Key words. Laplace-Beltrami operator, surface finite element method, superconvergence, gradient recovery method

Transcription

1 UPRCONVRGNC AND GRADINT RCOVRY OF LINAR FINIT LMNT FOR TH LAPLAC-BLTRAMI OPRATOR ON GNRAL URFAC HUAYI WI, LONG CHN, AND YUNQING HUANG Abstract. uperconvergence results and several gradient recovery methods of finite element methods in flat spaces are generalized to the surface linear finite element method for the Laplace- Beltrami equation on general surfaces with mildly structured triangular meshes. For a large class of practically useful grids, the surface linear finite element solution is proven to be superclose to an interpolant of the exact solution of the Laplace-Beltrami equation and as a result various post-processing gradient recovery, including simple and weighted averaging, local and global L 2 -projections, and Z-Z schemes are devised and proven to be a better approximation of the true gradient than the gradient of the finite element solution. Numerical experiments are presented to confirm the theoretical results. Key words. Laplace-Beltrami operator, surface finite element method, superconvergence, gradient recovery method AM subject classifications. 65N5, 65N5, 65N3. Introduction. The Laplace-Beltrami operator is a generalization of the Laplace operator in flat spaces to manifolds. Many partial differential equations (PDs) on two dimensional Riemannian manifolds, such as the mean curvature flow [25], surface diffusion flow [35], and Willmore flow [38] etc., are formulated using the Laplace- Beltrami operator. These PDs are frequently used in diverse applications such as image processing, surface processing, fluid dynamics, weather forecast, climate modeling, and so on [7, 8, 3, 36, 39]. In this paper, we study the discretization of the Laplace-Beltrami operator by the surface linear finite element method [22]. We shall prove a superconvergence result and use it to develop several post-processing schemes which improve numerical approximations significantly. Let be a two dimensional, compact, and closed C 3 -hypersurface in R 3, and =. Let f be a given data satisfying f dσ = where dσ is the surface measure, and let u be the solution of the Laplace-Beltrami equation: u = f on, (.) where is the Laplace-Beltrami operator on the surface. We require u dσ = in order to guarantee the uniqueness of the solution to (.). As an effective numerical method for solving PDs, the finite element method (FM) plays an important role in modern scientific and engineering computing. In particular, finite element methods for PDs defined on manifolds have been studied in the literatures; see [6, 6, 7, 2, 22, 23, 24, 28, 32, 34] and references therein. Hunan Key Laboratory for Computation and imulation in cience and ngineering, chool of Mathematics and Computational cience, Xiangtan Uinversity, Xiangtan 45, Hunan, P.R.China (huayiwei984@gmail.com). The first author is supported by Hunan Provincial Innovation Foundation for Postgraduate CX29B22. Department of Mathematics, University of California at Irvine, Irvine, CA (chenlong@math.uci.edu). The second author was supported in part by NF Grant DM-8272, and in part by NIH Grant P5GM7656 and RGM7539. Hunan Key Laboratory for Computation and imulation in cience and ngineering, chool of Mathematics and Computational cience, Xiangtan Uinversity, Xiangtan 45, Hunan, P.R.China (huangyq@xtu.edu.cn). The third author is supported in part by the NFC for Distinguished Young cholars 6256 and the National Basic Research Program of China under the grant 25CB327.

2 2 H. WI AND L. CHN AND Y. HUANG We briefly survey results related to our paper below. Dziuk [22] approximated the surface by a polyhedral surface h with triangular faces which avoids a local or global parametrization of, then discretized the equation (.) with the piecewise linear finite element and solved it on h. He also proved asymptotic error estimates in [22]. Recently, Demlow [6] defined higher-order analogues to the linear method in [22] and proved a priori error estimates in the L 2, H, and corresponding point-wise norms. Using superconvergence to improve the accuracy of finite element approximations has been an active research topic; see [, 3,, 3, 4, 29]. Again we briefly survey results related to our paper. Bank and Xu [3] developed superconvergence estimates for piecewise linear finite element approximations on meshes in which most pairs of adjacent triangles form O(h 2 ) approximate parallelograms. Following the idea in [3], Huang and Xu [29] investigated the superconvergence properties for quadratic triangular elements on such mildly structured grids. In this paper, we shall generalize theoretical results in [3] to the surface linear finite element. Passing from flat spaces to surfaces, there are two main difficulties. First, the surface is approximated by a polyhedral surface h with a union of triangular faces and thus an additional error to approximate the geometry is introduced. econd, in general two triangles sharing a common edge are not in the same plane, so we cannot use the techniques developed in [3] directly. To overcome these difficulties, we show that if the surface is C 3 and the union of the triangular faces of h are shape-regular and quasi-uniform of a diameter h, the geometry error from the polyhedral surface approximation is O(h 2 ). Then we show that the included angle between the unit outward normal vectors of h on any two neighboring triangles is O(h). Under the O(h 2σ ) irregular grids and C 3 surface assumptions, we show in Theorem 3.5 that h u h h ū I,h C h +min{,σ} ( u 3, + u 2,, ) + C 2 h 2 f,, (.2) where u h is the finite element approximation of u on h and ū I is the linear interpolation of ū, the extension of the exact solution u to h. In estimate (.2), we have included with C 2 h 2 f,, the effect of the additional approximation error of the geometry. Gradient recovery is designed to recover a better approximation of the true gradient than the gradient of the finite element solution does [3, 4, 5, 8, 2, 4, 42]. It is used to improve the numerical approximation and supply a posteriori error estimation for adaptive procedure. Bank and Xu [3] introduced a gradient recovery algorithm using the global L 2 -projection for O(h 2 ) approximated parallelogram meshes and proved the superconvergence of the recovered gradient. In [4], they developed a post-processing gradient recovery scheme on general unstructured and shape regular meshes using smoothing operators of the multigrid method. Together with Zheng, they successfully generalized such method to high order finite element methods [5]. Xu and Zhang [4] provided a general framework for the analysis of local recovery schemes including simple and weighted averaging, local L 2 -projections, and local discrete least-squares fitting (also known as Z-Z recovery [42]). These afore mentioned results are for planar meshes. For surfaces meshes, Du and Ju [2] developed a gradient recovery method for a finite volume approximation of linear convection-diffusion equations on spheres, and demonstrate numerically the superconvergence of their method. We shall prove the superconvergence of the simple and weighted averaging gradient recovery methods on polyhedral surface meshes. Due to the curvature of the surface, we cannot use directly the local L 2 -projection and local discrete least-squares

3 UPRCONVRGNC AND GRADINT RCOVRY ON URFAC 3 fitting methods on polyhedral surface meshes. Therefore we project the patch Ω i onto the tangent plane of at x i and modify these local recovery methods accordingly. We show that if the local patch Ω i of a vertex x i is O(h 2 )-symmetric on h, then (G h h ū I )(x i ) u(x i ) Ch 2 u 3,,P(Ω i), (.3) where G h is an appropriate defined recovery operator in section 4 and P (Ω i ) is the projection of Ω i on. The recovery method using the global L 2 -projection Q h developed by Bank and Xu [3] can be also generalized to the surface linear finite element method in a straightforward way. The analysis, however, is not a trivial generalization. Again the difficulty is the error introduced by the surface approximation. Let w h = Q h h u h and w h be the lift of w h to the surface. ince the difference between the tangent gradient operator on and h on h is only O(h), using the technique in [3], we can only get a first order approximation, i.e., u w h, Ch. Using a perturbation argument, we develop a new method to prove that if the mesh is O(h 2σ ) irregular: u w h, C h +min{,σ} u 3,, + C 2 h 2 f,. (.4) Our proof confirms that the symmetry of the element patch of vertices is the source of the superconvergence [37]. By the close relationship between FM and finite volume methods (FVM), we can use these global and local recovery methods to post-process the finite volume approximation and get similar superconvergence results. In this way, we give a theoretical justification of superconvergence observed numerically by Du and Ju [2]. We should mention that we assume all vertices of h lie on. In practical applications, the exact surface is unknown and h is only an approximation of. We hope that when the distance is O(h 2 ) between the vertices of h and, the same results obtained in this paper will still hold, which, of course, deserves a further studying. Also in the construction of the last four recovery operators, we assume the normal vectors of is known, which may not be true. In our future work, we will consider how to construct high order approximation of the normal vectors of from h. The paper is organized as follows. In section 2, we present some important definitions and preliminaries. In section 3, we discuss the superconvergence between the surface finite element solution and linear interpolation of the true solution. In section 4, we introduce several gradient recovery methods, and prove their better approximation property. We also generalize the results to FVM on surfaces. In ection 5, we present numerical tests in support of our theoretical results. We use x y to indicate x Cy and x y for x y and y x. 2. Preliminaries. In this section, at first, we introduce the notation about obolev spaces defined on surfaces, and then present the weak form of the Laplace- Beltrami equation (.) and a regularity result. After that, we discuss the obolev spaces on polyhedral surfaces and the relationship between functions defined on the surface and its polyhedral approximation. At the end of this section, we present the surface linear finite element method and the corresponding error estimates. 2.. obolev paces on mooth urfaces. ince is a closed surface and =, it partitions the space R 3 into three distinct sets: points inside the surface, points on the surface, and points outside the surface; denoted by Ω, Ω, and Ω +,

4 4 H. WI AND L. CHN AND Y. HUANG respectively. For any x R 3, let dist(x, ) = min y x y be the distance between x and, where is the standard uclidean distance. We can define a strip domain U := {x R 3 dist(x, ) < δ}, where δ > and is small enough such that we can define a unique signed distance function d : U R, satisfying the following properties: d C 3 (U); d(x) <, for all x Ω U; d(x) =, for all x Ω U = ; d(x) >, for all x Ω + U, and d(x) = dist(x, ), for all x U. We shall view the surface as the zero level set of the distance function. Let be the conventional gradient operator in R 3. o d(x) R 3 is the gradient of d(x) and H(x) := 2 d(x) R 3 3 is the Hessian matrix of d(x). For any x U, let y be the closest point of x on, i.e., d(x) = x y. ince d(x) is the signed distance function and is its zero level set, it is easy to show that d(x) is the unit outward normal vector of at y, namely, d(x) =. Then for any x U, let n(x) = d(x), we can define the following unique projection P : U P (x) := x d(x)n(x). (2.) For x U, differentiation on the identity d(x) d(x) = implies that H(x) d(x) = H(x)n(x) =. Therefore zero is an eigenvalues of H(x) and n(x) is the corresponding eigenvector. The other two eigenvalues of H(x) are denoted by κ (x) and κ 2 (x). When x, κ (x) and κ 2 (x) are the principal curvatures of at x. For v C (), since is C 3, we can extent v to C (U) and still denote the extension by v [33]. The tangential gradient of v on can be written as: v = v ( v n)n = (I nn t ) v = P v R 3, where P (x) = (I nn t )(x) is the projection operator to the tangent plane at a point x and therefore P 2 = P. Notice that, we use the extension of v to define the surface gradient. However it can be shown that v depends only on the value of v on but not on the extension. Namely is an intrinsic operator. imilarly, for a vector field v (C ()) 3, we can extend it to (C (U)) 3 and define the tangential divergence of v on as: v = v n t vn R. The Laplace-Beltrami operator on reads as follows v = ( v) = v ( v n)( n) n t 2 vn R, provided v C 2 () and 2 v is the Hessian matrix of v (suitably extended as a C 2 (U) function). Let α = (α, α 2, α 3 ) Z 3 + be a vector of nonnegative integers, and α = 3 i= α i. The α -th tangential derivatives of u on, D α u can be defined recursively from the above definition of the tangential gradient. We introduce the obolev spaces W m p () := {u L p () D α u L p (), α m}, where p and m is a nonnegative integer. For p <, the space Wp m () is equipped with the norm u m,p, := ( Du α p L p () )/p, α m

5 UPRCONVRGNC AND GRADINT RCOVRY ON URFAC 5 and semi-norm u m,p, := ( α =m D α u p L p () )/p, with standard modification for p =. For p = 2, we denote W m 2 () by H m () and the corresponding norm and seminorm by u m, = u m,2, and u m, = u m,2,, respectively The Laplace-Beltrami quation and A Regularity Result. The variational formulation of the Laplace-Beltrami equation (.) is: find u H () such that u v dσ = fv dσ, for all v H (). (2.2) The following well-posedness and regularity results on (2.2) can be found in [6]. Lemma 2.. Let f L 2 () satisfy f dσ =. Then there exists a unique weak solution u to (2.2) satisfying u dσ =, and u 2, C f, holds for a constant C depending only on the surface obolev paces on Polyhedral urfaces. Let be approximated by a polyhedral surface h which is a union of triangular faces. We assume that these triangular faces are shape-regular and quasi-uniform of a diameter h and their vertices lie on. Because h is C,, only H ( h ) is well defined [22, 27]. Let N h = {x i } be the set of vertices of h, T h = {τ h } the set of triangular faces, and h = {} the set of edges of h. For any τ h T h, let n h be the unit outward normal vector of h on τ h. For v h C( h ) and v h τh C (τ h ), we have h v h τh := v h ( v h n h )n h = (I n h n t h) v h = P h v h R 3, where P h = I n h n t h R3 3. Obviously, h v h (L 2 ( h )) 3. Restricting the projection P : U to h, we get a continuous differentiable bijection from h to, still denoted by P. For any τ h T h, we can get a surface triangle τ := P (τ h ) and denote the set consisting of all surface triangles by T. We then establish the relationships between functions defined on and h following [7]. Through the bijection map P, a function v : R induces uniquely a function v : h R, as v(x) = v(p (x)), for all x h. For any τ h T h and a function v C (P (τ h )), we have h v(x) = (P h (I dh)p )(x) v(p (x)), for all x τ h. (2.3) Conversely, a function v h : h R induces uniquely a function ṽ h : R, as ṽ h (x) = v h (P (x)), for all x. For any τ h T h and a function v h C (τ h ), let τ = P (τ h ), we then get ṽ h (x) = (I dh) (I n hn t n t ) h v h (P n h (x)), for all x τ. (2.4) Let dσ h and dσ be the surface measures of h and, respectively. related by dσ = J(x)dσ h with (see [7]) J(x) = ( d(x)κ (x))( d(x)κ 2 (x))n n h, for all x τ h. They are

6 6 H. WI AND L. CHN AND Y. HUANG We need the following approximation result [22]. Lemma 2.2. For any τ h T h, the following estimate holds: d(x),τh + J,τh + h n n h,τh + h P P h,τh h 2, (2.5) For the relationship between the smoothness of function v defined on and it is extension v on h, we have the following results [6, 22]. Lemma 2.3. Let τ h T h and τ = P (τ h ). If v W 3, (τ) H 3 (τ), then the following results hold: v,τh v,τ v,τh, (2.6) v,τh v,τ v,τh, (2.7) v k,τh v k,τ, k = 2, 3, (2.8) v k,,τh v k,,τ, k = 2, 3. (2.9) 2.4. Linear urface Finite lement Method. Given a surface and its polyhedral approximation h. For a triangle τ h T h, let {λ i } be the barycentric coordinates of τ h. Let V h be the continuous piecewise linear finite element space on h, namely, for any v h V h and τ h T h, v h is continuous on h and v h τh span{λ, λ 2, λ 3 }. We define the corresponding lifted spaces on : Ṽ h = {ṽ h ṽ h := v h P, where v h V h }. Recall that the bijection P : h is defined in (2.). For f L 2 (), let f h (x) = f(x) f dσh, (2.) h h where h is the area of h. Then h f h (x) dσ h =, and therefore there exists a unique finite element solution u h V h with h u h dσ h = to the following equation [22]: h u h h v h dσ h = h f h v h dσ h, h for all v h V h. (2.) By (2.3) and the fact P 2 = P, we can transform the equation (2.) posed on h to the surface A h ũ h ṽ h dσ = J f h ṽ h dσ, (2.2) where A h = J P (I dh)p h(i dh)p. ubtracting (2.2) from (2.2), we obtain the error equation ( u A h ũ h ) ṽ h dσ = (f J f h )ṽ h dσ, for all ṽ h Ṽh. (2.3) From (2.3), one can obtain the following estimates [22].

7 UPRCONVRGNC AND GRADINT RCOVRY ON URFAC 7 Theorem 2.4. When is C 2, one has the following results: f J f h h, 2 f,, (2.4) (A h I)P,τh h 2, for any τ h T h, (2.5) and consequently the a priori error estimate: u ũ h L2 () + h u ũ h H () h2 ( u 2, + f, ). (2.6) 3. uperconvergence. In this section, we generalize superconvergence results in [3] from planar meshes to surface meshes. We consider the effect of the additional discrete geometry error introduced by using a polyhedral surface h as an approximation of. Following [5] we introduce the following notations. For each edge h, let l denote its length and let Ω be the patch of, consisting of two triangles τ h and τ h sharing the edge ; see Figure 3.. For the element τ h Ω, θ denotes the angle opposite to the edge, l + and l denote the lengths of other two edges, and n h denotes the outwards normal of h on τ h. All triangles are orientated counterclockwise. The subscript + or is used for the next or previous edge in this orientation. Let t be the unit tangent vector of with counterclockwise orientation and n be the unit outward normal vector of in the supporting plane of τ h. Recall that n h is the unit outward normal vector of h on τ h, therefore n h n and n h t. An index will be added for the corresponding quantity in τ h. Note that t = t because of the orientation. Unlike the planar domain case, n n in general. By (2.5), however, we have n + n = n h n h n h n + n h n h. (3.) n h n h x 4 t l + n τ h θ l x 2 n τ h θ x 3 x Fig. 3.. The patch Ω of the edge. For Ω, we introduce the following definition which is first introduced in [9] and called strong regular conditions there. Definition 3.. The patch Ω is an O(h 2 ) approximate parallelogram if it satisfies x x 4 x 2 x 3 = O(h 2 ), x x 2 x 4 x 3 = O(h 2 ). x i3 x i2 x i4 x i x i

8 8 H. WI AND L. CHN AND Y. HUANG Notice that, the included angle of n h and n h differ only by O(h), namely, τ h and τ h are almost on the same plane. Therefore we still call Ω as an approximate parallelogram following the terminology in [3]. Remark 3.. The O(h 2 ) condition can be relaxed to O(h +ρ ) for ρ (, ) introduced in [4] and the analysis in our paper can be easily adapted to this case. Definition 3.2 ([3, 5, 4]). Let h = 2 denote the set of edges in T h. The triangulation T h is O(h 2σ ) irregular if for each, Ω forms an O(h 2 ) approximate parallelogram, while 2 Ω = O(h 2σ ). When the triangulation T h is O(h 2σ ) irregular, we can split T h into two parts: T h = T,h T 2,h, with T i,h = {τ T h, τ Ω, with i }. (3.2) We define the domain Ω i,h τ h, i =, 2. Then τ h T i,h Ω,h Ω 2,h = h, and Ω 2,h = O(h 2σ ). (3.3) For u C(), recall that ū = u P C( h ). We define two interpolations of ū on h. First, let ū I V h be the linear interpolation of ū on h, defined by ū I (x i ) = ū(x i ), for all x i N h, where N h be the vertex set of h. Let W h be the continuous and piecewise quadratic finite element space on h, namely, for any w h W h and τ h T h, w h is continuous on h and w h τh span{λ, λ 2, λ 3, λ λ 2, λ λ 3, λ 2 λ 3 }. We define ū Q W h to be the quadratic interpolation of ū satisfying ū Q (x i ) = ū(x i ), for all x i N h, and ū Q = ū, for all h. We can lift ū I and ū Q onto as u I = ū I P : R and u Q = ū Q P : R. On the flat triangle τ h, we have the following two important lemmas [5]. Lemma 3.3. Let ū I and ū Q be the linear and quadratic interpolations of ū defined above. For all v h V h, we have the local error expansion formula: τ h h (ū ū I ) h v h dσ h = τ h [α ( 2 ū Q t 2 v h t ) + β ( 2 )] ū Q v h, t n t where α = 2 cot θ (l 2 + l 2 ), β = 3 cot θ τ h. Lemma 3.4. For any edge h, we have α + α = O(h 2 ), β + β = O(h 2 ); (3.4) α α = O(h 3 ), β β = O(h 3 ), if ; (3.5) 2 ū t z v h t h u 2,, τ h h v h ; (3.6)

9 UPRCONVRGNC AND GRADINT RCOVRY ON URFAC 9 2 ū v h (h 2 t z t h ū + 3 h ū ) h v h ; (3.7) τ h 2 (ū ū Q ) t z v h 3 t h ū h v h ; (3.8) τ h where z is n or t. Our main result is the following superconvergence between the surface finite element solution and the linear interpolation of the extension of the true solution on the polyhedral surface mesh. Theorem 3.5. uppose the triangulation T h is O(h 2σ ) irregular. Let u be the solution of (.), and u h the linear finite element solution on h. If u H 3 () W 2 (), then for all v h V h, we have and h h (ū ū I ) h v h dσ h h +min{,σ} ( u 3, + u 2,, ) v h,h, (3.9) h u h h ū I,h h +min{,σ} ( u 3, + u 2,, ) + h 2 f,. (3.) Proof. Using the basic identity in Lemma 3.3 we get ( h (ū ū I ), h v h ) = where I i = [(α α ) i I 3 = h +β [ β τ h T h τ h = I + I 2 + I 3, 2 ū Q t 2 ( 2ū t n + 2 (ū Q ū) t n v h t [α ( 2 ū Q t 2 v h + (β β t ) ]. 2 ū t n ) vh t + β ) ( v h + β t 2 ū v h t n t 2 (ū Q ū) t n 2 )] ū Q v h t n t ], i =, 2, v h t (3.) To estimate I, we use Lemma 2.3, the estimates (3.5) and (3.7) to get I [ α α τ h h 2 h ū Q h v h + β β (h 2 h ū + 3 h ū ] ) h v h τ h ( h 2 2 h ū + h 3 h ū ) h v h τ h τ h T,h h 2 u 3, v h,h. (3.2)

10 H. WI AND L. CHN AND Y. HUANG To estimate I 2, we use Lemma 2.3, the estimates (3.4) and (3.6) to get I 2 2 h u 2,, Ω h v h h +σ u 2,, v h,h. get To estimate I 3, we use Lemma 2.3, (3.), the estimates (3.4), (3.7) and (3.8) to I 3 β h h 2 ū h (n + n t ) v h t + h 2 3 h ū h v h τ h T τ h h τ h T h ( ū 2,τh + h ū 3,τh + ū 3,τh ) v h,τh h 2 τ T u 3,τ v h,h h 2 u 3, v h,h. The inequality (3.9) then follows. We now prove (3.) as h (u h ū I ) h v h dσ h h = h (u h ū) h v h dσ h + h (ū ū I ) h v h dσ h h h = (A h ũ h u) ṽ h dσ + ( u A h u) ṽ h dσ + h (ū ū I ) h v h dσ h h =I 4 + I 5 + I 6. (3.3) The estimate of I 6 is given by (3.9). Let C = ṽhdσ. From (2.3) and (2.5), we can estimate I 4 as (A h ũ h u) ṽ h dσ = (A h ũ h u) (ṽ h C ) dσ = ( (3.4) J f h f)(ṽ h C ) dσ h 2 f, ṽ h, h 2 f, v h,h. From (2.7) and (2.5), we can estimate I 5 as ( u A h u) ṽ h dσ = (I A h ) P u ṽ h dσ (A h I)P, u, ṽ h, h 2 u, ṽ h, h 2 u, v h,h. (3.5) Combining (2.3), (2.5), (3.), (3.9) and (3.5), we proved (3.). Using the equivalence of H norm of u and ū (c.f. Lemma 2.3), we obtain the supercloseness between ũ h and u I. Corollary 3.6. Assume the same hypothesis in Theorem 3.5. Let ũ h and u I be the lifting of u h and ū I onto, respectively. Then ũ h u I, h +min{,σ} ( u 3, + u 2,, ) + h 2 f,. (3.6)

11 l + n θ l τ h x 2 τ h θ x UPRCONVRGNC AND GRADINT RCOVRY ON URFAC 4. Gradient Recovery chemes. In this section, we discuss how to generalize the gradient recovery methods, including local and global schemes, from planar meshes 2 to polyhedral meshes. 4.. Local Averaging chemes on urface Meshes. We first recall the local averaging schemes in the planar case. Give an element patch Ω i R 2 around the vertex x i, namely, Ω i = x i τ h τ h, let {x ij } m be the boundary vertices of Ω i which are orientated counterclockwise. Let τ j = x i x ij x i(j+), j =,..., m, and x i(m+) = x i ; see Figure 2 (a). x i3 x i2 x i4 x i x i3 x i4 x i x i x i5 x i2 x i6 x i x i5 x i6 (a) the patch Ω i of x i on the plane (b) the patch Ω i of x i on a surface Fig. 4.. Local patches on planar meshes and surface meshes. We consider the following two gradient recovery operators G h applied to a finite element function v h V h :. imple averaging: (G h v h )(x i ) = m v h τj (x i ). m 2. Weighted averaging: (G h v h )(x i ) = m τ j Ω i v h τj (x i ). Theorem 4. ([4]). Assume every two adjacent triangles in patch Ω i form an O(h 2 ) approximate parallelogram and u W 3 (Ω i ). Let u I be the nodal interpolation of u and (G h u I )(x i ) be produced by either the simple averaging, or the weighted averaging. Then (G h u I )(x i ) u(x i ) h 2 u 3,,Ωi. (4.) Theorem 4. can be found in [4]. In their proof, the authors utilized the result that the barycenter of a triangle is the superconvergence point for the derivative of the linear interpolation which is not true; see [] (page 285). We shall extend Theorem 4. to polyhedral surface meshes and provide a correct proof. Give an element patch Ω i h of the vertex x i, see Figure 2 (b). Let Ω i = P (Ω i ) be the patch lifted to the surface. For a finite element function v h V h ( h ), a natural generalization of (G h h v h )(x i ) on surface meshes is given below:. imple averaging: (G h h v h )(x i ) = m h v h τj (x i ). m

12 2 H. WI AND L. CHN AND Y. HUANG 2. Weighted averaging: (G h h v h )(x i ) = m τ j Ω i h v h τj (x i ). Here we simply change the conventional gradient operator in R 2 to the gradient operator h on polyhedral meshes. In principle, the symmetry of the element patch of a vertex is the source of the superconvergence [37]. We thus introduce the following concept. An element patch Ω i of a vertex x i is called O(h 2 )-symmetry if for any boundary vertex x ij of Ω i, there exits another boundary vertex x ij of Ω i, so that x ij 2x i + x ij = O(h 2 ). (4.2) which mean x ij and x ij is approximately symmetric about x i. Let n and n h τj be the unit outward normal vectors of at x i and h at the triangle τ j in the patch Ω i, respectively. ince the C 3 smoothness of and the O(h 2 )-symmetry of patch Ω i, for any triangle τ j in the patch Ω i, there exists another triangle τ j, so that their unit outward normal vectors n h and n h at x i satisfy and their areas τ j and τ j satisfy n h 2n + n h = O(h 2 ). (4.3) τ j τ j = O(h 3 ). (4.4) Theorem 4.2. Let u W 3 (), ū = u P and ū I be the linear interpolation of ū on Ω i ; let (G h h ū I )(x i ) be produced by either the simple averaging, or the weighted averaging on surface meshes. If Ω i is O(h 2 )-symmetric, then (G h h ū I )(x i ) u(x i ) h 2 u 3,,. (4.5) Proof. For the simple averaging, using the triangle inequality, substituting x = x i into (2.3), and noting P 2 = P, we have (G h h ū I )(x i ) u(x i ) m ( h ū I h ū) τj (x i ) m + m ( h ū τj P u ) (x i ) m m ( h ū I τj h ū ) τj (x i ) m + m ( (P h τj P ) u ) (x i ) m =I + I 2. We estimate the two terms as follows. First for I 2, by P h = I n h n t n, P = I nn t, and (4.3), we have I 2 m ( P h τj P ) (x i ) m u,, h2 u,,. For I, we use the following identity on the error expansion. For a triangle τ j = x i x ij x i(j+) in Ω i and ū W (τ 3 j ), we have the following identity []: h (ū I ū)(x) = 2 3 d 2 kū(x) h λ k (x) 2 k= 3 h λ k (x) k= d 3 kū(ς k )t 2 dt, (4.6)

13 UPRCONVRGNC AND GRADINT RCOVRY ON URFAC 3 where x is a point in τ j h, d k = (x x k ) T h, x = x i, x 2 = x ij, x 3 = x i(j+), ς k = x k + t(x x k ), and λ k is the barycenter coordinate of τ j respect to the vertex x k, k =,..., 3. Then by Lemma 2.3 and (4.6), we have I = m m m m [ 2 [ 2 3 d 2 kū(x i ) h λ k (x i ) 2 k= 3 k= 3 h λ k (x i ) k= d 2 kū(x i ) h λ k (x i )] + h2 u 3,, d 3 kū(x k + t(x i x k ))t 2 dt] For any d 2 kū(x i) h λ k (x i ) in one element τ j, by the O(h 2 )-symmetry and the gradient formulas of barycentric coordinates, there exists another d 2 k ū(x i) h λ k (x i ) in another element τ j, so that d 2 kū(x i ) h λ k (x i ) + d 2 k ū(x i) h λ k (x i ) = O(h 2 ), Then the conclusion (4.5) follows. The weighted averaging case is proved similarly by using Lemma 2.3, (4.2), (4.3), (4.4), and (4.6). Remark 4.. When the element patch Ω i is not O(h 2 )-symmetric, since h is shape-regular and quasi-uniform, it will be at least O(h)-symmetric. o using the same proof, we can get (G h h ū I )(x i ) u(x i ) h u 2,,. (4.7) 4.2. Local Least-quares Fitting on Tangent Planes. Let Ω i h be the element patch around x i. Let n i be the unit outward normal vector of at x i and M i be the tangent plane of at x i. Along the direction of n i, define the projection P (x) = x + (x i x, n i )n i M i, for any x Ω i. (4.8) Then we can project Ω i onto M i and get a planar patch Ω i = P (Ω i ). If Ω i is O(h 2 )- symmetric, since is C 3, it is easy to show that Ω i has the same property by Taylor expansion. In [2], the authors firstly projected the patch Ω i on the tangent plane M i of at x i, then used simple averaging method to recover the gradient at x i. We adopt this approach to develop several recovery schemes. Let { x ij x ij = P (x ij ), j =, m} be the boundary vertices of the lifted patch Ω i. Then we construct a local Cartesian coordinate system whose origin is x i, the direction of z-axis is the direction of n z := n i, the direction of x-axis is the direction of n x := ( x i x i )/ x i x i, and the direction of y-axis is the direction of n y := n z n x. Let x j := (( x ij x i ) n x, ( x ij x i ) n y ), j =, m. We then get a patch Ω i R2 whose center is x O := (, ) and boundary vertices are { x j }m. Let v h be a linear finite element function on Ω i. We define a linear finite element function v h on Ω i by setting v h (x O) = v h (x i ) and v h ( x j ) = v h(x ij ) for j =,, m. We use the the following two least-square fitting schemes:. Local L 2 -projection. We seek linear functions p l P ( Ω i ) (l =, 2), such that [p l (x) l v h(x)]q dx =, for all q P ( Ω i), l =, 2. (4.9) Ω i

14 4 H. WI AND L. CHN AND Y. HUANG 2. Local discrete least-squares fitting proposed by Zienkiewicz-Zhu [42]. Let c j be the barycenter of τ j Ω i. We seek linear functions p l P ( Ω i ) (l =, 2), such that m [p l (c j ) l v h(c j )]q(c j ) =, for all q P ( Ω i), l =, 2, (4.) Finally, we get G h h v h V h, the recovery gradient of v h, by setting (G h h v h )(x i ) = p (x O )n x + p 2 (x O )n y. Theorem 4.3. Let u W (), 3 Ω i = P (Ω i ) and u M (P ) = u(p ); let u M I be the linear interpolation of u M on Ω i and G h be one of the four gradient recovery operators defined on the tangent plane patches: the simple averaging, the weighted averaging, local L 2 -projection (4.9) and local discrete least-squares fitting (4.). If Ω i is O(h 2 )-symmetric, then u(x i ) G h Mi u M I (x i ) h 2 u 3,,. (4.) Proof. ince Mi u M (x i ) = u(x i ) ( u(x i ) n i )n i = u(x i ), we have u(x i ) G h Mi u M I (x i ) = Mi ū(x i ) G h Mi u M I (x i ), which transforms the surface patch problem into a plane patch problem. When G h is the simple or the weighted averaging, (4.) is a special case of Theorem 4.2. For the local L 2 -projection and local discrete least-squares fitting (ZZ), following the results of the simple and the weighted averaging and the approach of Theorem 3. in [4], we can prove (4.). Remark 4.2. Again if the patch is not O(h 2 ) symmetric, we will have estimate u(x i ) G h Mi u M I (x i ) h u 2,,. (4.2) Theorem 4.4. Let u be the solution of (2.2) and u h be the solution of (2.). Let G h be one of the six recovery operators: the simple averaging, the weighted averaging on the surface patches and on the tangent plane patches, respectively, and the local L 2 -projection and the local discrete least-squares fitting (ZZ) defined on the tangent plane patches. et w h = G h h u h. If the triangulation T h is O(h 2σ ) irregular and u W 3 (), then u w h, h +min{,σ} u 3,, + h 2 f,. Proof. Let u = ( u) P. By the norm equivalence (2.7), we only need to estimate u G h h u h,h. We decompose u G h h u h = u ( u) I + ( u) I G h h ū I + G h ( h ū I h u h ). For the first term, by the standard approximation theory, u ( u) I,h h 2 u 3,. (4.3)

15 UPRCONVRGNC AND GRADINT RCOVRY ON URFAC 5 To control the second term, we shall split the O(h 2σ ) irregular T h into two parts, slightly different than that in (3.2). We define N,h = {x i N h every two neighboring triangles in Ω i forms an approximated parallelogram}, N 2,h = N h \ N,h, and Ω i,h = xk N i,h Ω k for i =, 2. If we introduce xi = { h x i is one of the end points of }. Then x i N 2,h is equivalent to that there exists at least one edge xi such that 2. By the shape regularity of the mesh, we still have By (4.5) and (4.), in Ω,h, Ω 2,h 2 Ω = O(h 2σ ). ( u) I G h h ū,ω,h I τ h G h h ū I (z) u(z) 2 τ h T,h z N h τ h h 2 u 3,, Ω,h /2 h 2 u 3,,. /2 (4.4) On the other hand, by (4.7) and (4.2), in Ω 2,h, ( u) I G h h ū I,Ω2,h h u 3,, Ω 2,h /2 h +σ u 3,,. (4.5) Combining (4.4) with (4.5), we have ( u) I h ū I,h h +min{,σ} u 3,,. (4.6) ince G h is a linear bounded operator in L 2 norm [4], by Theorem 3.5, G h ( h ū I h u h ),h h (ū I u h ),h h +min{,σ} ( u 3, + u 2,, ) + h 2 f,. (4.7) The conclusion then follows by applying (4.3), (4.6), and (4.7) Global L 2 -projection. We discuss the post-processing operator using the global L 2 -projection Q h : L 2 ( h ) V h (Q h v, w h ) = (v, w h ), for all w h V h. The global L 2 -projection of a vector function v (L 2 ( h )) 3 is a vector function in (V h ) 3 whose k-th component is the global L 2 -projection of the k-th component of v. For the purpose of analysis, we introduce a perturbation of Q h. Let ϕ i V h be the nodal basis at the vertex x i. et V = (v,, v N ) t with v i = (v, ϕ i ) and M = (m ij ), with m ij = (ϕ i, ϕ j ). The matrix M is known as the mass matrix. Then the matrix realization of Q h will be Q h v = (ϕ,, ϕ N )M V. For piecewise continuous function v, we then define V = (v,, v N )t with v i = 3 m v τ j (x i ) τ j. Note that v i is an approximation of v i as v i = Ω i v(x i )ϕ i dσ h. We define Q h, a perturbation of the global L2 -projection, as : Q hv = (ϕ i,, ϕ N )M V.

16 6 H. WI AND L. CHN AND Y. HUANG Note that if v is piecewise constant on T h, then Q h v = Q h v. We first consider the error introduced by the in-exact evaluation of the integral. Lemma 4.5. uppose v W 2 () and the triangulation T h is O(h 2σ ) irregular. Let v = v P, then Q h v Q h v,h h +min{,σ} v 2,,. (4.8) Proof. By the definitions of Q h and Q h, we have Q h v Q h v 2 N, h =(V V ) t M MM (V V ) h 2 ( v i v i) 2. (4.9) We then apply the Taylor expansion to get v i v i m = ( v v(x i ))ϕ i dσ h τ j m [ = h v τj (x i ) (x x i ) + ] τ j 2 (x x i) 2 h v τj (x i + t(x x i ))(x x i ) ϕ i dσ h m v(x i ) (x x i )ϕ i dσ h τ j + m ( h v τj v)(x i ) (x x i )ϕ i dσ h τ j m + τ j 2 (x x i) 2 h v τj (x i + t(x x i ))(x x i )ϕ i dσ h =I + I 2 + I 3. i= (4.2) We estimate the three terms as follows. First for I 2, by (2.5) in Lemma 2.2, we have m I 2 = (P h τj P ) v(x i ) (x x i )ϕ i dσ h τ j h4 v,,. (4.2) For I 3, we have I 3 h 4 v 2,,P(Ω i). (4.22) For I, since (x i ) (x x i )ψ i is a quadratic function, using three-points (middle points of three edges) numerical integration formula, we have m I v,, Ω τ j i Ω i (x ij x i ) (4.23) Then if Ω i is O(h 2 )-symmetric, by (4.2), (4.2), (4.22) and (4.23), we have v i v i h 4 ( v,, + v 2,, ). (4.24)

17 UPRCONVRGNC AND GRADINT RCOVRY ON URFAC 7 Otherwise we only have v i v i h 3 ( v,, + v 2,, ). (4.25) We split the vertices of T h into two parts, N h = N,h N 2,h as in Theorem 4.4. Then we have N,h = O(h 2 ) and N 2,h = O(h 2+2σ ) by the O(h 2σ ) irregularity of T h, where denotes the cardinality of a set. Then Q h v Q h v 2, h (h 2 N,h h 8 + h 2 N 2,h h 6 )( v,, + v 2,, ) 2 from which (4.8) is proved. (h 4 + h 2+2σ )( v,, + v 2,, ) 2, We then consider the error introduced by the geometry approximation. Lemma 4.6. uppose u W () and the triangulation T h is O(h 2σ ) irregular. Let ū = u P and u = ( u) P, then Q h( u h ū),h h +min{,σ} u,,. (4.26) Proof. By the definition of Q h, the O(h2σ ) irregular of T h, P h = I n h n t n, P = I nn t and (4.3), we can follow the same proof pattern of (4.8) in Lemma 4.5, to get Q h ( u h ū) 2, h N m τ j ( ) u ( h ū) τj (xi ) h 2 i= i= N m =h 2 τ j ( ) Ω i u ( h ū) τj (xi ) Ω i i= N m =h 2 τ j Ω i Ω i (P P h τj ) u(x i ) (h 4 + h 2+2σ ) u 2,,. The conclusion (4.26) then follows. Lemma 4.7. uppose u W 3 () and the triangulation T h is O(h 2σ ) irregular. Let ū = u P and ū I be the linear interpolation of ū on h, then Q h( h ū h ū I ) h +min{,σ} u 3,,. (4.27) Proof. By the definition of Q h, the O(h2σ ) irregular of T h and the identity (4.6),

18 8 H. WI AND L. CHN AND Y. HUANG we can follow the same proof pattern of (4.8) in Lemma 4.5, to get Q h( h ū h ū I ) 2, h N m τ j ( h ū h ū I ) τj (x i ) 3 h 2 h 2 i= N 3 Ω i i= m τ j Ω i ( h ū h ū I ) τj (x i ) 2 2 (4.28) (h 4 + h 2+2σ ) u 2 3,, The conclusion (4.27) then follows. Now we are in the position to present our main result on the global L 2 -projection. Theorem 4.8. Let u W 3 () be the solution of (2.2) and u h be the solution of (2.). et w h = Q h h u h. If the triangulation T h is O(h 2σ ) irregular, then u w h, h +min{,σ} u 3,, + h 2 f,. (4.29) Proof. Let u = ( u) P. By the norm equivalence (2.7), we only need to estimate u Q h h u h,h. We decompose u Q h h u h =(I Q h ) u + (Q h Q h) u + Q h( u h ū) +Q h( h ū h ū I ) + Q h( h ū I h u h ). We bound the first term using the standard approximation property u Q h u,h h 2 u 3,, (4.3) the second term by (4.8) in Lemma 4.5, the third term by (4.26) in Lemma 4.6, and the fourth term by (4.27) in Lemma 4.7. For the last one, by (3.) in Theorem 3.5 and the boundedness of Q h, we have Q h( h ū I h u h ),h = Q h ( h ū I h u h ),h h ū I h u h,h h +min{,σ} ( u 3, + u 2,, ) + h 2 f,. Finally conclusion (4.29) is followed by the triangle inequality The Recovery chemes of Finite Volume Methods. We discuss the superconvergence between the approximations from FM and FVM of the Laplace- Beltrami equation (.). Given a triangulation T, we construct a dual mesh B as follows: for each triangle τ T, select a point c τ τ. The point c τ can coincide with one of the mid-points of edges, but not the vertices of triangles (to avoid the degeneracy of the control volume). In each triangle, we connect c τ to three mid-points on the edges of τ. This will divide each triangle in T into three regions. For each vertex x i of T, we collect all regions containing this vertex and define it as ω i.

19 UPRCONVRGNC AND GRADINT RCOVRY ON URFAC 9 The vertex-centered finite volume method of (.) is: find u B h V h such that h u B h n ωi ds = f h dσ h, for all ω i, (4.3) ω i ω i where n ωi is the outward unit normal vector of ω i on the supporting plane of corresponding triangles. The following identity can be found in [2, 3, 26, 4]. For completeness, we include a simple proof from [4] here. Lemma 4.9. Let u h be a linear finite element function. One has h u h n ωi ds = h u h h ϕ i dσ h, (4.32) ω i τ h τ h Ω i where ϕ i V h is the hat basis function at x i. Therefore the stiffness matrix of finite volume methods is the same as that from finite element methods. Proof. Let us consider a triangle τ h = x i x j x k in Ω i orientated counterclockwise. Let c be the interior point in τ h of ω i and m and m 2 the mid-points of x i x j and x i x k, respectively. o segments m c and cm 2 are the parts of ω i. By the divergence theorem, the fact h u h is piecewise constant, and ϕ i is linear on edges x i x j and x i x k and vanished on x j x k, we have = 2 ( c m2 + m c ( xj xi + x i x k ) h u h n ωi ds = ( m xi + x i m 2 ) h u h n ds = ϕ i h u h n ds = τ h ) h u h n ds τ h h u h h ϕ i dσ h. Theorem 4.. Let u G h be the standard Galerkin approximation and ub h the box (finite volume) approximation of the Laplace-Beltrami equation (.). Assume f H (), then u G h u B h,h Ch 2 f,. (4.33) Proof. Let V,B be the piecewise constant space on B. We can define a mapping Π h : V h V,B, as Π h v h = N i= v h(x i )χ ωi, where χ ωi is the characteristic function about ω i. ince we use barycenters as the vertices of control volumes, we have v h = Π hv h. τ h τ h Let f c be the L 2 projection of f h (see (2.)) into the piecewise constant function space on h, then (f h, v h Π v h ) = (f h f c, v h Π v h ) f h f c,h v h Π v h h Ch 2 f, v h,h. Here we use average type Poincaré inequality for f h and v h on each τ h. By Lemma 4.9 and (4.4), we have h u B h h v h dσ h = v h (x i ) h u B h n ωi ds = (f h, Π hv h ). h ω i ω i

20 2 H. WI AND L. CHN AND Y. HUANG Then for any v h V h, we have h (u G h u B h ) h v h dσ h =(f h, v h ) h h h u B h h v h dσ h =(f h, v h Π hv h ) Ch 2 f, v h,h. The estimate (4.33) is obtained by taking v h = u G h ub h. o we reach to the conclusion that the gradient recovery methods for FM can also be used for FVM approximations and result superconvergence. 5. Numerical Tests. In this section, we present three numerical examples to support our theoretical results. The first two are equations on a two-sphere and a torus, for which structured grids can be easily constructed. The third is on a more general surface triangulated into unstructured grids. Let u be the true solution of the Laplace-Beltrami equation (.) on the surface, ū = u P and u = ( u) P. Let u h be the surface linear finite element approximation of u and ū I the linear interpolation of ū on h. For the easy of computation, we report the following errors on discrete surface h : I = h ū I h u h,h, h = u Q h h u h,h, i = u G i h h u h,h, i =,..., 6, where G h and G2 h represent the directly simple and weighted averaging recovery operators on h, respectively, and G 3 h, G4 h, G5 h and G6 h represent the corresponding simple averaging, weighted averaging, local L 2 -projection and local discrete least-squares fitting (ZZ) recovery operators on the tangent plane, respectively. All the errors are computed on the polyhedral surface h using the 9-th order quadrature rules. Our numerical programs are based on the package ifm [2]. Let {Th i}k i= be a sequence of meshes of a surface by uniform refinement and (Th i i ) be the error associated to Th. ince the mesh size h i 2h i, we determine the experimental order of convergence by ln i (Th ) (T i h ) / ln 2, i = 2,..., k. Also instead of using h, we list number of degree of freedoms in the error tables. 5.. xample on the unit two-sphere. We consider the unit sphere whose signed distance function is d(x, y, z) = x 2 + y 2 + z 2. (5.) We choose f such that u(x, y, z) = xy is the true solution of the Laplace-Beltrami equation (.) defined on the unit spherical surface = {(x, y, z) R 3 d(x, y, z) = }. The initial mesh T is the projection of an icosahedron on the unit two-sphere. Then we obtain a sequence of meshes by the regular refinement of T, that is each triangle is divide into four conjugate small triangles by connecting mid-points of edges, and the new nodes are projected to the surface. The corresponding results are given in Table and the numerical approximation of u(x, y, z) = xy with 2562 degree of freedoms (dofs) is shown in Fig. 5..

21 UPRCONVRGNC AND GRADINT RCOVRY ON URFAC 2 Table 5. xample : rror table for surface linear finite element approximation and several gradient recovery methods on the unit sphere. N I order h order order 2 order 2 2.7e e e e e e e e e e e e e e e e e e e e e-3.83 N 3 order 4 order 5 order 6 order e e e e e e e e e e e e e e e e e e e e Z Y.5.5 X Fig. 5.. The numerical approximation of u(x, y, z) = xy with 2562 dofs on the unit two-sphere xample 2 on a torus. We consider a torus surface whose signed distance function is d(x, y, z) = (4 x 2 + y 2 ) 2 + z 2. (5.2) We choose f such that u(x, y, z) = x y is the true solution of the Laplace-Beltrami equation (.) on this torus. Through the parametrization of the torus, we can get a series of uniform meshes on its parameter space, and then map the mesh onto the torus. The corresponding results are given in Table 2 and the numerical approximation of u(x, y, z) = x y with 32 dofs is shown in Fig. 5.2.

22 22 H. WI AND L. CHN AND Y. HUANG Table 5.2 xample 2: rror table for surface linear finite element approximation and several gradient recovery methods on the torus. N I order h order order 2 order e e e e e e e e e e e e e e e e-2 2. N 3 order 4 order 5 order 6 order e e e e e e-.99.3e-.98.8e-.98.2e e e e e e e e e Z Y 5 5 X 6 Fig The numerical approximation of u(x, y, z) = x y with 32 dofs on the torus xample 3 on a general surface. We consider the example from [22]. We solve the Laplace-Beltrami equation (.) on a general surface whose level set function is φ(x, y, z) = (x z 2 ) 2 + y 2 + z 2. (5.3) We choose f such that the true solution is u(x, y, z) = xy. We start from a crude mesh with eighteen nodes to approximate this surface, then refine this mesh and project the new nodes on the surface. We use the first-order projection algorithm in [7]. The corresponding results are given in Table 3 and the numerical approximation of u(x, y, z) = xy with 26 dofs is shown in Fig. 5.3.

23 UPRCONVRGNC AND GRADINT RCOVRY ON URFAC 23 Table 5.3 xample 3: rror table for surface linear finite element approximation and several gradient recovery methods on the general surface. N I order h order order 2 order e e e e e e e e-.2 4.e e e e e e e e e-2.67 N 3 order 4 order 5 order 6 order e e e e e e e e e-.64.5e-.58.2e-.69.6e e e e e Z Y.5.5 X Fig The numerical approximation of u(x, y, z) = xy with 26 dofs on the general surface. From the above numerical tests, we can see clearly the existence of superconvergence phenomena in the surface linear FM. Like the planar case, the mesh quality is a key factor for the superconvergence order. In xample 2, the meshes are almost uniform, so its convergence order is the best possible among the three examples. In future work, we shall develop mesh smoothing schemes on surfaces, such as methods based centroid Voronoi tessellation [9, 2] and optimal Delaunay triangulation [4, ] to improve the mesh quality. The error of the global L 2 -projection recovery method seems better than local recovery methods on h. Although the global L 2 -projection requires an inversion of a mass matrix, it can be solved efficiently by preconditioned conjugate gradient method since the mass matrix is well conditioned. It may need more computational time than local methods. Among six local methods, the local recovery methods on the tangent

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