A Novel Finite Element Discretization of Domains with Spheroidal. Geometry

Transcription

1 A Novel Finite Element Discretization of Domains with Spheroidal Geometry Except where reference is made to the work of others, the work described in this dissertation is my own or was done in collaboration with my advisory committee. This dissertation does not include proprietary or classified information. Certificate of Approval: Necibe Tuncer Paul Schmidt Professor Mathematics and Statistics Amnon J. Meir, Chair Professor Mathematics and Statistics Georg Hetzer Professor Mathematics and Statistics Richard Zalik Professor Mathematics and Statistics George T. Flowers Interim Dean Graduate School

2 A Novel Finite Element Discretization of Domains with Spheroidal Geometry Necibe Tuncer A Dissertation Submitted to the Graduate Faculty of Auburn University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Auburn, Alabama May 0, 2007

3 A Novel Finite Element Discretization of Domains with Spheroidal Geometry Necibe Tuncer Permission is granted to Auburn University to make copies of this dissertation at its discretion, upon the request of individuals or institutions and at their expense. The author reserves all publication rights. Signature of Author Date of Graduation iii

4 Dissertation Abstract A Novel Finite Element Discretization of Domains with Spheroidal Geometry Necibe Tuncer Doctor of Philosophy, May 0, 2007 (M.A.,Dokuz Eylul University, 200) (B.S., Dokuz Eylul University, 999) 66 Typed Pages Directed by Amnon J. Meir We describe and analyze a new finite element discretizations for domains with spheroidal geometry. In particular, we describe how the method can be used to approximate solutions as well as eigenvalues and eigenfunctions of partial differential equations posed on the sphere, ellipsoidal shell, and cylindrical shell. These novel, so-called, radially projected finite elements are particularly attractive for numerical simulations since the resulting finite element discretization is logically rectangular and may be easily implemented or incorporated into existing finite element codes. iv

5 Style manual or journal used Journal of Approximation Theory (together with the style known as aums ). Bibliograpy follows van Leunen s A Handbook for Scholars. Computer software used The document preparation package TEX (specifically L A TEX) together with the departmental style-file aums.sty. v

6 Table of Contents List of Figures vii Introduction. Overview Background Our Contribution The Finite Element Method Discretization of the Domain and Basis Elements Error Estimates Smooth Manifolds Eigenvalue Problem The Radial Projection Properties of the Radial Projection Some Examples of Mapping Mesh on the Sphere Meshes on Ellipsoids Mesh on the Cylinder Mesh on the Disc Analysis Finite Element Construction Error Analysis Numerical Experimetns Finite elements on the sphere Example Example Example Conclusion 57 Bibliography 58 vi

7 List of Figures. A uniform triangulation of the square Linear basis element A transition map Mesh on the box and the sphere Faces of the box F F + and P (F + ) Geodesic distance between a and b Geodesic and Euclidean distance of a and b Mesh on the ellipsoid Mesh on the cylinder Mesh on the disc Spherical triangle, K and planar triangle K h Basis functions on the sphere Numerical approximation to µ(a) = cosa vii

8 Chapter Introduction. Overview Mathematical models of many engineering and scientific problems usually consist of partial differential equations and some given conditions. The solution of such problems can be approximated by numerical solution techniques. Scientific computing gained significant importance in studies of physical problems. The Finite Element Method (FEM) is a numerical approach to approximate the solution of a partial differential equation. Compared to the Finite Difference Method, the FEM is a fairly new method. FEM has become a favorite method immediately, since it is very easy to use even for complex domains, for general boundary value problems, and also has a strong theoretical basis which yields easy derivation of error estimates. The FEM is very nicely structured so that a general computer program can be used to approximate solutions of various problems easily. Despite all the advantages of the FEM, it is mainly convenient for domains that have a polyhedral geometry. Thus, it has mainly been applied to the polygonal and polyhedral domains. Many studies have been performed for FEM disretization of curved domains or domains with spherical geometry. However, the results of these studies have focused on approximating the domain by a polygonal domain, rather than applying FEM to these exact domains. Recently, there has been growing interest in finite element discretization of partial differential equations defined on the sphere, since these equations have many applications in areas such as climate modeling or weather forecast and various engineering problems.

9 .. Background Mathematical weather models or climate models consist of partial differential equations posed on a domain which is assumed to be a sphere for the obvious reason. Lately, the study of numerically approximating the solution of partial differential equations defined on the sphere has become a favorite subject. However, this is a challenging problem no matter which numerical method is being applied to approximate the solutions of the equations because of the geometry of the sphere. Finite difference methods, spectral methods, finite element methods, finite volume methods are all numerical techniques that have been applied to approximate the solutions of the partial differential equations on the sphere. Grid generation is an important part of approximating solutions of partial differential equations, since the accuracy of numerical solutions depend on the quality of the grid. Various researchers have used different methods for mesh generations on the sphere, such as, constructing the finite element basis directly on spherical triangles by using barycentric coordinate systems [0], approximating, not just the sphere, but any surface with a polyhedral surface, [9] also [6] and [7]. Both of these approaches, described in [0],[9], yield the optimal convergence rate (which is 2 if linear finite elements are used to approximate the solution, the error is measured in the L 2 norm, and the solution is smooth enough). Although all these approaches result in a nice mesh, they suffer from some weaknesses, such as not discretizing the sphere exactly or requiring multiple computations for each refinement step. Our work aims to overcome these difficulties by developing a new method that discretizes the sphere exactly and that can also be easily implemented and incorporated into existing finite element codes. 2

10 ..2 Our Contribution Our work is motivated by a desire for exact finite element discretization of the sphere which is yielding a conforming finite element method. We develop and analyze a method which can also be easily implemented and incorporated into an existing code. In our approach, we disceretize the sphere exactly, rather than approximate it, which has not been done before. The method is also very easy to implement if one knows how to implement finite element methods on planar domains. It requires very little computational effort since all computations are done on a logically rectangular domain. Our primary objective is to develop a finite element discretization of a sphere or a ball. Our method can easily be generalized to other domains, such as cylinders, ellipsoids or tori, and can be used for problems posed on domains with spherical, cylindrical, or toroidal holes. In this work we present our approach for the discretization of the sphere, and the same method can be also used for cylinders, ellipsoids and tori. Basically, one can apply this discretization to any domain that has a spherical, cylindrical, ellipsoidal, or toroidal geometry, which we call spheroidal geometry. We develop and analyze a novel finite element discretization of domains with spheroidal geometry.we present some computational examples related to some of these domains. We first give an introduction to finite element method in section.2, introduce smooth manifolds and Laplace Beltrami operator in section.3. In chapter 2 we introduce our new method, in chapter 3 we analyze the method, derive error estimates, and in chapter 4 we present numerical results from computer experiments. 3

11 .2 The Finite Element Method In this section, we introduce the finite element method and illustrate it with the classical Laplace s equation on a bounded, connected, open domain Ω in R n. Consider the following boundary value problem, u = f in Ω (.) u = 0 on Ω (.2) which is also known as Poisson s equation. In (.), is the Laplacian, Ω is the boundary of the domain Ω, and u is the unknown function. The right hand side f is a real valued function defined on the open subset Ω of R n, we say that f C r (Ω), if all the k th order (classical) partial derivatives exist and are continuous for k r, where r, k are positive integers. We call any function u C 2 (Ω) that satisfies the differential equation and the boundary condition a classical solution. Consider the Lebesgue measure of R n, let Ω be a Lebesgue measurable domain, and let f be a Lebesgue-integrable function, then we denote the Lebesgue integral by fdω Ω where dω denotes the Lebesgue measure. We say that f is in L p (Ω), for p < if ( /p f dω) p <. The set, L p (Ω) is a Banach space, with the following norm, Ω ( /p f L p (Ω) := f dω) p. Ω 4

12 The Sobolev spaces W m,p (Ω) are defined to be the set of all functions f L p (Ω) that have weak derivatives in L p (Ω) up to order m. W m,p (Ω) := {f L p (Ω) : D α f L p (Ω) α m}, where D α is a multi index notation for partial derivatives (in the weak sense), α = (α,α 2,...,α n ) is an n-tuple of non-negative integers, and the length of α is denoted by α = α + α α n. The Sobolev space W m,p (Ω) equipped with the norm f W m,p (Ω) := D α f p L p (Ω) α m /p, is a Banach Space. We denote by W m,p 0 (Ω) the closure of C m c (Ω) in W m,p (Ω). We denote by C m c (Ω) the space of functions u Cm (Ω) that have compact support. Also, we denote by H m (Ω), the spaces H m (Ω) = W m,2 (Ω), these are the Hilbert spaces. Note that H 0 (Ω) is the subspace of H (Ω) and consists of the functions that are zero on Ω in the sense of traces. A variational formulation of problem (.), can be derived by multiplying the both sides of (.) by functions v in H0 (Ω). Then by integration by parts we get, Ω u vdω = Ω f vdω. 5

13 Now the solution u is required to only be in H0 (Ω), which means we look for a solution in a bigger set. In this case we say that u is a weak solution of (.). The new problem: Find u H 0 (Ω) such that Ω u vdω = Ω f vdω v H0 (Ω) (.3) This is called a weak or variational formulation of (.). We define the bilinear form a(u,v) : H0 (Ω) H 0 (Ω) R as a(u,v) := u vdω, which is also known as the energy Ω inner product. Similarly, (f, v) := f vdω, is the standard inner product in L 2 (Ω). We Ω say that a bilinear form a(u,v) : H0 (Ω) H 0 (Ω) R is elliptic if there exists α > 0 such that a(v,v) α v 2 H (Ω) v H 0 (Ω). We say that the the bilinear form is continuous or bounded if there exists a constant C > 0 such that a(u,v) C u H 0 (Ω) v H 0 (Ω). Clearly, a(, ) induces a norm, which is equivalent to the norm of the Hilbert space H 0 (Ω). This norm v a = a(v,v) is called the energy norm. The question we need to answer is the relationship between (.) and its variational problem (.3). Let f L 2 (Ω) and u H 2 (Ω) then u satisfies (.3) if and only if u is a solution of (.). Within the theory of functional analysis, one can prove the following theorem, which is known as Riesz Representation theorem in the literature [2]. 6

14 Theorem. Let H be a Hilbert space and let L be a continuous linear functional L : H R, s.t. L(u) = (u,v) u H then L is an isomorphism. Theorem.2 There exists a unique solution to (.3). Proof: The bilinear form a(, ) is a bounded, elliptic such that (H0 (Ω),a(, ))is a Hilbert space. Then the theorem follows as a consequence of Riesz representation theorem. To approximate the solution of (.3), we construct the discrete variational problem in a finite dimensional subspace of H 0 (Ω). Let χ be the finite dimensional subspace of H 0 (Ω), then the discrete analog of problem (.3) is : Find u h χ such that u h v h dω = Ω Ω f v h dω v h χ. (.4) It is worth noting that by (.4), we construct a discrete scheme for approximating the solution of (.). Also, at this point we need to remark that (.4) has a unique solution, since χ together with the bilinear form, a h (u h,v h ) := Ω u h v h dω is a Hilbert space..2. Discretization of the Domain and Basis Elements Finite element method can be used to solve the discrete problem (.4). One important step is to define, χ, the finite dimensional subspace of H0 (Ω). We define χ by first partitioning the domain. For simplicity, hereinafter let Ω be a convex polygonal domain in R 2. Let K, K 2,...,K n be triangles or quadrilaterals such that they form a partition of Ω, i.e n Ω = K i and K i Kj =. i= 7

15 For any intersection of two triangles or quadrilaterals K i and K j, if the intersection consists of one point then it is a common vertex, if the intersection consists of more than one point then it is a common edge. Let T j = {K j, Kj 2,...,Kj n j },j N denote the family of triangulation where j denotes the refinement step. We say that the mesh size of the triangulation T at the j th refinement step is h j where h j = max h K j K j i T j i, and h K j i = diam(k j i ), here diam(k j i ) = sup x y. x,y K j i The quantity h j is a measure of how refined the mesh is. The smaller h j is, the finer the mesh. Let ρ K j i denote the radius of inscribed circle in K j i. We say that a family of triangulation T j, j N is shape regular if there exists a constant κ > 0, independent of j such that for each K j i T j, we have h K j i ρ K j i κ. Figure. shows a uniform triangulation of the square. Usually we choose the finite dimensional subspace to be the space of piecewise polynomials on the domain Ω. Piecewise linear or quadratic functions are widely used in defining these subspaces. Figure.2 shows the linear basis element. For instance, define χ to be the space of piecewise linear polynomials, that is: χ = {ω : ω is piecewise linear continuous polynomials and ω = 0 on Ω}. 8

16 Figure.: A uniform triangulation of the square Let {ω i } n i be a basis for χ, then for any u h χ we have u h = reduces to a linear system of equations of the following form n u i ω i. Thus, (.4) i Au = f where A is a n n matrix whose entries are A ij = a h (ω j,ω i ), and u and f are n vectors of the form u(i) = u i, f(i) = (f,ω i ). Since the bilinear form a h (, ) is symmetric and elliptic, the matrix A is symmetric and positive definite, which assures the existence and uniqueness of a solution of (.4). The matrix A is also a sparse matrix, if the basis functions are chosen to have compact support. In the following section we talk about the error estimates in the finite element method. 9

17 Figure.2: Linear basis element.2.2 Error Estimates Let u h be the solution of (.4) and u be the exact solution of (.3), we first observe the orthogonality between u u h and u h in terms of the energy inner product. Simply, subtracting, (.4) from (.3), we get, a(u u h,v h ) = 0, v h χ. (.5) The following lemma, which is known as Cea s lemma, plays a significant role in establishing error estimates in finite element method. Lemma. (Cea s Lemma) Let u h be the solution of (.4) and u be the exact solution of (.3), then u u h H (Ω) C α inf v h χ u v h H (Ω). 0

18 Proof: The bilinear form is elliptic, a(u u h,u u h ) α u u h 2 H (Ω) and a(u u h,u u h ) = a(u u h,u v h ), since a(u u h,u h v h ) = 0 from (.5) and the bilinear form is continuous, that is, a(u u h,u v h ) C u u h H (Ω) u v h H (Ω) α u u h 2 H (Ω) C u u h H (Ω) u v h H (Ω) u u h H (Ω) C α u v h H (Ω). Cea s Lemma states that the solution of (.4) is a best approximation to the solution of (.3) by an element of χ. Since, Ω is a convex polygonal domain in R 2, by Sobolev Embedding theorem, H 2 (Ω) is continuously imbedded in C( Ω). Thus, for every u H 2 (Ω), there exists a uniquely defined interpolation operator, Iu, associated with the nodal points. In the theory of finite elements one can prove the following theorem which states the convergence rates of the finite element method []. Theorem.3 Let u H 2 (Ω) and T be a shape regular triangulation of Ω, and I be an interpolation operator I : H 2 (Ω) P t, where P t denotes the space of piecewise polynomial functions of degree t, then there exists a constant c such that u Iu H m (K i ) ch t m u H t (K i ) for u H 2 (Ω), 0 m t, t = 2.

19 .3 Smooth Manifolds We are interested in finite element discretization of spheres, ellipsoids, cylinders and tori or geometries that we generally call spheroidal geometries. Spheres are smooth manifolds in R 3, or genereally in R n. In this section, we give a brief introduction to manifolds and define the Laplace-Beltrami operator on smooth manifolds. Before introducing smooth manifolds, let s first define topological manifolds. Definition. A set X together with the collection of its subsets, J, form a topology if X, J, union of any sets that are in J is also in J the intersection of any two sets that are in J is also in J. We call these sets that are in J the open sets. A special case of topological spaces is a Hausdorff space. It is a topological space with the property that for every two distinct points x, y in X, there exists an open neighborhood U of x and an open neighborhood V of y, which are distinct as well. The definition of k-dimensional topological manifold, M, M R n, is given in [] as a Haussdorff, second countable topological space that is locally homeomorphic (one-to-one, onto, continuous and has a continuous inverse) to R k. In other words, topological manifolds are the spaces that locally look like Euclidean spaces. Example The unit sphere, S 2, in R 3 is a 2-dimensional topological manifold. The unit sphere, S 2, is the set of unit vectors in R 3, that is S 2 = {x = (x,x 2,x 3 ), x R 3 : x = }. 2

20 The unit sphere, S 2, is by definition a topological subspace of R 3. Thus, it is Haussdorff and second countable. We need to show that it is locally Euclidean. To show that it is locally homeomorphic to R 2, we need to show that for every point p in S 2, there exists a homeomorphism, ϕ : U S 2 V R 2 between the open neighborhood U of p and a open subset V of R 2. Let U + i denote the subset of S 2 such that i th coordinate is positive, for i =,2,3. Similarly, let U i denote the subset of S 2 such that i th coordinate is negative, for i =,2,3, U + i = {(x,x 2,x 3 ) S 2 : x i > 0} i =,2,3, U i = {(x,x 2,x 3 ) S 2 : x i < 0} i =,2,3. Let ϕ U + i : U i + R 2 denote the homeomorphisms from U i + to ϕ U + (U i + ) R2, for i =,2,3, i which are defined as ϕ U + ϕ U + 2 ϕ U + 3 (x,x 2,x 3 ) = (x 2,x 3 ), ϕ (x U + 2,x 3 ) = ( (x,x 2,x 3 ) = (x,x 3 ), ϕ (x U +,x 3 ) = (x, 2 (x,x 2,x 3 ) = (x,x 2 ), ϕ U + 3 ϕ U (x,x 2,x 3 ) = (x 2,x 3 ), ϕ (x U 2,x 3 ) = ( ϕ U (x,x 2,x 3 ) = (x,x 3 ), ϕ (x 2 U,x 3 ) = (x, 2 (x x2 3 ),x 2,x 3 ), (x 2 + x2 3 ),x 3), (x,x 2 ) = (x,x 2, (x 2 + x2 2 )), (x x2 3 ),x 2,x 3 ), (x 2 + x2 3 ),x 3), (x 2 + x2 2 )). ϕ U (x,x 2,x 3 ) = (x,x 2 ), ϕ (x 3 U,x 2 ) = (x,x 2, 3 Thus, the unit sphere in R 3 is 2-dimensional topological manifold. 3

21 Let U be an open subset of M and let ϕ : U ϕ(u) R k be a homeomorphism then we call the pair (U,ϕ) a chart. If p is a point in U we say that the pair (U,ϕ) is a chart at p, and we call U the coordinate neighborhood of p, and we call the Cartesian coordinates of p, ϕ(p) = (ϕ (p),ϕ 2 (p),...,ϕ k (p)) = (x,x 2,...,x k ) R k the coordinates of p. Definition.2 A function f : M R is continuous at p M, if for some chart (U,ϕ) at p, f ϕ : ϕ(u) R is continuous at ϕ(p). The function f : M R is continuous on the open set U M, if it is continuous at all points p in U. If M is a topological manifold, the notion of a continuous function f : M R makes sense, but the notion of a differentiable function does not, since differentiablity property of a function is not invariant under homeomorphism. We need to add more features to a topological manifold and get a smooth manifold. We say that f is smooth, if f C (U), that is if f C r (U) for every r. The function f is said to be a diffeomorphism, if f and its inverse f are smooth. Definition.3 Let (U,ϕ) and (V,ψ) be two charts, such that U V, the charts (U,ϕ) and (V,ψ) are called C -compatible if the transition map ψ ϕ : ϕ(u V ) ψ(u V ) is a diffeomorphism. If we index the charts, (U i,ϕ i ), i I where i ranges over some index set I, such that the sets U i cover M, then we call such a collection of charts an atlas and denote it by A. An atlas A is called a differential structure if (U,ϕ) is a chart in M such that it is compatible with every chart in M, then (U,ϕ) is in A. Definition.4 The pair (M, A) is a smooth manifold, where M is a topological manifold and A is a differential structure [2]. 4

22 M U V ϕ ψ ϕ(u) R n ψ ϕ ψ(v ) R n Figure.3: A transition map We usually omit the differential structure and say M is a smooth manifold. A smooth manifold is a topological manifold whose transitions maps are all smooth. For more detailed information on smooth manifolds see [2]. Example The unit sphere, S 2, in R 3 is a 2-dimensional smooth manifold. Since, it is clear from the previous example that {U i +,U i,i =,2,3} is an open cover for S2. For simplicity, let s consider, U + U = {(x,x 2,x 3 ) S 2 : x > 0,x 2 > 0}, then the transition maps, ϕ U + ϕ U + 2 = ( (x 2 + x2 2 ),x 3) and ϕ U + 2 ϕ U + = ( (x x2 3 ),x 3) are compatible. Any other pair can be shown to be compatible, in a similar way. Furthermore, the unit sphere is a compact smooth manifold. Definition.5 A function f : M R is smooth (C (M)) if and only if f is continuous and for every p in M, and there exists a chart (U,ϕ) at p such that f ϕ : ϕ(u) R is smooth (C ((ϕ(u)). 5

23 This definition is well defined, since it is independent of the choice of the the chart at p. We can generalize this definition easily to smooth maps between smooth manifolds. Let M, M R n, be k-dimensional smooth manifold and N, N R n be l-dimensional smooth manifold, the mapping F : M N is a smooth mapping, if and only if F is continuous and for every p M, there exists a chart (U,ϕ) in M and a chart (V,ψ) in N such that F(U) V, and the mapping ψ F ϕ : ϕ(u) ψ(v ) is smooth. Tangent Spaces Hereinafter M denotes k-dimensional smooth submanifold of R n. We first, define the partial derivatives of a differentiable function f : M R with respect to the coordinate system (U, ϕ). At this point, it is worth recalling the limit definition of partial derivatives of the function f : R k R. Let x = (x,...,x k ) R k, then we denote by D i f(x,....x k ), the number lim h 0 f(x,...,x i + h,...,x k ) f(x,....x k ). h For a function f : M R and local charts (U,ϕ), we define D i f(p) = D i ( f ϕ )(ϕ(p) ). (.6) Thus, D i f(p) is the number lim h 0 f ( ϕ (x),...,ϕ i (x) + h,...,ϕ h k (x)) f(ϕ (x)). 6

24 If we define the curve γ i : ( ǫ,ǫ) M by γ i (h) = (ϕ (x),...,ϕ i (x) + h,...,ϕ k (x)), then (.6) is just (f γ i ) f(γ i (h)) f(γ i (0)) (0) = lim. h 0 h Define the collection of all such C curves in M by C p = {γ C ((ǫ,ǫ),m),ǫ > 0,γ(0) = p}. We say that x R n is tangent to M at the point p, if and only if there exists a γ C p such that x = γ (0). Definition.6 We define the tangent space at p as T p M = {(γ (0)) : γ C p }, and tangent bundle TM as TM := T p M. An element of TM is an ordered pair (p,x) with p p M M x T p M. The projection map π : TM M maps each vector in T p M to the point at which it is tangent as π(p,x) = p. Remark. For any p M, T p M is a k-dimensional vector space, and if (U,ϕ) is chart containing p, then (D ϕ,d 2 ϕ,...,d k ϕ ) forms a basis for T p M. Remark.2 Tangent bundle TM is a 2k-dimensional manifold and the π : TM M is a smooth map. Definition.7 The tangent map TF : TM TN for a C -map F : M N is defined by TF(γ(0),γ (0)) = (γ(0),(f γ) (0)). 7

25 The differential of F, is a linear mapping DF : T p M T F(p) N which is defined as: F(γ (0)) = (F γ) (0). Define g p (, ) : T p M T p M R as g p (x,y) = x y, where x T p M y T p M and x y is the inner product T p M inherited from R n. Recall that (D ϕ,d 2 ϕ,...d k ϕ ) is a basis for T p M, and define the metric tensor G = g ij as g ij = D i ϕ D j ϕ and g = det(g) is called the Gramm determinant and is strictly positive. Let f : M R be a differentiable function, then the differential of f is a linear mapping T p M to R, thus it is a functional on T p M. So, by the Riesz Representation (Theorem.), there exists a unique vector in T p M, call it gradient of f, and denote it by f, such that Df(p)v = f v for every v T p M. Remark.3 Let f : M R and (U,ϕ) a chart containg p, then f(p) is defined as k k f(p) = g ij D j (f ϕ )(ϕ(p)). (.7) i= j= Definition.8 A vector field is a continuous map X : M TM, such that π X = Id M, that is X(p) = (p,x(p)) p M such that x(p) is a tangent vector to p. 8

26 k Let X be C -vector field on M with local representation X ϕ = X i i ϕ, define a scalar function X : M R as i= ( X) ϕ = k ( i X i ). g i= For any C 2 function f : M R define Laplace-Beltrami operator as f = f. Let ϕ be a local chart then the local representation is ( f) ϕ = ( f) ϕ = g k j ( g j= k g ij i (f ϕ)) (.8) i= where g ij are the components of g. The Laplace Beltrami Operator on the Sphere Let ϕ(θ,θ 2 ) = (sin θ cos θ 2,sin θ sin θ 2,cos θ ) 0 θ π, 0 θ 2 < 2π be a local chart of the sphere, then g = θ ϕ θ ϕ θ ϕ θ2 ϕ θ2 ϕ θ ϕ θ2 ϕ θ2 ϕ = 0 0 sin 2 θ and also, g = 0 0 sin 2 θ and g = sin 2 θ. 9

27 We obtain the local representation of the Laplace-Beltrami S f of any C 2 function f on the sphere as ( S f) ϕ = = = g 2 j ( g j= 2 g ij i (f ϕ)) i= [ θ (sin θ θ ) + θ2 ( θ2 )](f ϕ) (.9) sin θ sin θ θ (sin θ θ ) + sin θ sin 2 θ2 ( θ2 )f ϕ. θ.4 Eigenvalue Problem Let S be the Laplace operator on the sphere, then the eigenvalue problem is to find a scalar λ such that S µ = λµ. If the local representation of Laplace-Beltrami operator is obtained using polar coordinate charts (.9), then the eigenvalue problem is given by θ (sin θ θ µ) + sin θ sin 2 θ2 ( θ2 µ) = λµ. (.0) θ Writing µ in separable form µ(θ,θ 2 ) = ν(θ )ρ(θ 2 ), we obtain the following two eigenvalue problems and d 2 ρ + ζρ = 0 ρ(0) = ρ(2π), dθ 2 2 dρ (0) = dρ (2π) (.) dθ 2 dθ 2 sin θ d dθ (sin θ dν dθ ) + (λsin 2 θ ζ)ν = 0, lim ν <, lim ν <. (.2) θ 0 θ π 20

28 Eigenvalues and the corresponding eigenfunctions of the problem (.) are ζ m = m 2, ρ(θ 2 ) = A m cos mθ 2 + B m sin mθ 2, m = 0,,2.... Substituting the eigenvalues ζ m and t = cos θ into (.2), and setting P(t) = ν(θ (t)) we get the associated Legendre equation ( t 2 ) d dt [ ( t 2 ) dp ] + [ ( t 2 )λ m 2] P = 0 t, m = 0,,2,.... (.3) dt The associated Legendre Equation (.3) has solutions that are bounded everywhere if and only if the eigenvalues λ are of the form, λ n = n(n+), n = 0,,2,.... The eigenfunctions associated with such an eigenvalue λ n are the Legendre functions P m n of degree n order m. Thus eigenvalues λ n of (.0) are λ n = n(n + ), n = 0,,... and the corresponding eigenfunctions are S m n (θ,θ 2 ) = (A m cos mθ 2 + B m sinmθ 2 )P m n (cos θ ). The weak formulation of (.0) is, find λ R, and 0 µ H (S) such that a(µ,ν) = λ(µ,ν) (.4) holds for every ν H (S), where a(, ) : H (S) H (S) R is a continuous, elliptic bilinear form given by a(µ,ν) = S S µ S ν. The problem (.4) has a sequence of eigenvalues, λ 0 λ, = λ,2 = λ,3... λ k, = λ k,2 =... = λ k,2k+..., 2

29 and the corresponding eigenfunctions, which are spherical harmonics, µ 0,µ,.... These eigenfunctions are orthogonal in the energy inner product a ( µ i,µ j ) = λ i (µ i,µ j ) = λ i δ ij, where δ ij is the Kronecker-delta function. Let µ with µ a = denote an eigenfunction corresponding to λ and let M(λ) denote the space of eigenfunctions corresponding to λ. We are interested in approximating eigenvalues and eigenfunctions of (.4) by the finite element method. We consider the following eigenvalue problem: Find λ h R and 0 µ h χ such that a h (µ h,ν h ) = λ h (µ h,ν h ) (.5) for every ν h χ where χ is a finite dimensional subspace χ H (S). Problem (.5) has a sequence of eigenvalues, λ h 0 λh... λh n n = dimχ and corresponding eigenfunctions µ h 0,µh,...,µh n which are also orthogonal in the energy inner product. a ( µ h i,µh j ) = λ i(µ h i,µh j ) = λ iδ ij i,j =,... n. 22

30 Let µ h k,,µh k,2,...,µh k,2k+ with µh k = denote the finite element approximate eigenfunctions of λ h k,,λh k,,... λh k,2k+ respectively. The eigenpairs (λh,µ h ) of (.5) are approximates of the eigenpairs (λ,µ) of (.4). The eigenvalues λ k and their approximates λ h k satisfy the following well-known minmax principles. λ k = min max a(µ,µ) U k H (S) µ S k (µ,µ) λ h k = min S k X max µ h S k a(µ h,µ h ) (µ h,µ h ). The minimum is taken over all k-dimensional subspaces, U k, and S k of H (S), and X respectively. It follows immediately from the minmax principles that every eigenvalue is approximated from above by 2k + of the approximated eigenvalues [5]. λ k λ h k, λh k,2...λh k,2k+ λ k λ h k,,λh k,2,... λh k,2k+. It is well known that, λ h k,q λ k C sup µ M(λ) inf µ ν h a q =,...,2k +. (.6) ν h χ Here a denotes the energy norm [3],[4],[5]. 23

31 Chapter 2 The Radial Projection The aim of our work is to develop an exact finite element discretization of the sphere. This new method can be easily applied to other domains, e.g. cylinders, ellipsoids, and tori. The principal idea behind our new method is to radially project the finite elements constructed on the surface of the cube onto the sphere. The cube is used as a tool in our approach, so that all the calculations are done on logically rectangular grids. We call these spherical triangles and the associated finite element functions formed by this radial projection as radially projected finite elements. The radial projection is a mapping from the cube to the ball, or can be modified such that it is also a mapping from the surface of the cube (which we will call box) to the sphere, cylinder, ellipsoid, etc. For definiteness we illustrate the method on the sphere. In order to make our terminology clearer, we will use box to indicate the surface of the cube. The relation between the box and the cube is similar to the relation between the sphere and the ball. We denote the sphere centered at the origin with radius r by S r and the box centered at the origin with a side length 2d by B d. We denote points on the box by x and y and points on the sphere by a and b. Functions defined on B d will be denoted by u or v, and similarly, functions defined on the S r will be denoted by µ or ν. We now define the box B d, cube C d, sphere S r, and ball B r, to emphasize our terminology. 24

32 B d C d S r B r = {x = (x,x 2,...,x n ) R n : x = d}, = {x = (x,x 2,...,x n ) R n : x d}, = {a = (a,a 2,...,a n ) R n : a 2 = r}, = {a = (a,a 2,...,a n ) R n : a 2 r}, where a 2 = a 2 + a a2 n is the Euclidean norm and x = max( x, x 2,..., x n ) is the maximum norm. Unless otherwise mentioned, hereinafter will denote 2 ; and S will denote the unit sphere in R 3 centered at the origin and B will denote the box in R 3 with d = centered at the origin. The radial projection from a box to a sphere is denoted by P : B d S r and is given by x P(x) = r, d x 2 and the inverse projection is P : S r B d, is P a (a) = d. r a In section 2. we list some properties of the radial projection and in section 2.2 we generate meshes on spheres, ellipsoids and cylinders. 25

33 2. Properties of the Radial Projection The radial projection P is a one-to-one mapping from the box onto the sphere. In the following Lemma we show that both P and P are Lipschitz continuous mappings. Lemma 2. The radial projection P and its inverse P satisfy the inequalities P(x) P(y) 2r x y (2.) x and P (a) P (b) 2d a a b. (2.2) Proof: P(x) P(y) = r x x r y y r = y x x y x y r ( ) = x y y (x y) + y y x r [ ] y x y + y y x x y 2r x y x Similary, P (a) P (b) 2d a a b. 26

34 z 0 z y x y x Figure 2.: Mesh on the box and the sphere. Corollary 2. The radial projection P and its inverse P, are globally Lipschitz continuous mappings, and P(x) P(y) 2r x y, (2.3) d P (a) P (b) 2dn a b. (2.4) Proof: For any x R n, we have x x n x, and for any a S we have n a, and similarly, for any x B x n. Thus, the corollary is direct consequence of the lemma

35 2.2 Some Examples of Mapping 2.2. Mesh on the Sphere Let B denote the box in R 3 centered at the origin with side length 2, that is B = {x = (x,x 2,x 3 ) R 3 : x = }, and let S denote the unit sphere in R 3 centered at the origin, that is S = {a = (a,a 2,a 3 ) R 3 : a = }. The mapping from the box B to the sphere S is given by, P(x) = x x, and the inverse mapping from sphere S to B is, P (a) = a a. Figure 2. shows the finite element triangulation on B and the corresponding triangulation on S. We can think of the box, B, as a union of 6 faces. Let F ±i, i =... 3 denote the faces of the box such that, F +i corresponds to the x i = plane, and F i corresponds to the x i = plane. Since on each face one of the variables is constant, the mesh generation on these faces is the same as mesh generation on a 2-dimension planar square. The radial projection of any of these faces F ±i i =... 3 onto the sphere is given as P F±i (x,x 2,x 3 ) = ( x x 2 + x 2 2 +, x2 3 x 2 x 2 + x 2 2 +, x2 3 x 3 x 2 + x x2 3 ) i =,...,3 28

36 x 3 F +3 F + F +2 x 2 x Figure 2.2: Faces of the box and the inverse projection onto these faces are given as P F ± (a,a 2,a 3 ) = P F ±2 (a,a 2,a 3 ) = P F ±3 (a,a 2,a 3 ) = ( ±, ± a 2, ± a ) 3, a a ( ± a, ±, ± a ) 3, a 2 a 2 ( ± a, ± a ) 2, ±. a 3 a 3 Figure 2.3 and 2.4 show the triangulation on F + and its radial projection onto the sphere. Let K = {K, K 2,..., K n } be a triangulation of the box, then T = P(K), is a triangulation of the sphere. Let T = {T, T 2,..., T n } denote the corresponding triangulation of the sphere where {T i : T i = P(K i ), i =,...,n} are the spherical triangles covering the sphere, n such that S = T i. i= 29

37 Figure 2.3: F +. Figure 2.4: F + and P (F + ). a b α r d(a,b) = αr Figure 2.5: Geodesic distance between a and b. 30

38 We measure the distance between two points on the sphere by the geodesic distance. Let a and b be two points on S r, then the geodesic distance is defined by d(a,b) = r cos ( a b ). (2.5) r2 The geodesic distance between two points, a and b on a sphere is the length of the arc of the great circle connecting them. This great circle lies on a plane, figure 2.5 shows a 2D cross section of the sphere. Let α be the angle between a, the center of the sphere O and b as shown in the figure, then d(a,b) = αr. Proposition 2. Let a and b be two distinct points on the unit sphere centered at the origin in R 3, then we have the following estimate, d(a,b) a b d(a,b). (2.6) 2 Proof: The geodesic distance between a and b is d(a,b) = α, which is the distance along the great circle connecting them. Figure (2.6) shows that a b = 2sin ( α 2). Clearly, a b d(a,b) = 2sin ( α 2). α Let f(α) = 2sin( α 2) α, then f (α) = αcos( α 2) 2sin( α 2). The function f is a decreasing function α 2 on the interval α [0,π], since f (α) < 0 α (0,π) which is equivalent to x tan(x) < 0 x (0,π/2). Clearly, f(α) has a maximum at α = 0, f(0) = and a minimum at α = π, f(π) = 2 π, that is 2 a b d(a,b). 3

39 d(a, b) a a b b r r α 2 α 2 O Figure 2.6: Geodesic and Euclidean distance of a and b With the following proposition, we prove that radially projected finite elements form a shape regular triangulation of the sphere. Proposition 2.2 Let T j, j N denote the family of triangulations of the sphere S, and let K j, j N denote the family of triangulations of the box B, such that P(K j ) = T j. If K j is a shape regular triangulation of B, then T j is a shape regular triangulation of S. Proof: Let T j = {T, T 2,...,T nj }, j N denote the triangulation of the sphere and let K j = {K, K 2,..., K nj }, j N be a triangulation of the box such that P(K i ) = T i, i =,...,n j. Let a and b be two points on the sphere such that h Ti = d(a,b). Since d(a,b) 2 a b 2 x y, where x = P (a) and y = P (b) we get that h Ti 2h Ki. Similarly, let ρ Ki denote the radius of inscribed circle in the triangle K i and consider two points x and y in the triangle K i, such that ρ Ki = x y. Let a and b be two pints on the sphere such that a = P(x) and b = P(y), then ρ Ti d(a,b). Since, d(a,b) a b 6 x y, we conclude that ρ Ti 6 ρ K i. 32

40 z y x Figure 2.7: Mesh on the ellipsoid Meshes on Ellipsoids Let B denote the box in R 3 with side length 2, that is B = {x R 3 : x = }, and let S denote the sphere in R 3 with radius, that is S = {a R 3 : a = } and let E denote the ellipsoid in R 3 such that E = {b = (b,b 2,b 3 ) R 3 : b2 k 2 + b2 2 l 2 + b2 3 m 2 = }. The mapping M from box B to ellipsoid E is the composition of the radial projection P of B onto sphere S with the linear mapping L from sphere S to the ellipsoid E. The mapping L : S E is given as L(a) = (ka,la 2,ma 3 ), thus M : B E is and M : E B is ( k x M(x) = x x, l x x x 2, m x ) x x 3, M (b) = (, kb 2, kb ) 3, lb mb if L (b) = b k. 33

41 Let K = {K, K 2,..., K n } be the triangulation of the box, then T = M(K), is the triangulation of the ellipsoid. Let T = {T, T 2,..., T n } denote the corresponding triangulation of the ellipsoid where {T i : T i = M(K i ), i =,...,n} are the triangles covering the n ellipsoid, such that E = T i. i= Mesh on the Cylinder Let B denote the surface of the cube in R 3 with side length 2 and centered at the origin, that is B = {x R 3 : x = }, and let C denote the surface of the circular right cylinder in R 3 of height 2 and radius. By cylinder we mean the following: C = {a = (a,a 2,a 3 ), a R 3 : a 2 + a2 2 =, < a 3 <, and a 2 + a2 2,a 3 = ±}. In other words it is a cylinder with top and bottom faces. The mapping from box B to sphere C is given by, P(x) = ( x x x, x ) x x 2,x 3. and the inverse mapping from the cylinder C to the box B is given by, P (a) = ( ) a a a, a 2,a 3. a a Let K = {K, K 2,..., K n } be the triangulation of the box, then T = P(K), is the triangulation of the cylinder. Let T = {T, T 2,..., T n } denote the corresponding triangulation of the cylinder where {T i : T i = P(K i ), i =,...,n} are the triangles covering the cylinder, n such that C = T i. One may also prove that, mesh generated on the cylinder in such way, i= is also a shape regular mesh. 34

42 z y x Figure 2.8: Mesh on the cylinder Mesh on the Disc Let B denote the square in R 2 such that B = {x = (x,x 2 ) R 2 : x }, and let D denote the unit disc in R 2, such that D = {a = (a,a 2 ) R 2 : a }, then the mapping from the square B to the unit disc D is given by P(x) = x x x and the inverse mapping is given as P (a) = a a a. 35

43 Figure 2.9: Mesh on the disc 36

44 Chapter 3 Analysis In this chapter, we consider the problem of determining error estimates in L 2 and H norms. We estimate the difference (µ µ h ) where µ is the solution of a second order elliptic problem and µ h is the discrete solution obtained using the proposed finite element method. In section (3.) we analyze our finite element discretization and in section (3.2) we derive error estimates. 3. Finite Element Construction We describe our method in the context of the Laplace-Beltrami equation defined on the sphere S. We consider the following second order elliptic equation on the sphere, S µ + µ = f, (3.) where S is the Laplace-Beltami operator S = S S. Here S is the tangential gradient which is defined as S µ = µ ( µ n)n 37

45 where n is the unit outer normal to the sphere at the point a and µ is a smooth extension of µ to a neighborhood of S. Since S is a smooth compact manifold without boundary, there are no boundary conditions needed. We use the standard notation L p (S), W m,p (S) for Sobolev spaces defined on the sphere. L p (S) = {µ : µ p < } S W m,p (S) = {µ L p (S) : α S µ p <, for 0 < α m}, S where α = (α,α 2,α 3 ) is a multi index notation with norm α = α + α 2 + α 3. W m,p (S) is a Banach space equipped with the norm µ W m,p (S) = 0< α m α S µ p L p (S) p p <. Let H m (S) denote W m,2 (S), then the semi-norms on H m (S) are µ H m (S) = p α S µ 2 L 2 (S). Let f L 2 (S), then there exist an exact solution µ H 2 (S) α =m to (3.). There exist a constant c, such that[6], µ H 2 (S) c f L 2 (S). A weak formulation of (3.): find a solution µ H (S) such that the following holds, S µ S ν + µ ν = S S f ν ν H (S). (3.2) 38

46 Let f L 2 (S) and µ H 2 (S), then µ is a solution of (3.2) iff µ is a solution of (3.). Let a(, ) : H (S) H (S) R be the bilinear form defined by a(µ,ν) := S µ S ν + µ ν. (3.3) S Notice that a(, ) is continuous, elliptic bilinear form, since µ H (S) = a(µ,µ), and (f,ν) := f ν S is the standard inner product in L 2 (S). Theorem 3. There exists a unique solution to (3.2). Proof: The symmetric, elliptic bilinear form a(, ) (3.3) induces a norm on H (S). The space (H (S),a(, )) is a Hilbert space, so the conclusion is a direct consequence of the Riesz Representation theorem. We approximate the solution µ by µ h χ, where χ is in a finite dimensional subspace of H (S). We solve the following discrete problem: find a function µ h χ such that S µ h S ν h + µ h ν h = S S f ν h ν h χ. (3.4) Let a h (, ) : χ χ R denote the bilinear form defined as a h (µ h,ν h ) := S µ h S ν h + µ h ν h, S 39

47 and let (f, ) : χ R denote the linear functional defined as (f,ν h ) := S f ν h, then the discrete problem is: find µ h χ such that a h (µ h,ν h ) = (f,ν h ) for every ν h χ. Define χ as χ = {µ h : µ h = u P, u is a piecewise linear continuous polynomial}. (3.5) This dicrete problem has a unique solution. Since χ is a finite dimensional subspace of n H (S), let {ϕ i } n i= be a basis for χ. Letting µ h = µ hi ϕ i, leads to following linear system of equations i= Aµ = f. (3.6) Here, A ij = a(ϕ j,ϕ i ) is a sparse, symmetric, positive definite matrix, (µ) i = µ hi is the unknown vector, (f) i = (f,ϕ i ) is the right hand side vector. 3.2 Error Analysis Let µ be the solution of (3.2) and µ h be the solution of (3.4). We want to get an error estimate of the form, µ µ h H k (S) ch p, k = 0,, where p is the order of convergence for the method and depends on the the regularity of the solution, degree of the finite elements used (on the planar triangles in the construction of the finite elements) and the Sobolev norm used to measure the error. If we let T = {K, K 2,..., K n } denote the partition of the sphere, such that S = n i= K i, then ( n /2 µ H k (S) = µ 2 H k (K i )). i= 40

48 Cea s Lemma has a significant role in describing error estimates in finite element analysis, since it reduces the problem of estimating the error µ µ h H (S) to a problem of estimating the distance between the function µ H (S) and a function µ h in its subspace χ H (S). By Cea s Lemma we know that µ µ h H (K) C inf ξ χ µ ξ H (K). Thus, µ h is the best approximation of µ with respect to H -norm within the space χ. We can replace ξ on right hand side by the interpolant of the exact solution Iµ and we get, µ µ h H (K) C µ Iµ H (K). We wish to consider the size of the error in µ µ h L 2 (K). To estimate the error µ µ h in L 2 -norm, we use so called duality argument thus, µ µ h L 2 (K) Ch µ µ h H (K). Therefore, rather than estimating the difference µ µ h L 2 (K), µ µ h H (K), we will be estimating µ Iµ L 2 (K), µ Iµ H (K). Consider a spherical triangle K in T and consider a planar triangle K h such that for every x K h x x K. In other words, K is the radial projection of K h onto the sphere, i.e., P(K h ) = K. Moreover, suppose that the spherical triangle, K, and the planar triangle K h share the same nodes. Figure (3.) shows the planar triangle K h and the spherical triangle K. The union of the spherical triangles covers the sphere, and the union of planar triangles 4

49 K K h Figure 3.: Spherical triangle, K and planar triangle K h. forms a polygonal surface Ω, i.e., S = n i= K i and Ω = n i= K h i. If {η i } m i= denote the nodes of spherical triangles covering the sphere, then Ω S = {η i } m i=. Define U = {x = (x,x 2,x 3 ) R 3 : 2 x 3 2 }, so that both K U, and K h U, then we may extend a function µ defined on the S uniquely to the three dimensional domain U by Eµ(x) = µ( x ) for every x U. (3.7) x Clearly Eµ(a) = µ(a) if a S and note that Eµ is constant along the normal direction to the sphere, that is Eµ does not depend on r, if spherical coordinates are used, so Eµ n = 0, where n = (a,a 2,a 3 ) is the normal to the sphere S at the point a. 42

50 The tangential gradient of the real valued function µ on S is the orthogonal projection of the gradient of µ onto the tangent plane at the point a. Let n = (a,a 2,a 3 ) denote the unit outward normal at the point a on S, then tangential gradient, S µ = Eµ ( Eµ n)n, can be expressed as: S µ = A µ (3.8) where A is the 3 3 matrix matrix with a 2 i i = j (A) ij = a i a j i j. (3.9) Since S µ is the projection of µ on to the tangent plane, S µ n = 0, we have S µ 2 = µ 2 n µ 2. Thus, S µ µ. Proposition 3. We have the following identities for the second order tangential derivatives of µ and 2 S µ = ( Sµ) + S µn, (3.0) 2 Sµn = A S µ, (3.) where A is given in (3.9). Note that unlike the Hessian, 2 S µ is not symmetric. 43

51 Proof: Since a S, µ + a 2 S,2 µ + a 3 S,3 µ = 0, (3.2) taking the gradient of both sides of (3.2), we have a D i S, µ + a 2 D i S,2 µ + a 3 D i S,3 µ = D i Si µ i =,2,3. (3.3) (3.0) is direct consequence of (3.3). Let ν be a real valued function differentiable at a S, then S (µν) = µ S ν + ν S µ. Taking tangential derivative of both sides of (3.2), we get a S ( S, µ) + a 2 S ( S,2 µ) + a 3 S ( S,3 µ) = S a S µ + S a 2 S µ + S a 3 S µ a 2 a a 2 a 3 a Since, S a = a a 2, S a 2 = a 2, and 2 S a 3 = a 3 a 2, from which a a 3 a 2 a 3 a 2 3 (3.) follows immediately. Proposition 3.2 Let Eµ : U R be defined by (3.7) for x U, let a = x x have the following identities; S then we S µ = a Eµ, (3.4) x Eµ = x Sµ, (3.5) x ( S µ) = x A a ( S µ), (3.6) 2 S µ = A a ( S µ). (3.7) 44

52 Here x = (D x,d x2,d x3 ) denotes the ordinary gradient with respect to the variable x and similarly a = (D a,d a2,d a3 ) denotes the ordinary gradient with respect to the variable a. Proof: For any a S, we have S µ(a) = S Eµ(a) = a Eµ(a) ( a Eµ(a) n)n = a Eµ(a). Since Eµ(x) = Eµ( x x ), by applying the chain rule we get, Eµ x i = Eµ a a x i + Eµ a 2 a 2 x i + Eµ a 3 a 3 x i i =,2,3. So, x Eµ(x) = J T Eµ(a), (3.8) where J T is the transpose of Jacobian matrix of the projection P(x) = x x and, J T = x x2 x x 2 x x 3 x 3 x 3 x 3 x 2x x 3 x x2 2 x 2x 3 x 3 x 3 x 3x x 3 x 3x 2 x 3 x x2 3 x 3 = x A Thus x Eµ(x) = x Sµ(a). Since D x Eµ = x D a Eµ(a) and a = P(x), by chain rule we have the following D xi ( Daj Eµ(a) ) = D a (D aj Eµ) a x i + D a2 (D aj Eµ) a 2 x i + D a3 (D aj Eµ) a 3 x i i,j =,2,3. (3.9) 45

53 In matrix form, (3.9) and (3.6) are the same. Applying the tangential gradient twice to the real valued function µ we get (3.7). Let χ h denote the finite element space on a polygonal domain, Ω, consisting of piecewise linear continuous functions, i.e; χ h = {v : v is piecewise linear continuous polynomial}. Let {φ i } n i= be basis for χ h, such that φ i (η j ) = δ ij where δ ij is the Kronecker-delta. Let χ be the finite element space in (3.5), define ϕ i = φ i P, i =,...,n, then {ϕ i } n i= is a basis for χ. For any piecewise linear function v on Ω, we have n v(x) = c i φ(x) i= x Ω U thus, n µ(a) = c i ϕ(a) = i= n c i φ(p (a)) a S. i= Let Eµ be defined by (3.7), then Eµ(x) = v(x), x Ω. Clearly, if x K h, then a = x x K. Lemma 3. Let µ and v be real valued functions, then for some constants c i, i =,...,5 we have the following relations for the equivalence of norms, c v L 2 (K h ) µ L 2 (K) c 2 v L 2 (K h ) c 3 v H (K h ) µ H (K) c 4 v H (K h ) 46

54 v H 2 (K h ) c 5 µ H 2 (K). Proof: Let J denote the Jacobian matrix of the projection P, then there exists constants c > 0, c 2 > 0 such that J c and J c 2.Thus µ 2 L 2 (K) = µ 2 K = µ P 2 J K h c v 2 K h = c v 2 L 2 (K h ). Similarly, v 2 L 2 (K h ) c 2 µ 2 L 2 (K). For any x K h, there exists constants such that d x d 2, then by (3.4) we get d a Eµ(a) x Eµ(x) d 2 a Eµ(a). Thus, K S µ 2 = Eµ 2 K = Eµ P) 2 J K h c v 2. d K h So, there exists some constants c 3 > 0, and c 4 > 0 such that c 3 v H (K h ) µ H (K) c 4 v H (K h ). 47