ζ ch k+1, we get uζ) = uz) + r 0, with r 0 ch k+1 u H 2 ). We insert this into 10.6) and rearrange the terms to obtain ˆPx)Dp h x) T Dpy) T ũz) uz) ) = ˆPx) Dpx) T Dp h x) T Dpy) T ) Pz) uz) + ˆPx)Dp h x) T r 0 =: r ) For the matrix in first term on the right hand-side in 10.7) we have ˆPx) Dpx) T Dp h x) T Dpy) T ) Pz) = ˆPx) Dpx) T Dp h x) T ) Pz) + ˆPx)Dp h x) T I Dpy) T ) Pz) =: A 0 + A 1. Using Pz) Px) = Pz) Pζ) c z ζ ch k+1 and 8.9) we obtain A 0 ch k+1. Differentiating the relation py) = y dy)npy)) one obtains Dpy) = I + dy)hz) ) 1Pz), with Hz) = Dnz). From 7.7) one obtains dy) ch k+1, and this yields Dpy) = Pz) + Oh k+1 ). 10.8) Thus we get A 1 c Pz)I Dpy)) ch k+1. Using the bounds for A 0, A 1 in 10.7) we get ũz) ) A uz) = r1, r 1 ch k+1 u H 2 ), A := ˆPx)Dp h x) T Dpy) T Pz) )

23 We now analyze the matrix A. From 8.3) and d h x) ch 2 we get Dp h x) = Qy) + Oh 2 ) ) Combining this with 10.8) yields A = ˆPx)Qy) T + E 1 )Pz), with E 1 ch 2. We consider the matrix ˆPx)Qy) T. Using 8.19) we get ˆPx)Qy) T I) 1 αy) ˆPx) n h y) ch. Using Pz) Pζ) ch k+1, Pζ) = Px), 9.7) gives ˆPx) Pz) ch. Hence, ˆPx)Qy) T Pz) ˆPx)Qy) T I) + ˆPx) Pz) ch. From this we get A = I + E 2 )Pz), with E 2 ch. Using this in 10.9) we get, for h sufficiently small, ũz) uz) ) = I + E2 ) 1 r 1 c r 1 ch k+1 u H 2 ), which implies ũ u) L2 ) ch k+1 u H 2 ). Combining this with the result in 10.4) completes the proof. 11. Geometric error. We study the terms I A )P L ) h v h L 2 h ) and Ẽf := f 1 µ l h f l h L 2 ) that occur in the Strang Lemma, cf. 9.10). Theorem Let u h, S h be as in 10.3). For h sufficiently small the following estimates hold: I A )P L ) h u h, L 2 h ) ch k+1 u H 2 ), 11.1) f 1 µ l fh l L 2 ) ch k+1 f H 1 U). h 11.2) Proof. Using d L h ) ch k+1, 1 µ h 1 L h ) ch k+1, cf. 2.3) and 2.5), in 9.3) yields A px)) = Px) Px)Px) + Oh k+1 ), x h. From Ppx)) = Px) and the identity P PP P = PP P) P P)P we obtain I A )P L ) c P PP P L h ) + ch k+1 c P P 2 L h ) + chk+1 ch k ) The last estimate above follows from 2.3). Using 9.5), 5.9) and 10.2) we get h u h, L 2 h ) I c ˆh h u L 2 ˆ h ) c u e ˆh L 2 ˆ h ) + u ) H 2 ) c 11.4) u L 2 ) + u H )) 2 c u H 2 ). Combination of 11.3) and 11.4) yields the proof of 11.1). We consider the data extension f h x) = fx) c f, x h with c f as in 5.3). For the constant c f we get, using the smoothness of f, dist, h ) ch k+1, 2.5), and f ds = 0: c f 1 h f f p) ds h + 1 h h f p ds h h c f L U)h k h f 1 µ l ds 11.5) h c f L U)h k+1 + c 1 µ l 1 L ) f L 1 ) ch k+1 f H 1 U). h 22

24 With this we obtain f 1 µ l fh l L2 ) c f fh l L2 ) + ch k+1 f L2 ) h c f e f h L 2 h ) + ch k+1 f L 2 ) c f e f L 2 h ) + c c f + ch k+1 f L 2 ) c f L U) x px) L h ) + ch k+1 f H 1 U) ch k+1 f H 1 U), which proves the result in 11.2). 12. Quadrature error. In this section we analyze the quadrature error, i.e., we derive bounds for the third and fourth term in the Strang Lemma. Recall from section 5 the affine mapping from the unit reference triangle T to T F h, given by x = M T x = B T x + b T, x T, x T. Note that B T R 3 2. Furthermore B T ch holds. Correspondence of functions on T and T is given by ũ x) = ûb T x + b T ) = ûx). Note that for n N, ũ C n T ) and ξ i R 2, 1 i n we have D n ũ x)ξ 1,... ξ n ) = D n ûx)b T ξ 1,..., B T ξ n ) = DT nûˆx)b T ξ 1,..., B T ξ n ) where D T denotes the tangential derivative along T ), and thus as in Theorem 15.1 in [3] we obtain, for n N, p [1, ], ũ H n p T ) c B T n T 1/p û H n p T ) ch n T 1/p û H n p T ) for ũ H n p T ). 12.1) In the seminorm û H n p T ) only the derivatives of order n are involved and these derivatives are the tangential ones along the triangle T. We note that an estimate in the other direction, i.e. bounding derivatives of û by those of ũ, causes problems, because the triangle ˆT may have arbitrary small angles. Thus the smallest singular value of B T cannot be bounded from below by ch with a uniform w.r.t. T and h) constant c > 0. The quadrature error for the quadrature rule 5.12) is defined by E T φ) = φ d x Q T φ), E T ˆφ) = ˆφ dˆx Q T ˆφ). 12.2) Note that E T ˆφ) = T E T φ) holds. T Smooth approximation of G. In the bilinear form a h u h, v h ) the quadrature rule Q T is applied to the function û G ˆh h ˆv ˆh h. On each triangle T ˆ h the vector functions û ˆh h and ˆv ˆh h are polynomials and thus have C smoothness. The matrix G = Gx) = T h x) 1 ˆµ h x), x T, however, contains derivatives of the function p h, which is only Lipschitz. Hence, G is not even continuous. In this section we show that, on T, this matrix function can be approximated with accuracy Oh k+1 ) by a smooth matrix function, denoted by G s. The components T 1 h = W W T ) 1 and ˆµ h of G are treated in the lemmas 12.1 and 12.2 below. Recall the definition W x) = I ˆQx) + ˆPx)Dp h x) T, cf. Lemma 5.1. From 5.8), 8.2), 8.4), and the definition of W, we get for almost all x ˆ h : T ˆPx)W x) = ˆPx)Dp h x) T = ˆPx)Dp h x) T Qy) T = ˆPx)Dp h x) T Py) = W x) Py), y = ph x). 12.3) This implies the commutator relations note that W is invertible, cf. 9.6)): W x)w x) T ˆPx) = ˆPx)W x)w x) T, W x)w x) T ) 1 ˆPx) = ˆPx) W x)w x) T ) ) 23