Isoparametric finite element approximation of curvature on hypersurfaces

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1 Isoparametric finite element approximation of curvature on ypersurfaces C.-J. Heine Abstract Te discretisation of geometric problems often involves a point-wise approximation of curvature, altoug te discretised surfaces are globally only of class C 0,1. We define a discrete second fundamental form by means of an L -projection in te context of isoparametric Lagrange-elements of polynomial degree γ. We prove tat te order of convergence wit respect to te L -norm for tis discrete second fundamental form is γ 1. 1 Introduction In many interesting matematical, pysical and engineering problems te curvature of surfaces plays an important role. Often tese surfaces are free boundaries and are to be determined as one of te major unknowns; some examples are mean-curvature flow, dendrite growt or equilibrium figures of freely rotating drops. Te numerical treatment of tese problems can be difficult if te curvature enters te governing equations explicitly. Often it is possible to arrive at a weak formulation wic does no longer explicitly contain iger derivatives of te free boundary (Dziuk, 1991, for example). Wen suc a weak formulation is not known (Scmidt, 1996; Heine, 006, for example) one needs a pointwise definition of curvature for te discretised surface. Tis paper is concerned wit te approximation of te second fundamental form of a C k -ypersurface R n+1 in te context of an approximation of wit isoparametric Lagrange finite elements. Using te coordinates of te ambient space R n+1 we can write II = ν (1.1) for te Weingarten map, were ν denotes te outer unit normal field of and its tangential gradient. In Section 3 we use a weak formulation of (1.1) to define a discrete second fundamental form II for te C 0,1 -surface. Section 4 establises L -estimates for II for te case n 3. Finally Section 5 presents te results of experimental convergence tests wit respect to te L - and L -norm. Abteilung für Angewandte Matematik, Universität Freiburg, Hermann-Herder-Straße 10, Freiburg i.br., Germany. 1

2 c.-j. eine Notation and Fundamental Definitions.1 Differential geometry In te following R n+1 is an n-dimensional isometric C -immersion into R n+1. We identify te tangent space T P at a point P wit te affine yperplane of R n+1 wic is tangential to at P. We denote te (local) outer unit normal field by ν = (ν 1,..., ν n+1 ) : R n+1. For f C 1 () we define te tangential gradient of f on by f := f ( f ν ) ν R n+1, were f denotes te usual gradient of R n+1. Above definition makes sense only if f is extended to a neigbourood of. However, f does not depend on te extension but only on te values of f on. Likewise, for w = (w i ) n+1 i=1 ( C 1 () ) n+1 we define its tangential divergence by w := w ( w i ν ) ν i and we define te Laplace-Beltrami operator of g C () by g := g. Te Weingarten-map II = (II 1,..., II n+1 ) : R (n+1) (n+1) is defined by II i := ν i, 1 i n + 1. II is te usual second fundamental form but written in te coordinate frame of te ambient space R n+1, wit zero-extension to (T P ). For P te eigenvalues κ i, 1 i n, of te second fundamental form II(P ) TP T P are called principal curvatures of at P. Furter, we define te mean curvature H, te curvature vector Y and te Gauss curvature K at P by H := 1 n n κ i, Y := n H ν, K := i=1 n κ i. (.1) Elementary computation yields te formula id = Y for te curvature vector Y. Finally, for = te formula for integration by parts on is given by f do = f Y do, f C 1 (). (.). Function spaces For Ω R n open we denote by L p (Ω) and H m p (Ω) te usual Lebesgue- and Sobolevspaces wit te corresponding norms ( u Lp(Ω) = Ω u p dx ) 1 p i=1 ( m, u H m p (Ω) = D k u p L p(ω) wit te usual modification for p =. For p = we simply write H m (Ω) := H m (Ω). By C ( m,α Ω ) and C m,α (Ω) we denote te usual Hölder spaces and k=0 ) 1 p,

3 finite element approximation of curvature on ypersurfaces 3 teir associated norms. Using local coordinate-carts it is possible to carry over te definitions of Sobolev- and Hölder-spaces to a differential manifold. Te resulting spaces are independent from te special coice of te coordinate system up to norm-equivalence. In particular, Wloka (198, pp. 9) sows tat te spaces H m () are well defined for a C m 1,1 -manifold for any m N, wic covers especially te case of te C 0,1 -manifolds constructed by te finite element approac of Section.3 below. For an n-dimensional C 0,1 -immersion R n+1 we coose u H 1 () := u u do for te H 1 semi-norm..3 Isoparametric finite elements For simplicity we consider a compact closed C k, 1 -ypersurface R n+1 were k 1. We note tat te results could be extended to immersions but tis would involve some notational complications. Because te sectional curvature of is bounded tere is δ > 0 suc tat te decomposition x = π(x) + d(x) ν(x), wit π : U, is unique for x U := {x R n+1 dist(x, ) < δ}, were d(x) C k (U) is a signed distance function and ν(x) := d π(x). It follows tat ν(x) is normal to at π(x) wit ν(x) 1. Figure.1 Construction of γ π( Q i ) γ 1 Q i To construct an isoparametric finite element approximation γ of of given polynomial degree γ we first approximate by a simplicial polytope 1 wit associated triangulation T 1 = {T 1 T 1 is a face of 1 }. Te polytope 1 must be contained in te δ-strip U were te projection π is well-defined. Ten, for any simplex T 1 T 1, we take te images π(q i) of te canonical Lagrange-nodes Q i of T 1 and define an isoparametric simplex T γ by applying Lagrange-interpolation of degree γ to te coordinates of te projected Lagrange-nodes π(q i ), see Figure.1. Te result is a discrete surface γ of class C0,1 and associated isoparametric triangulation T γ were T γ := Φγ ( T ) T γ T 1 is an image of a reference simplex T R n under a polynomial parametrisation Φ γ : T T γ T 1 of degree γ. By construction 1 interpolates γ between its vertex Lagrange-nodes. Tis will be a elpful feature in Section 4 below.

4 4 c.-j. eine On γ we define parametric finite elements wic again are piece-wise polynomial of degree γ or over te reference simplex T : W γ (γ ) := {v C 0,1 ( γ ) Φγ T 1 v := (v Φ γ ) P T 1 γ ( T ) T γ T γ }. (.3) Te finite element space W γ (γ ) is virtually isoparametric as te local parametrisation of γ over te reference simplex is piecewise a polynomial of degree γ; so we ave in particular id γ (W γ (γ ))n+1. Because π γ is by construction a bijection we can lift v W γ (γ ) to v C 0,1 () by v (y) := (π 1 v )(y) = v (π 1 (y)), x γ, y = π(x). We call a family of isoparametric triangulations {T γ } >0 quasi-uniform if tere is c > 0 independent from wit c ρ(t ) T T γ wit = max (T ), T T γ were for T T γ we denote by (T ) te diameter of T and by ρ(t ) te radius of te largest ball contained in T. 3 Definition of te discrete curvature In te following always denotes a C 0,1 -approximation of a closed compact C immersion R n+1 of dimension n wit W H 1 ( ) suc tat id (W ) n+1. Definition 3.1 (Discrete Curvature Vector) Under te given assumptions te discrete curvature vector Y (W ) n+1 is defined as te solution of (Y, ψ ) L( ) = ( id, ψ ) L( ) ψ (W ) n+1. (3.1) Tis definition follows Scmidt (1993) and is motivated by te identity Y ψ do = id ψ do = id ψ do ψ (H 1 ()) n+1, wit Y := n H ν and R n+1 at least C 1, 1, see Section.1. Te experimental convergence tests for te case of a spere presented in Scmidt (1993) sow approximate linear convergence in te L and L norms for quadratic Lagrange elements and no convergence for linear elements. For te definition of te scalar mean and Gaussian curvature we ave a look at te Weingarten map II, see Section.1. To derive a weak formulation for II = (II 1,..., II n+1 ) R (n+1) (n+1) we multiply te identity II j = (ν ) j by a test-function ψ H 1 (), integrate over and take te integration by parts formula (.) on page into account. Carried over to and W tis yields:

5 finite element approximation of curvature on ypersurfaces 5 Definition 3. (Discrete Weingarten map) Te discrete Weingarten map II = (II 1,..., IIn+1 ) (W ) (n+1) (n+1) is defined as te solution of ( II j, ψ )L = ( ν j ( ), ψ )L ( ν j ( ), Y ) ψ ψ L ( ) (W ) n+1, (3.) were Y is te discrete curvature vector defined in (3.1) and ν = (ν 1,..., νn+1 ) is te (discontinuous) outer normal field of. We use II to define te discrete mean curvature H and discrete Gaussian curvature K in te obvious way, see Section.1. 4 Error estimates Teorem 4.1 Let R n+1 (n 3) be a closed compact embedded ypersurface of class C m, 1 wit m 1 and 1 γ m. Let {T γ } >0 be a quasi-uniform family of isoparametric triangulations of of polynomial degree γ. Ten it olds for te discrete curvature vector and te discrete Weingarten map Y, II W γ (γ ) tat were c = c(σ, γ, ) does not depend on. Y Y L() c γ 1, (4.1) II II L() c γ 1, (4.) Remark 4. Te regularity assumption for ensures tat Y, II H m 1, (). Remark 4.3 Our proof of Teorem 4.1 is based on Dziuk (1988) wo proves error estimates for te discretisation of te Poisson equation on arbitrary surfaces using linear finite elements. Te numerical experiments performed by Scmidt (1996) and our own numerical experiments (see Section 5) are in accordance wit tis proposition. We continue wit some intermediate results before we give a proof of Teorem 4.1. Lemma 4.4 For u W γ (γ ) it olds: Proof. See Dziuk (1988). 1 c u L( ) u L() c u L( ), 1 c u L( ) u L() c u L( ). (4.3) Lemma 4.5 (Inverse Inequality) Let T T γ. Ten it olds u L(T ) c σ(t )n/+1 u (T ) L(T γ ) u W γ (γ ). (4.4)

6 6 c.-j. eine Proof. Te proof is a one-to-one copy of te proof for non-parametric finite elements. Following Dziuk (1988) we introduce te following abbreviations: P u := u u ν u ν, P ik := δ ik ν i ν k, H ik := d xix k = ν i,xk = ν k,xi. We ave tat u (x) = (P (x) d(x) H(x)) u x d(x) ν(x) and P H = H P = H. Te defining equations (3.1) for Y can be rewritten using integrals over instead of : 1 Y ϕ do = (P (I d H) id ) : (P (I d H) ϕ ) 1 do, (4.5) wit := do do. Obviously we ave 1 c c <. By defining te abbreviation A := 1 P (I d H)P (I d H)P we arrive at 1 ( ) (4.5) Y ϕ do = A id : ϕ do. (4.6) Alternatively, we can use te following formulation: 1 1 Y ϕ do = (P (I d H) ) ϕ do = B : ϕ do, (4.7) wit B := 1 P (I d H)P. To cope wit te error introduced by lifting W γ (γ ) to we need te following estimates: Lemma 4.6 Let R n+1 be a closed compact embedded ypersurface of class C m, 1 wit m 1. Let T γ be a quasi-uniform isoparametric triangulation of of polynomial degree 1 γ m. Ten were c = c(σ, γ, ). d c γ+1, (4.8) 1 c γ+1, (4.9) (A I) P c γ+1, (4.10) (B I) P c γ, (4.11) (4.8) and (4.9) are already mentioned witout proof in Nédélec (1976) for surfaces in R 3. Dziuk (1988) proves (4.10) for te linear case (γ = 1) for surfaces in R 3. We neverteless give a detailed proof of (4.9) because we tink tat te result is not obvious, toug te calculations are elementary. Proof Lemma 4.6. For any isoparametric surface γ we also ave naturally a simplicial polytope 1 wic interpolates te Cm, 1 -surface as well as γ linearly in te vertex nodes of te triangulation T γ. Let T γ T γ be an isoparametric

7 finite element approximation of curvature on ypersurfaces 7 element, T = π(t γ ) its associated lifted element and T 1 te corresponding triangle wic interpolates T γ and T in te vertex Lagrange-nodes (Figure 4.1). Witout loss of generality we can assume tat T 1 Rn {0}. We use te following notations for te coordinates on R n+1, T 1, T γ and T : R n+1 : x = (x 1,..., x n+1 ). T 1 : η = (η 1,..., η n ). Witout loss of generality we can assume η i = x i, 1 i n. We parametrise T γ and T over T 1. T γ : ξ = (ξ 1(η),..., ξ n+1 (η)). Te coordinate functions ξ i (η) of T γ are polynomials of degree γ over T 1. T : π(ξ(η)) = (π 1 ξ(η),..., π n+1 ξ(η)) = ξ(η) d(ξ(η)) ν(ξ(η)). We remark tat te coordinate functions ξ i (η) of γ are ordinary (i.e. nonparametric) finite element interpolations of polynomial degree γ of te functions π i ξ(η) H m+1, (T 1 ). Tis means tat te usual estimates for Lagrange-interpolation apply wit respect to te interpolation of π i by ξ i. Equation (4.8) is ten clear because c γ+1 π i (ξ(η)) ξ i (η) = d(ξ(η)) ν i (ξ(η)), 1 i (n + 1). Te estimate (4.9) for te surface element is more complicated. For any (n + 1)- tuple (a 1,..., a n+1 ) we denote by a b i te n-tuple were te i-t component as been omitted (e.g. a b 1 = (a,..., a n+1 )). In tis notation te surface elements do on T γ and do on T are n do = π n+1 + π bk n dη, do = x n+1 + ξ bk dη. So we ave 1 do do k=1 x n+1 π n+1 + n ξ bk k=1 k=1 π bk. (4.1) Figure 4.1 Parametrisation of over T γ over T 1 π (ξ(η)) d( ξ ( η)) ν( ξ( η)) T ξ(η) T γ η T 1

8 8 c.-j. eine For estimating te individual determinants we employ Leibniz formula, using te notation k i := i for i < k, k i := i + 1 for i k: ξ bk = n ξ ki sgn(α), α S n i=1 α(i) π bk = n sgn(α) α S n i=1 π ki α(i). (4.13) First we note tat T 1 interpolates T γ as well as T linearly between teir vertices. So we ave te following interpolation estimates for te derivatives of te coordinate functions (1 j n): c ξ i i = ξ i δ ij, c π i δ ij (1 i n), (4.14) c ξ n+1, c π n+1. (4.15) It follows tat ξ b i c, π b i c (1 i n), (4.16) because eac of te summands in Leibniz formula (4.13) contains a factor ξn+1 respectively πn+1. Furter, te polynomials ξ i (η) interpolate π i (η), so we ave π k i ξ bk ξ k i c γ. Togeter wit (4.16) tis yields π bk It remains to estimate ξ n+1 = ξ bk π bk ξ bk + π bk c γ+1 (1 k n). }{{}}{{} =O( γ ) =O() π n+1 = ( + O()) (4.14) ξ n+1 (4.17) π n+1. (4.18) To find an estimate we ave to consider te differences ξi πi = d ν i d νi for 1 i, j n. From te approximation properties of ξ i (η) we first get c γ ξ k π k = d ν k + d ν k (1 k n + 1) = d c γ a.e., because d = O( γ+1 ) and ν k is bounded as is C 1, 1. (4.19) Furter, we ave ν i = O() for 1 i n. To see tis we apply te cain rule to d(ξ(η)) i = d x n+1 ξ(η) ξ n+1 i + n j=1 d x j ξ(η) ξ j i

9 finite element approximation of curvature on ypersurfaces 9 So we get ν i = d x i = O(), 1 i n, by estimating eac term according to (4.14), (4.15) and (4.19). Terefore it olds because νi yields ξ i π i = d ν i d ν i = O( γ+1 ), 1 i n, (4.0) is bounded as is at least C 1, 1. Togeter wit (4.13) and (4.18) tis ξ n+1 π n+1 = c x n+1 π n+1 c γ+1, wic completes te proof of (4.9). We quote te proof for (4.10) from Dziuk (1988) were te result is proved for te linear case (γ = 1). Te generalisation for γ > 1 is just a consequence of (4.9): (A I) P = 1 P ((I d H) P (I d H) P I) P (4.9) = P (I d H) P (I d H) P I) P + O( γ+1 ) (4.8) = P P P P + O( γ+1 ) = P ν ν t P + O( γ+1 ). It follows from (4.17) and (4.0) tat ν = ν + O( γ ). So we ave For (4.11) we similarly get (A I) P P ν + c γ+1 c γ+1, (B I) P = 1 P ((I d H) P I) P (4.9) = P (I d H) P I) P + O( γ+1 ) (4.8) = P P P + O( γ+1 ) = ν ν t P + O( γ+1 ) = O( γ ). Proof Teorem 4.1. Let ϕ W γ (γ ) be an arbitrary test-function. For brevity we set := L() and := L (). We ave Y Y (Y Y ) (ϕ Y ) do } {{ } :=I 1 + (Y Y ) (Y ϕ ) do. (4.1) } {{ } :=I

10 10 c.-j. eine To use te weak formulation (4.6) wit I 1 we ave to compensate for te factor 1 introduced by projecting te finite element functions to te smoot surface : I 1 = (4.7) (Y + Y Y Y ) (ϕ Y ) do (Y Y ) (ϕ Y ) do + 1 Y ϕ Y + Lem. 4.6 c γ+1 ϕ Y + c γ (ϕ Y ) 1 Lem. 4.5 c γ 1 ϕ Y. (Y 1 Y ) (ϕ Y ) do (I B ) P : (ϕ Y ) do So we finally get from (4.1): Y Y Y Y Y ϕ + c γ 1 ϕ Y ϕ W γ (γ ). = Y Y c γ 1, by coosing ϕ = I γ 1 Y, using tat Y H m 1, (). Tis proves equation (4.1) of Teorem 4.1 on page 5. Te proof of te estimate (4.) for te discrete second fundamental form II works very similar. As for te proof of (4.1) we start by transforming te integrals in te defining equation (3.) for II into surface integrals over ; we get j 1 II j ϕ do = ν ( A (id ) ) : ϕ do ν j 1 Y ϕ do. (4.) Terefore j II j II j j j (II j II ) (IIj ϕ ) do + (II j II ) (ϕ II ) do j j II j II II j ϕ + 1 II j ϕ II + (II j 1 j j II ) (ϕ II ). µ } {{} :=I 3 (4.3)

11 finite element approximation of curvature on ypersurfaces 11 We transform I 3 by employing te weak formulation (4.): I 3 (ν j id ν j j A id ) : ( ϕ II ) do + (ν j Y ν j j Y ) (ϕ II ) do ( (A I) P id + (id id ) ) (ϕ II j ) + ν j ν j id (ϕ II j ) + µ 1 Y Y II j ϕ + ( µ 1 ν j ν j Y + 1 µ 1 ) j Y II ϕ c γ 1 II j ϕ (see (4.1), (4.9), (4.10), Lemma 4.5). Te remaining terms in (4.3) are of iger order so tat we finally arrive at j II j II c γ 1, wic completes te proof of Teorem 4.1 on page 5. 5 Numerical experiments We ave implemented our definition of te discrete curvature (see Section 3) for isoparametric finite elements in te context of te finite element toolbox ALBERT (Scmidt and Siebert, 1998). Table 5.1 to 5.4 on pages sow te results of our experimental convergence tests wit piecewise polynomial discretisations of degree 1,, 3 and 4 for te following model problems (see Figure 5.1 on te next page): (x z ) + y + z 1 = 0, (5.1a) (x z ) + (y z ) + z 1 = 0, (5.1b) x + ( y) + ( 3 z) 1 = 0, x + y + z 1 = 0. (5.1c) (5.1d) Table 5. on page 14 corresponding to surface (5.1b) as already been publised in Heine (006). Te underlying macro triangulation for te discretisation of all model-problems is an icosaedron. Te discrete surface is just te interpolation of te exact sape wit Lagrange-polynomials. At first te mes was refined adaptively until te L error wit respect to te exact surface was below Afterwards te meses were refined globally by doubly bisecting te elements in eac refinement step. Te tables 5.1 to 5.4 on pages sow te maximal mes widt, te relative L and L errors and te experimental order of convergence (EOC) of te discrete mean and Gaussian curvatures H and K derived from te discrete second fundamental form as defined in Definition 3. on page 5.

12 1 c.-j. eine Figure 5.1 Surface meses for testing our definition of te discrete curvature. Te colour indicates te magnitude of te discrete Gaussian curvature. Te defining equations for te surfaces are (5.1b)-(5.1d). A picture for (5.1a) as been omitted as it is of te same type as te surface sown in figure (a). (a) Equation (5.1b) (b) Equation (5.1c) (c) Equation (5.1d) For te normalisation of te relative errors we use te curvature of te exact surfaces defined by (5.1a) to (5.1d): E C,r := C C Lr( ), were C {K, H}, r {, }. C Lr( ) In order to compare functions defined on wit tose defined on we were using an approximate ortogonal lift from onto wit respect to an averaged normal field on. For a sequence of isoparametric triangulations T j te experimental order of convergence (eoc) is defined by EOC j := (log Ej E j+1 )/(log j j+1 ), were E j := C C j denotes te error between te discrete solution C j and te exact solution C in te appropriate norm. Wile it is not surprising tat te discrete curvatures originating from Definition 3. on page 5 do not converge for linearly interpolated surfaces, te experimental order of convergence for iger order parametrisations is surprising. Te tables 5.1 to 5.4 on pages suggest tat for Lagrange-interpolates of polynomial degree k te experimental order of convergence for te discrete curvature is greater tan k 1, provided tat k > 1. We would ave expected values for te EOC of at most k 1, but not greater. However, one notes tat te results for te less complicated surfaces (5.1c) an ellipsoid and (5.1d) a spere are not as suggestive as te results for te more complicated surfaces: for tem table 5.3 on page 15 and 5.4 on page 16 sow a decreasing EOC as te number of unknowns increases. Tis migt indicate tat te values for te EOC as sown in te tables 5.1 on te next page and 5. on page 14 just did not stabilise yet.

13 finite element approximation of curvature on ypersurfaces 13 Table 5.1 Discrete curvature, surface (5.1a), relative errors Polynomial degree: Polynomial degree: Polynomial degree: Polynomial degree: e e e

14 14 c.-j. eine Table 5. Discrete curvature, surface (5.1b), relative errors Polynomial degree: Polynomial degree: Polynomial degree: Polynomial degree:

15 finite element approximation of curvature on ypersurfaces 15 Table 5.3 Discrete curvature, surface (5.1c), relative errors Polynomial degree: Polynomial degree: Polynomial degree: Polynomial degree: e e e e e e e 05 3.

16 16 c.-j. eine Table 5.4 Discrete curvature, surface (5.1d), relative errors Polynomial degree: Polynomial degree: e e e e e e e 05. Polynomial degree: e e e e Polynomial degree: e e e e e e e e e e e e e e

17 finite element approximation of curvature on ypersurfaces 17 References G. Dziuk. Finite elements for te Beltrami operator on arbitrary surfaces. In S. Hildebrandt and R. Leis, editors, Partial Differential Equations & Calculus of Variations, volume 1357 of Lecture Notes in Mat., pages Springer, Berlin, G. Dziuk. An algoritm for evolutionary surfaces. Numer. Mat., 58(6): , Te algoritm uses Dziuk (1988). Claus-Justus Heine. Computations of form and stability of rotating drops wit finite elements. IMA J. Numer. Anal., 6(4):73 751, 006. ISSN J.-C. Nédélec. Curved finite element metods for te solution of singular integral equations on surfaces in R 3. Comput. Metods Appl. Mec. Engrg., 8(1):61 80, A. Scmidt. Computation of tree dimensional dendrites wit finite elements. Journal of Computational Pysics, 15:93 31, A. Scmidt. Die Berecnung dreidimensionaler Dendriten mit Finiten Elementen. PD tesis, Albert-Ludwigs-Universität Freiburg i. Br., A. Scmidt and K.G. Siebert. ALBERT 0. Adaptive multilevel finite element toolbox using Bisectioning refinement and Error control by Residual Tecniques, August Scmidt and Siebert (1999). A. Scmidt and K.G. Siebert. Abstract data structures for a finite element package: Design principles of ALBERT. Zeitscrift für Angewandte Matematik und Mecanik, 79:49 5, J. Wloka. Partielle Differentialgleicungen. B.G. Teubner, Stuttgart, 198.