Approximation by C^1 Splines on Piecewise Conic Domains
aa r X i v : . [ m a t h . NA ] M a r Approximation by C Splines on Piecewise ConicDomains
Oleg Davydov and Wee Ping Yeo
Abstract
We develop a Hermite interpolation scheme and prove error bounds for C bivariate piecewise polynomial spaces of Argyris type vanishing on the boundary ofcurved domains enclosed by piecewise conics. Spaces of piecewise polynomials defined on domains bounded by piecewise alge-braic curves and vanishing on parts of the boundary can be used in the Finite El-ement Method as an alternative to the classical mapped curved elements [7, 11].Since implicit algebraic curves and surfaces provide a well-known modeling tool inCAGD [1], these methods are inherently isogeometric in the sense of [14]. More-over, this approach does not suffer from the usual difficulties of building a globally C or smoother space of functions on curved domains (see [4, Section 4.7]) sharedby the classical curved finite elements and the B-spline-based isogeometric analysis.In particular, a space of C piecewise polynomials on domains enclosed by piece-wise conic sections has been studied in [11] and applied to the numerical solutionof fully nonlinear elliptic equations. These piecewise polynomials are quintic on theinterior triangles of a triangulation of the domain, and sextics on the boundary tri-angles (pie-shaped triangles with one side represented by a conic section as well asthose triangles that share with them an interior edge with one endpoint on the bound-ary) and generalize the well-know Argyris finite element. Although local bases forthese spaces have been constructed in [11] and numerical examples demonstrated Oleg DavydovDepartment of Mathematics, University of Giessen, Department of Mathematics, Arndtstrasse 2,35392 Giessen, Germany, e-mail: [email protected]
Wee Ping YeoFaculty of Science, Universiti Brunei Darussalam, BE1410, Brunei Darussalam, e-mail: [email protected] the convergence orders expected from a piecewise quintic finite element, no errorbounds have been provided.In this paper we study the approximation properties of the spaces introduced in[11]. We define a Hermite-type interpolation operator and prove an error bound thatshows the convergence order O ( h ) of the residual in L -norm, and order O ( h − k ) in Sobolev spaces H k ( Ω ) . This extends the techniques used in [7] for C splines toHermite interpolation.The paper is organized as follows. We introduce in Section 2 the spaces S , d , ( △ ) of C piecewise polynomials on domains bounded by a number of conic sections,with homogeneous boundary conditions, define in Section 3 our interpolation op-erator in the case d =
5, and investigate in Section 4 its approximation error forfunctions in Sobolev spaces H m ( Ω ) , m = ,
6, vanishing on the boundary. C piecewise polynomials on piecewise conic domains We make the same assumptions on the domain and its triangulation as in [7, 11], asoutlined below.Let Ω ⊂ R be a bounded curvilinear polygonal domain with Γ = ∂Ω = S nj = Γ j , where each Γ j is an open arc of an algebraic curve of at most second order( i.e., either a straight line or a conic). For simplicity we assume that Ω is simplyconnected, so that its boundary Γ is a closed curve without self-intersections. Let Z = { z , . . . , z n } be the set of the endpoints of all arcs numbered counter-clockwisesuch that z j , z j + are the endpoints of Γ j , j = , . . . , n , with z j + n = z j . Furthermore,for each j we denote by ω j the internal angle between the tangents τ + j and τ − j to Γ j and Γ j − , respectively, at z j . We assume that ω j ∈ ( , π ) for all j . Hence Ω is aLipschitz domain.Let △ be a triangulation of Ω , i.e., a subdivision of Ω into triangles, whereeach triangle T ∈ △ has at most one edge replaced with a curved segment of theboundary ∂Ω , and the intersection of any pair of the triangles is either a commonvertex or a common (straight) edge if it is non-empty. The triangles with a curvededge are said to be pie-shaped . Any triangle T ∈ △ that shares at least one edgewith a pie-shaped triangle is called a buffer triangle, and the remaining trianglesare ordinary . We denote by △ , △ B and △ P the sets of all ordinary, buffer and pie-shaped triangles of △ , respectively, such that △ = △ ∪△ B ∪△ P is a disjoint union,see Figure 1. Let V , E , V I , E I , V ∂ , E ∂ denote the set of all vertices, all edges, interiorvertices, interior edges, boundary vertices and boundary edges, respectively.For each j = , . . . , n , let q j ∈ P be a polynomial such that Γ j ⊂ { x ∈ R : q j ( x ) = } , where P d denotes the space of all bivariate polynomials of total degree at most d . By changing the sign of q j if needed, we ensure that q j ( x ) is positive for pointsin Ω near the boundary segment Γ j . For simplicity we assume in this paper that allboundary segments Γ j are curved. Hence each q j is an irreducible quadratic polyno-mial and ∇ q j ( x ) = x ∈ Γ j . (1) pproximation by C Splines on Piecewise Conic Domains 3
Fig. 1
A triangulation of a curved domain with ordinary triangles (green), pie-shaped triangles(pink) and buffer triangles (blue).
We assume that △ satisfies the following conditions:(A) Z = { z , . . . , z n } ⊂ V ∂ .(B) No interior edge has both endpoints on the boundary.(C) No pair of pie-shaped triangles shares an edge.(D) Every T ∈ △ P is star-shaped with respect to its interior vertex v .(E) For any T ∈ △ P with its curved side on Γ j , q j ( z ) > z ∈ T \ Γ j .(F) No pair of buffer triangles shares an edge.It can be easily seen that (B) and (C) are achievable by a slight modification of agiven triangulation, while (D) and (E) hold for sufficiently fine triangulations. Theassumption (F) is made for the sake of simplicity of the analysis. Note that thetriangulation shown in Figure 1 does not satisfy (F).For any T ∈ △ , let h T denote the diameter of T , and let ρ T be the radius ofthe disk B T inscribed in T if T ∈ △ ∪ △ B or in T ∩ T ∗ if T ∈ △ P , where T ∗ denotes the triangle obtained by joining the boundary vertices of T by a straightline, see Figure 2. Note that every triangle T ∈ △ is star-shaped with respect to B T .In particular, for T ∈ △ P this follows from Condition (D) and the fact that the conicsdo not possess inflection points. conic q = 0 PSfrag replacements v v v B T T ∗ conic q = 0 PSfrag replacements v v v B T T ∗ v v v B T T ∗ Fig. 2
A pie-shaped triangle with a curved edge and the associated triangle T ∗ with straight sidesand vertices v , v , v . The curved edge can be either outside (left) or inside T ∗ (right). Oleg Davydov and Wee Ping Yeo We define the shape regularity constant of △ by R = max T ∈△ h T ρ T . (2)For any d ≥ S d ( △ ) : = { s ∈ C ( Ω ) : s | T ∈ P d , T ∈ △ , and s | T ∈ P d + , T ∈ △ P ∪ △ B } , S , d , I ( △ ) : = { s ∈ S d ( △ ) : s is twice differentiable at any v ∈ V I } , S , d , ( △ ) : = { s ∈ S , d , I ( △ ) : s | Γ = } . We refer to [11] for the construction of a local basis for the space S , , ( △ ) and itsapplications in the Finite Element Method.Our goal is to obtain an error bound for the approximation of functions vanishingon the boundary by splines in S , , ( △ ) . This is done through the construction of aninterpolation operator of Hermite type. Note that a method of stable splitting wasemployed in [6, 9, 10] to estimate the approximation power of C splines vanish-ing on the boundary of a polygonal domain. C finite element spaces with a stablesplitting are also required in B¨ohmer’s proofs of the error bounds for his methodof numerical solution of fully nonlinear elliptic equations [2]. A stable splitting ofthe space S , , I ( △ ) will be obtained if a stable local basis for a stable complement of S , , ( △ ) in S , , I ( △ ) is constructed, which we leave to a future work. We denote by ∂ α f , α ∈ Z + , the partial derivatives of f and consider the usualSobolev spaces H m ( Ω ) with the seminorm and norm defined by | f | H m ( Ω ) = ∑ | α | = m k ∂ α f k L ( Ω ) , k f k H m ( Ω ) = m ∑ k = | f | H k ( Ω ) ( H ( Ω ) = L ( Ω )) , where | α | : = α + α . We set H ( Ω ) = { f ∈ H ( Ω ) : f | ∂Ω = } .In this section we construct an interpolation operator I △ : H ( Ω ) ∩ H ( Ω ) → S , , ( △ ) and estimate its error for the functions in H m ( Ω ) ∩ H ( Ω ) , m = ,
6, in thenext section.As in [7] we choose domains Ω j ⊂ Ω , j = , . . . , n , with Lipschitz boundary suchthat(a) ∂Ω j ∩ ∂Ω = Γ j ,(b) ∂Ω j \ ∂Ω is composed of a finite number of straight line segments,(c) q j ( x ) > x ∈ Ω j \ Γ j , and(d) Ω j ∩ Ω k = /0 for all j = k . pproximation by C Splines on Piecewise Conic Domains 5
In addition we assume that the triangulation △ is such that(e) Ω j contains every triangle T ∈ △ P whose curved edge is part of Γ j ,and that q j satisfy (without loss of generality)(f) max x ∈ Ω j k ∇ q j ( x ) k ≤ k ∇ q j k ≤ , for all j = , . . . , n ,where ∇ q j denotes the (constant) Hessian matrix of q j .Note that (e) will hold with the same set { Ω j : j = , . . . , n } for any triangulationsobtained by subdividing the triangles of △ .The following lemma can be shown following the lines of the proof of [13, The-orem 6.1], see also [7, Theorem 3.1]. Lemma 1.
There is a constant K depending only on Ω , the choice of Ω j , j = , . . . , n, and m ≥ , such that for all j and u ∈ H m ( Ω ) ∩ H ( Ω ) , | u / q j | H m − ( Ω j ) ≤ K k u k H m ( Ω j ) . (3)Given a a unit vector τ = ( τ x , τ y ) in the plane, we denote by D τ the directionalderivative operator in the direction of τ in the plane, so that D τ f : = τ x D x f + τ y D y f , D x f : = ∂ f / ∂ x , D y f : = ∂ f / ∂ y . Given f ∈ C α + β ( △ ) , α , β ≥
0, any number η f = D ατ D βτ ( f | T )( z ) , where T ∈ △ , z ∈ T , and τ , τ are some unit vectors in the plane, is said to be a nodal value of f , and the linear functional η : C α + β ( △ ) → R is a nodal functional ,with d ( η ) : = α + β being the degree of η .For some special choices of z , τ , τ , we use the following notation: • If v is a vertex of △ and e is an edge attached to v , we set D α e f ( v ) : = D ατ ( f | T )( v ) , α ≥ , where τ is the unit vector in the direction of e away from v , and T ∈ △ is one ofthe triangles with edge e . • If v is a vertex of △ and e , e are two consecutive edges attached to v , we set D α e D β e f ( v ) : = D ατ D βτ ( f | T )( v ) , α , β ≥ , where T ∈ △ is the triangle with vertex v and edges e , e , and τ i is the unit vectorin the e i direction away from v . • For every edge e of the triangulation △ we choose a unit vector τ ⊥ (one of twopossible) orthogonal to e and set D α e ⊥ f ( z ) : = D ατ ⊥ f ( z ) , z ∈ e , α ≥ , Oleg Davydov and Wee Ping Yeo provided f ∈ C α ( z ) .On every edge e of △ , with vertices v ′ and v ′′ , we define three points on e by z je : = v ′ + j ( v ′′ − v ′ ) , j = , , . For every triangle T ∈ △ with vertices v , v , v and edges e , e , e , we define N T to be the set of nodal functionals corresponding to the nodal values D α x D β y f ( v i ) , ≤ α + β ≤ , i = , , , D e ⊥ i f ( z e i ) , i = , , , see Figure 3 (left), where the nodal functionals are depicted in the usual way bydots, segments and circles as for example in [5].Let T ∈ △ P . We define N PT to be the set of nodal functionals corresponding tothe nodal values D α x D β y f ( v ) , ≤ α + β ≤ , D α x D β y f ( v i ) , ≤ α + β ≤ , i = , , D α x D β y f ( c T ) , ≤ α + β ≤ , where v the interior vertex of T , v , v are boundary vertices, and c T is the centerof the disk B T , see Figure 4.Let T ∈ △ B with vertices v , v , v . We define N B , T to be the set of nodal func-tionals corresponding to the nodal value f ( c T ) , c T : = ( v + v + v ) / . Also we define N B , T to be the set of nodal functionals corresponding to the nodalvalues f ( z e i ) , i = , , , D α x D β y f ( v i ) , ≤ α + β ≤ , i = , , , D e ⊥ i f ( z je i ) , j = , , i = , , , where v is the boundary vertex and v , v are the interior vertices of T . We set N BT : = N B , T ∪ N B , T , see Figure 3 (right).We define an operator I △ : H ( Ω ) ∩ H ( Ω ) → S , , ( △ ) of interpolatory type. Let u ∈ H ( Ω ) ∩ H ( Ω ) . By Sobolev embedding we assume without loss of generalitythat u ∈ C ( Ω ) . For all T ∈ △ ∪△ P we set I △ u | T = I T ( u | T ) , with the local operators I T defined as follows. pproximation by C Splines on Piecewise Conic Domains 7PSfrag replacements v v v PSfrag replacements v v v v v v Fig. 3
Nodal functionals corresponding to N T (left) and N BT (right).PSfrag replacements v v v PSfrag replacements v v v v v v Fig. 4
Nodal functionals corresponding to N PT . If T ∈ △ , then p : = I T u is the polynomial of degree 5 that satisfies the followinginterpolation conditions: η p = η u , for all η ∈ N T . This is a well-known Argyris interpolation scheme, see e.g. [15, Section 6.1], whichguarantees the existence and uniqueness of the polynomial p .Let T ∈ △ P with the curved edge on Γ j . Then I T u : = pq j , where p ∈ P satisfiesthe following interpolation condition: η p = η ( u / q j ) , for all η ∈ N PT . (4)The nodal functionals in N PT are well defined for u / q j even though the vertices v , v of T lie on the boundary Γ j because u / q j ∈ H ( Ω j ) by Lemma 1 and hence u / q j may be identified with a function ˜ u ∈ C ( Ω j ) by Sobolev embedding. Theinterpolation scheme (4) defines a unique polynomial p ∈ P , which will be justifiedin the proof of Lemma 3. In addition, we will need the following statement. Lemma 2.
The polynomial p defined by (4) satisfiesD α x D β y ( pq j )( v ) = D α x D β y u ( v ) , ≤ α + β ≤ , Oleg Davydov and Wee Ping Yeo where v is any vertex of the pie-shaped triangle T .Proof.
By (4), p ( v ) q j ( v ) = ˜ u ( v ) q j ( v ) = u ( v ) , where ˜ u ∈ C ( Ω j ) is the above func-tion satisfying u = ˜ uq j . Moreover, ∇ ( pq j )( v ) = ∇ p ( v ) q j ( v ) + p ( v ) ∇ q j ( v )= ∇ ˜ u ( v ) q j ( v ) + ˜ u ( v ) ∇ q j ( v )= ∇ ( ˜ uq j )( v ) = ∇ u ( v ) . Similarly, if v is the interior vertex of T , then ∇ ( pq j )( v ) = ∇ p ( v ) q j ( v ) + ∇ p ( v )( ∇ q j ( v )) T + ∇ q j ( v )( ∇ p ( v )) T + p ( v ) ∇ q j ( v )= ∇ ˜ u ( v ) q j ( v ) + ∇ ˜ u ( v )( ∇ q j ( v )) T + ∇ q j ( v )( ∇ ˜ u ( v )) T + ˜ u ( v ) ∇ q j ( v )= ∇ u ( v ) . If v is one of the boundary vertices, then q j ( v ) =
0, and hence ∇ ( pq j )( v ) = ∇ p ( v )( ∇ q j ( v )) T + ∇ q j ( v )( ∇ p ( v )) T + p ( v ) ∇ q j ( v )= ∇ ˜ u ( v )( ∇ q j ( v )) T + ∇ q j ( v )( ∇ ˜ u ( v )) T + ˜ u ( v ) ∇ q j ( v )= ∇ u ( v ) . ⊓⊔ It is easy to deduce from Lemma 2 that the interpolation conditions for p at theboundary vertices v , v of T can be equivalently formulated as follows: For i = , p ( v i ) = ∂ u ∂ n i ( v i ) . ∂ q j ∂ n i ( v i ) , ∂ p ∂ n i ( v i ) = ∂ u ∂ n i ( v i ) . ∂ q j ∂ n i ( v i ) , ∂ p ∂τ i ( v i ) = ∂ u ∂ n i ∂τ ( v i ) . ∂ q j ∂ n i ( v i ) , (5)where n i and τ i are the normal and the tangent unit vectors to the curve q j ( x ) = v i . Finally, assume that T ∈ △ B with vertices v , v , v where v is a boundary vertex.Then I T u = p ∈ P satisfies the following interpolation conditions: η p = η u , for all η ∈ N B , T , and η p = η I T i u , for all η ∈ N i ⊂ N B , T , i = , , , where T is a triangle in △ sharing an edge e = h v , v i with T and N correspondsto the nodal values f ( z e ) , D e ⊥ f ( z ie ) , i = , , D α x D β y f ( v i ) , ≤ α + β ≤ , i = , pproximation by C Splines on Piecewise Conic Domains 9 T is a triangle in △ P sharing an edge e = h v , v i with T and N corresponds tothe nodal values f ( z e ) , D e ⊥ f ( z ie ) , i = , , D α x D β y f ( v ) , ≤ α + β ≤ T is a triangle in △ P sharing an edge e = h v , v i with T and N correspondsto the nodal values f ( z e ) , D e ⊥ f ( z ie ) , i = , . Since N B , T = N ∪ N ∪ N and N BT = N B , T ∪ N B , T is a well posed interpolationscheme [16] for polynomials of degree 6, it follows that p is uniquely defined by theabove conditions. Theorem 1.
Let u ∈ H ( Ω ) ∩ H ( Ω ) . Then I △ u ∈ S , , ( △ ) .Proof. By the above construction I △ u is a piecewise polynomial of degree 5 on alltriangles in △ and degree 6 on the triangles in △ P ∪ △ B . Moreover, I △ u vanisheson the boundary of Ω .To see that I △ u ∈ S , , ( △ ) we thus need to show the C continuity of I △ u acrossall interior edges of △ . If e is a common edge of two triangles T ′ , T ′′ ∈ △ , then the C continuity follows from the standard argument for C Argyris finite element, see[4, Chapter 3] and [15, Section 6.1].Next we will show the C continuity of I △ u across edges shared by buffer tri-angles with either ordinary or pie-shaped triangles. Let T ∈ △ B and T ′ ∈ △ ∪ △ P with common edge e ′ = h v ′ , v ′′ i , and let p = I T u and s = I T ′ u . Consider the uni-variate polynomials p | e ′ and s | e ′ and let q = p | e ′ − s | e ′ . Assuming that the edge e ′ is parameterized by t ∈ [ , ] , Then q is a univariate polynomial of degree 6 withrespect to the parameterization v ′ + t ( v ′′ − v ′ ) , t ∈ [ , ] . Similarly, we consider theorthogonal/normal derivatives D e ′⊥ p | e ′ and D e ′⊥ s | e ′ and let r = D e ′⊥ p | e ′ − D e ′⊥ s | e ′ ,then r is a univariate polynomial of degree 5 with respect to the same parameter t .The C continuity will follow if we show that both q and r are zero functions.If T ′ = T ∈ △ , then using the interpolation conditions corresponding to N ⊂ N B , T , we have q ( ) = q ′ ( ) = q ′′ ( ) = q ( / ) = q ( ) = q ′ ( ) = q ′′ ( ) = , r ( ) = r ′ ( ) = r ( / ) = r ( / ) = r ( ) = r ′ ( ) = , which implies q ≡ r ≡ T ′ = T ∈ △ P , then the interpolation conditions corresponding to N ⊂ N B , T imply q ( ) = q ′ ( ) = q ′′ ( ) = q ( / ) = , r ( ) = r ′ ( ) = r ( / ) = r ( / ) = , In view of Lemma 2, we have D α x D β y s ( v ) = D α x D β y u ( v ) = D α x D β y p ( v ) , ≤ α + β ≤ , which implies q ( ) = q ′ ( ) = q ′′ ( ) = , r ( ) = r ′ ( ) = , and hence q ≡ r ≡ T ′ = T ∈ △ P , then the interpolation conditions corresponding to N ⊂ N B , T imply q ( / ) = , r ( / ) = r ( / ) = , whereas Lemma 2 gives q ( ) = q ′ ( ) = q ′′ ( ) = , r ( ) = r ′ ( ) = , q ( ) = q ′ ( ) = q ′′ ( ) = , r ( ) = r ′ ( ) = , which completes the proof. ⊓⊔ In follows from Lemma 2 that I △ u is twice differentiable at the boundary vertices,and thus I △ u ∈ { s ∈ S ( △ ) : s is twice differentiable at any vertex and s | Γ = } . Moreover, I △ u satisfies the following interpolation conditions: D α x D β y I △ u ( v ) = D α x D β y u ( v ) , ≤ α + β ≤ , for all v ∈ V , D e ⊥ I △ u ( z e ) = D e ⊥ u ( z e ) , for all edges e of △ , D α x D β y I △ u ( c T ) = D α x D β y u ( c T ) , ≤ α + β ≤ , for all T ∈ △ P , I △ u ( c T ) = u ( c T ) , for all T ∈ △ B , where c T denotes the center of the disk B T inscribed into T ∗ if T is a pie-shapedtriangle, and the barycenter of T if T is a buffer triangle. In view of (5), I △ u ∈ S , , ( △ ) is uniquely defined by these conditions for any u ∈ C ( Ω ) . In this section we estimate the error k u − I △ u k H k ( Ω ) for functions u ∈ H m ( Ω ) ∩ H ( Ω ) , m = ,
6. Similar to [7, Section 3], we follow the standard finite elementtechniques involving the Bramble-Hilbert Lemma (see [4, Chapter 4]) on the ordi-nary triangles, and make use of the estimate (3) on the pie-shaped triangles. Sincethe spline I △ u on the buffer triangles is constructed in part by interpolation and inpart by the smoothness conditions, the estimate of the error on such triangles relies pproximation by C Splines on Piecewise Conic Domains 11 in particular on the estimates of the interpolation error on the neighboring ordinaryand buffer triangles.
Lemma 3.
If p ∈ P and T ∈ △ P , then k p | T ∗ k L ∞ ( T ∗ ) ≤ max η ∈ N PT h d ( η ) T ∗ | η p | , (6) where T ∗ is the triangle obtained by replacing the curved edge of T by the straightline segment, and h T ∗ is the diameter of T ∗ . Similarly, if p ∈ P and T ∈ △ B , then k p | T k L ∞ ( T ) ≤ max η ∈ N BT h d ( η ) T | η p | , (7) where h T is the diameter of T .Proof. To show the estimate (6) for T ∗ , we follow the proof of [8, Lemma 3.9].We note that we only need to show that the interpolation scheme for pie-shapedtriangles is a valid scheme, that is, we need to show that N PT is P -unisolvent, andthe rest of the proof can be done similar to that of [8, Lemma 3.9]. Recall that a set offunctionals N is said to be P d - unisolvent if the only polynomial p ∈ P d satisfying η p = η ∈ N is the zero function.Let T ∗ = h v , v , v i , where v is the interior vertex. Set e : = h v , v i , e : = h v , v i , e : = h v , v i , see Figure 4. The interpolation conditions along e , e im-ply that s vanishes on these edges. After splitting out the linear polynomials fac-tors which vanish along the edges e , e we obtain a valid interpolation scheme forquadratic polynomials with function values at the three vertices, and function andgradient values at the the barycenter c of B T ⊂ T ∗ . The validity of this scheme canbe seen by looking at a straight line ℓ through c and any one of the vertices of T ∗ .Along the line ℓ , a function value is given at the vertex and a function value togetherwith the first derivative are given at the point c , and this set of data is unisolvent forthe univariate quadratic polynomials, which means s must vanish along ℓ . After fac-toring out the respective linear polynomial, we are left with function values at threenon-collinear points, which defines a valid interpolation scheme for the remaininglinear polynomial factor of s .To show the estimate (7) for T ∈ △ B , the proof is similar. We need to show theset of functionals N BT is P -unisolvent but this follows from the standard schemeof [16] for polynomials of degree six.We note that the argument of the proof of [8, Lemma 3.9] applies to affine invari-ant interpolation schemes, that is the schemes that use the edge derivatives. As ourscheme relies on the standard derivatives in the direction of the x , y axes, we need toexpress the edge derivatives as linear combinations of the x , y derivatives as follows.Assume that e , e are two edges that emanate from a vertex v . Let τ i = ( τ i , τ i ) bethe unit vector in the direction of e i away from v , i = ,
2. Then we can easily obtainthe following identities D e i f ( v ) = τ i D x f ( v ) + τ i D y f ( v ) , D e i f ( v ) = τ i D x f ( v ) + τ i τ i D x D y f ( v ) + τ i D y f ( v ) , D e D e f ( v ) = τ τ D x f ( v ) + ( τ τ + τ τ ) D x D y f ( v ) + τ τ D y f ( v ) . ⊓⊔ Lemma 4.
Let T ∈ △ P and its curved edge e ⊂ Γ j . Then k I T u k L ∞ ( T ) ≤ C max ≤ ℓ ≤ h ℓ + T | u / q j | W ℓ ∞ ( T ) if u ∈ H ( Ω ) ∩ H ( Ω ) , (8) where C depends only on h T / ρ T . Moreover, if ≤ m ≤ , then for any u ∈ H m ( Ω ) ∩ H ( Ω ) , k u − I T u k H k ( T ) ≤ C h m − kT | u / q j | H m − ( T ) , k = , . . . , m − , (9) | u − I T u | W k ∞ ( T ) ≤ C h m − k − T | u / q j | H m − ( T ) , k = , . . . , m − , (10) where C , C depend only on h T / ρ T .Proof. We will denote by ˜ C constants which may depend only on h T / ρ T and on Ω . Assume that u ∈ H ( Ω ) ∩ H ( Ω ) and recall that by definition I T u = pq j , where p ∈ P satisfies the interpolation conditions (4). Since u ∈ H ( Ω j ) ∩ H ( Ω j ) , itfollows that u / q j ∈ H ( Ω j ) by Lemma 1, and hence u / q j ∈ C ( Ω j ) by Sobolevembedding. From Lemma 3 we have k p k L ∞ ( T ∗ ) ≤ max η ∈ N PT h d ( η ) T ∗ | η p | , (11)and hence k p k L ∞ ( T ∗ ) ≤ max η ∈ N PT h d ( η ) T ∗ | η ( u / q j ) | ≤ ˜ C max ≤ ℓ ≤ h ℓ T | u / q j | W ℓ ∞ ( T ) . As in the proof of [7, Theorem 3.2], we can show that for any polynomial of degreeat most 6, k s k L ∞ ( T ) ≤ ˜ C k s k L ∞ ( T ∗ ) and k s k L ∞ ( T ∗ ) ≤ ˜ C k s k L ∞ ( T ) . (12)By using (f) it is easy to show that k q j k L ∞ ( T ) ≤ h T , and hence k I T u k L ∞ ( T ) = k pq j k L ∞ ( T ) ≤ h T k p k L ∞ ( T ) , which completes the proof of (8).Moreover, since the area of T is less than or equal π h T and ∂ α ( I T u ) ∈ P − k if | α | = k , it follows that k ∂ α ( I T u ) k L ( T ) ≤ √ π h T k ∂ α ( I T u ) k L ∞ ( T ) ≤ ˜ Ch T k ∂ α ( I T u ) k L ∞ ( T ∗ ) . By Markov inequality (see e.g. [15, Theorem 1.2]) we get furthermore k ∂ α ( I T u ) k L ∞ ( T ∗ ) ≤ ˜ C ρ − kT k I T u k L ∞ ( T ∗ ) , pproximation by C Splines on Piecewise Conic Domains 13 and hence in view of (12) | I T u | H k ( T ) ≤ ˜ Ch − kT k I T u k L ∞ ( T ) . In view of (8) we arrive at | I T u | H k ( T ) ≤ ˜ C max ≤ ℓ ≤ h ℓ + − kT | u / q j | W ℓ ∞ ( T ) , if u ∈ H ( Ω ) ∩ H ( Ω ) . (13)Let m ∈ { , } , and let u ∈ H m ( Ω ) ∩ H ( Ω ) . It follows from Lemma 1 that u / q j ∈ H m − ( T ) . By the results in [4, Chapter 4] there exists a polynomial ˜ p ∈ P m − suchthat k u / q j − ˜ p k H k ( T ) ≤ ˜ Ch m − k − T | u / q j | H m − ( T ) , k = , . . . , m − , | u / q j − ˜ p | W k ∞ ( T ) ≤ ˜ Ch m − k − T | u / q j | H m − ( T ) , k = , . . . , m − . (14)Indeed, a suitable ˜ p is given by the averaged Taylor polynomial [4, Definition 4.1.3]with respect to the disk B T , and the inequalities in (14) follow from [4, Lemma 4.3.8](Bramble-Hilbert Lemma) and an obvious generalization of [4, Proposition 4.3.2],respectively. It is easy to check by inspecting the proofs in [4] that the quotient h T / ρ T can be used in the estimates instead of the chunkiness parameter used there.Since u − I T u = ( u / q j − ˜ p ) q j − I T ( u − ˜ pq j ) , we have for any norm k · k , k u − I T u k ≤ k ( u / q j − ˜ p ) q j k + k I T ( u − ˜ pq j ) k . In view of (f) and (14), for any k = , . . . , m − | ( u / q j − ˜ p ) q j | W k ∞ ( T ) ≤ h T | u / q j − ˜ p | W k ∞ ( T ) + k u / q j − ˜ p k W k − ∞ ( T ) ≤ ˜ Ch m − k − T | u / q j | H m − ( T ) , and for any k = , . . . , m − k ( u / q j − ˜ p ) q j k H k ( T ) ≤ ˜ Ch T k u / q j − ˜ p k H k ( T ) + ˜ C k u / q j − ˜ p k H k − ( T ) ≤ ˜ Ch m − kT | u / q j | H m − ( T ) . Furthermore, by the Markov inequality, (8), (13) and (14), | I T ( u − ˜ pq j ) | W k ∞ ( T ) ≤ ˜ C max ≤ ℓ ≤ h ℓ + − kT | u / q j − ˜ p | W ℓ ∞ ( T ) ≤ ˜ Ch m − k − T | u / q j | H m − ( T ) , k I T ( u − ˜ pq j ) k H k ( T ) ≤ ˜ C max ≤ ℓ ≤ h ℓ + − kT | u / q j − ˜ p | W ℓ ∞ ( T ) ≤ ˜ Ch m − kT | u / q j | H m − ( T ) . By combining the inequalities in the five last displays we deduce (9) and (10). ⊓⊔ We are ready to formulate and prove our main result.
Theorem 2.
Let ≤ m ≤ . For any u ∈ H m ( Ω ) ∩ H ( Ω ) , (cid:16) ∑ T ∈△ k u − I △ u k H k ( T ) (cid:17) / ≤ Ch m − k k u k H m ( Ω ) , k = , . . . , m − , (15) where h is the maximum diameter of the triangles in △ , and C is a constant depend-ing only on Ω , the choice of Ω j , and the shape regularity constant R of △ .Proof. We estimate the norms of u − I T u on all triangles T ∈ △ . The letter C standsbelow for various constants depending only on the parameters mentioned in theformulation of the theorem.If T ∈ △ , then s | T is a macro element as defined in [15, Chapter 6]. Furthermore,by [15, Theorem 6.3] the set of linear functionals N T give rise to a stable localnodal basis, which is in particular uniformly bounded. Hence by [12, Theorem 2]we obtain a Jackson estimate in the form k u − I T u k H k ( T ) ≤ Ch m − kT | u | H m ( T ) , k = , . . . , m , (16)where C depends only on h T / ρ T . If T ∈ △ P , with the curved edge e ⊂ Γ j , then theJackson estimate (9) holds by Lemma 4.Let T ∈ △ B , p : = I △ u | T and let ˜ p ∈ P be the interpolation polynomial thatsatisfies η ˜ p = η u for all η ∈ N BT . Then η ( ˜ p − p ) = ( η ∈ N B , T , η ( u − I T ′ u ) if η ∈ N B , T , where T ′ = T ′ η ∈ △ ∪ △ P . Hence, by Markov inequality and (7) of Lemma 3, weconclude that for k = , . . . , m , k ˜ p − p k H k ( T ) ≤ Ch − kT k ˜ p − p k L ∞ ( T ) , with k ˜ p − p k L ∞ ( T ) ≤ C max { h ℓ T | u − I T ′ u | W ℓ ∞ ( T ′ ) : 0 ≤ ℓ ≤ , T ′ ∈ △ ∪ △ P , T ′ ∩ T = /0 } , whereas by the same arguments leading to (16) we have k u − ˜ p k H k ( T ) ≤ Ch m − kT | u | H m ( T ) , with the constants depending only on h T / ρ T . If T ′ ∈ △ ∪ △ P , then by (10) andthe analogous estimate for T ′ ∈ △ , compare [4, Corollary 4.4.7], we have for ℓ = , , | u − I T ′ u | W ℓ ∞ ( T ′ ) ≤ Ch m − ℓ − T ′ ( | u | H m ( T ′ ) if T ′ ∈ △ , | u / q j | H m − ( T ′ ) if T ′ ∈ △ P , where C depends only on h T ′ / ρ T ′ . By combining these inequalities we obtain anestimate of k u − I T u k H k ( T ) by C ˜ h m − k times the maximum of | u | H m ( T ) , | u | H m ( T ′ ) for pproximation by C Splines on Piecewise Conic Domains 15 T ′ ∈ △ sharing edges with T , and | u / q j | H m − ( T ′ ) for T ′ ∈ △ P sharing edges with T .Here C depends only on the maximum of h T / ρ T and h T ′ / ρ T ′ , and ˜ h is the maximumof h T and all h T ′ for T ′ ∈ △ ∪ △ P sharing edges with T .By using (16) on T ∈ △ , (9) on T ∈ △ P and the estimate of the last paragraphon T ∈ △ B , we get ∑ T ∈△ k u − I △ u k H k ( T ) ≤ Ch ( m − k ) (cid:16) ∑ T ∈△ ∪△ B | u | H m ( T ) + ∑ T ∈△ P | u / q j ( T ) | H m − ( T ) (cid:17) , where j ( T ) is the index of Γ j containing the curved edge of T ∈ △ P . Clearly, ∑ T ∈△ ∪△ B | u | H m ( T ) ≤ | u | H m ( Ω ) ≤ k u k H m ( Ω ) , whereas by Lemma 1, ∑ T ∈△ P | u / q j ( T ) | H m − ( T ) ≤ n ∑ j = | u / q j | H m − ( Ω j ) ≤ K k u k H m ( Ω ) , where K is the constant of (3) depending only on Ω and the choice of Ω j . ⊓⊔ Acknowledgements
This research has been supported in part by the grant UBD/PNC2/2/RG/1(301) fromUniversiti Brunei Darussalam.
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