Finite Element Interpolation

Transcription

1 Finite Element Interpolation This chapter introduces the concept of finite elements along with the corresponding interpolation techniques. As an introductory example, we study how to interpolate functions in one dimension. Finite elements are then defined in arbitrary dimension, and numerous examples of scalar- and vector-valued finite elements are presented. Next, the concepts underlying the construction of meshes, approximation spaces, and interpolation operators are thoroughly investigated. The last sections of this chapter are devoted to the analysis of interpolation errors and inverse inequalities.. One-Dimensional Interpolation The scope of this section is the interpolation theory of functions defined on an interval ]a, b[. For an integer k 0, P k denotes the space of the polynomials in one variable, with real coefficients and of degree at most k... The mesh A mesh of Ω = ]a, b[ is an indexed collection of intervals with non-zero measure {I i = [x,i, x 2,i ]} 0 i N forming a partition of Ω, i.e., N Ω = i=0 I i and I i I j = for i j. (.) The simplest way to construct a mesh is to take (N +2) points in Ω such that a = x 0 < x <... < x N < x N+ = b, (.2) and to set x,i = x i and x 2,i = x i+ for 0 i N. The points in the set {x 0,..., x N+ } are called the vertices of the mesh. The mesh may have a variable step size

2 4 Chapter. Finite Element Interpolation and we set h i = x i+ x i, 0 i N, h = max 0 i N h i. In the sequel, the intervals I i are also called elements (or cells) and the mesh is denoted by T h = {I i } 0 i N. The subscript h refers to the refinement level...2 The P Lagrange finite element Consider the vector space of continuous, piecewise linear functions P h = {v h C 0 (Ω); i {0,..., N}, v h Ii P }. (.3) This space can be used in conjunction with Galerkin methods to approximate one-dimensional PDEs; see, e.g., Chapters 2 and 3. For this reason, Ph is called an approximation space. Introduce the functions {ϕ 0,..., ϕ N+ } defined elementwise as follows: For i {0,..., N + }, h i (x x i ) if x I i, ϕ i (x) = h i (x i+ x) if x I i, (.4) 0 otherwise, with obvious modifications if i = 0 or N +. Clearly, ϕ i Ph. These functions are often called hat functions in reference to the shape of their graph; see Figure.. Proposition.. The set {ϕ 0,..., ϕ N+ } is a basis for P h. Proof. The proof relies on the fact that ϕ i (x j ) = δ ij, the ronecker symbol, for 0 i, j N +. Let (α 0,..., α N+ ) T R N+2 and assume that the continuous function w = N+ i=0 α iϕ i vanishes identically in Ω. Then, for 0 i N +, α i = w(x i ) = 0; hence, the set {ϕ 0,..., ϕ N+ } is linearly independent. Furthermore, for all v h Ph, it is clear that v h = N+ PSfrag replacements i=0 v h(x i )ϕ i since, on each element I i, the functions v h and N+ i=0 v h(x i )ϕ i are affine and coincide at two points, namely x i and x i+. ϕ ϕ i ϕ N+ a x x 2 x i x i x i+ x N b Fig... One-dimensional hat functions.

3 .. One-Dimensional Interpolation 5 I hv v PSfrag replacements a Fig..2. Interpolation by continuous, piecewise linear functions. b Definition.2. Choose a basis {γ 0,..., γ N+ } for L(Ph ; R); henceforth, the linear forms in this basis are called the global degrees of freedom in Ph. The functions in the dual basis are called the global shape functions in Ph. For i {0,..., N + }, choose the linear form γ i : C 0 (Ω) v γ i (v) = v(x i ) R. (.5) The proof of Proposition. shows that a function v h Ph is uniquely defined by the (N +2)-uplet (v h (x i )) 0 i N+. In other words, {γ 0,..., γ N+ } is a basis for L(Ph ; R). Choosing the linear forms (.5) as the global degrees of freedom in Ph, the global shape functions are the functions {ϕ 0,..., ϕ N+ } defined in (.4) since γ i (ϕ j ) = δ ij, 0 i, j N +. Consider the so-called interpolation operator I h : C 0 (Ω) v N+ i=0 γ i (v)ϕ i P h. (.6) For a function v C 0 (Ω), Ih v is the unique continuous, piecewise linear function that takes the same value as v at all the mesh vertices; see Figure.2. The function Ih v is called the Lagrange interpolant of v of degree. Note that the approximation space Ph is the codomain of I h. When approximating PDEs using finite elements, it is important to investigate the properties of Ih in Sobolev spaces; see Appendix B. In particular, recall that for an integer m, H m (Ω) denotes the space of square-integrable functions over Ω whose distributional derivatives up to order m are squareintegrable. We use the following notation: v 0,Ω = v L2 (Ω), v,ω = v 0,Ω, v,ω = ( v 2 0,Ω + v 2 0,Ω ) 2, v 2,Ω = v 0,Ω, etc. Lemma.3. P h H (Ω). Proof. Let v h P h. Clearly, v h L 2 (Ω). Furthermore, owing to the continuity of v h, its first-order distributional derivative is the piecewise constant function w h such that

4 6 Chapter. Finite Element Interpolation I i T h, w h Ii = v h(x i+ ) v h (x i ) h i. (.7) Clearly, w h L 2 (Ω); hence, v h H (Ω). Proposition.4. I h is a linear continuous mapping from H (Ω) to H (Ω), and I h L(H (Ω);H (Ω)) is uniformly bounded with respect to h. Proof. () In one dimension, a function in H (Ω) is continuous. Indeed, for v H (Ω) and x, y Ω, v(y) v(x) y x v (s) ds y x 2 v,ω, (.8) owing to the Cauchy Schwarz inequality (this can be justified rigorously by a density argument). Furthermore, taking x to be a point where v reaches its minimum over Ω, the above inequality implies v L (Ω) b a 2 v 0,Ω + b a 2 v,ω, (.9) since v(x) b a 2 v 0,Ω. Therefore, Ih v is well-defined for v H (Ω). Moreover, Lemma.3 implies Ih v H (Ω); hence, Ih maps H (Ω) to H (Ω). (2) Let I i T h for 0 i N. Owing to (.7), (Ih v) I i = h i (v(x i+ ) v(x i )); hence, using (.8) yields the estimate Ih v,i i v,ii. Therefore, Ih v,ω v,ω. Moreover, since Ih v 0,Ω b a 2 Ih v L (Ω) and Ih v L (Ω) v L (Ω), we deduce from (.9) that Ih v 0,Ω c v,ω where c is independent of h (assuming h bounded). The conclusion follows readily. Proposition.5. For all h and v H 2 (Ω), v I hv 0,Ω h 2 v 2,Ω and v I hv,ω h v 2,Ω. (.0) Proof. () Consider an interval I i T h. Let w H (I i ) be such that w vanishes at some point ξ in I i. Then, owing to (.8) we infer w 0,Ii h i w,ii. (2) Let v H 2 (Ω), let i {0,..., N}, and set w i = (v Ih v) I i. Note that w i H (I i ) and that w i vanishes at some point ξ in I i owing to the meanvalue theorem. Applying the estimate derived in step to w i and using the fact that (Ih v) vanishes identically on I i yields v Ih v,i i h i v 2,Ii. The second estimate in (.0) is then obtained by summing over the mesh intervals. To prove the first estimate, observe that the result of step can also be applied to (v Ih v) I i yielding v I hv 0,Ii h i v I hv,ii h 2 i v 2,Ii. Conclude by summing over the mesh intervals.

5 .. One-Dimensional Interpolation 7 Remark.6. (i) The bound on the interpolation error involves second-order derivatives of v. This is reasonable since the larger the second derivative, the more the graph of v deviates from the piecewise linear interpolant. (ii) If the function to be interpolated is in H (Ω) only, one can prove the following results: h, v I hv 0,Ω h v,ω and lim h 0 v I hv,ω = 0. The proof of Proposition.5 shows that the operator Ih is endowed with local interpolation properties, i.e., the interpolation error is controlled elementwise before being controlled globally over Ω. This motivates the introduction of local interpolation operators. Let I i = [x i, x i+ ] T h and let Σ i = {σ i,0, σ i, } where σ i,0, σ i, L(P ; R) are such that, for all p P, σ i,0 (p) = p(x i ) and σ i, (p) = p(x i+ ). (.) Note that Σ i is a basis for L(P ; R). The triplet {I i, P, Σ i } is called a (onedimensional) P Lagrange finite element, and the linear forms {σ i,0, σ i, } are the corresponding local degrees of freedom. The functions {θ i,0, θ i, } in the dual basis of Σ i (i.e., σ i,m (θ i,n ) = δ mn for 0 m, n ) are called the local shape functions. One readily verifies that θ i,0 (t) = t xi h i and θ i, (t) = t xi h i. (.2) Finally, introduce the family {II i } Ii T h that, for i {0,..., N}, of local interpolation operators such I I i : C 0 (I i ) v σ i,m (v)θ i,m. (.3) m=0 The proof of Propositions.4 and.5 can now be rewritten using the local interpolation operators I I i. In particular, the key properties are, for 0 i N and v H 2 (I i ), v I I i v 0,Ii h 2 i v 2,Ii and v I I i v,ii h i v 2,Ii...3 P k Lagrange finite elements The interpolation technique presented in..2 generalizes to higher-degree polynomials. Consider the mesh T h = {I i } 0 i N introduced in... Let P k h = {v h C 0 (Ω); i {0,..., N}, v h Ii P k }. (.4) To investigate the properties of the approximation space Ph k and to construct an interpolation operator with codomain Ph k, it is convenient to consider Lagrange polynomials. Recall the following:

6 8 Chapter. Finite Element Interpolation Definition.7 (Lagrange polynomials). Let k and let {s 0,..., s k } be (k + ) distinct numbers. The Lagrange polynomials {L k 0,..., L k k } associated with the nodes {s 0,..., s k } are defined to be L k l m m(t) = (t s l) l m (s, 0 m k. (.5) m s l ) The Lagrange polynomials satisfy the important property L k m(s l ) = δ ml, 0 m, l k. Figure.3 presents families of Lagrange polynomials with equi-distributed nodes in the reference interval [0, ] for k =, 2, and 3. For i {0,..., N}, introduce the nodes ξ i,m = x i + m k h i, 0 m k, in the mesh interval I i ; see Figure.4. Let {L k i,0,..., Lk i,k } be the Lagrange polynomials associated with these nodes. For j {0,..., k(n+)} with j = ki + m and 0 m k, define the function ϕ j elementwise as follows: For m k, { L k i,m ϕ ki+m (x) = (x) if x I i, 0 otherwise, and for m = 0, Fig..3. Families of Lagrange polynomials with equi-distributed nodes in the reference interval [0, ] and of degree k = (left), 2 (center), and 3 (right). mesh vertices PSfrag replacements k = k = 2 k = 3 Fig..4. Mesh vertices and nodes for k =, 2, and 3.

7 .. One-Dimensional Interpolation 9 L k i,k (x) if x I i, ϕ ki (x) = L k i,0 (x) if x I i, 0 otherwise, with obvious modifications if i = 0 or N +. The functions ϕ j are illustrated in Figure.5 for k = 2. Note the difference between the support of the functions associated with mesh vertices (two adjacent intervals) and that of the functions associated with cell midpoints (one interval). Lemma.8. ϕ j P k h. Proof. Let j {0,..., k(n+)} with j = ki + m. If m k, ϕ j (x i ) = ϕ j (x i+ ) = 0; hence, ϕ j C 0 (Ω). Moreover, the restrictions of ϕ j to the mesh intervals are in P k by construction. Therefore, ϕ j Ph k. Now, assume m = 0 (i.e., j = ki) and 0 < i < N +. Clearly, ϕ ki is continuous at x i by construction and ϕ ki (x i ) = ϕ ki (x i+ ) = 0; hence, ϕ ki Ph k. The cases i = 0 and i = N + are treated similarly. Introduce the set of nodes {a j } 0 j k(n+) such that a j = ξ i,m where j = ik + m. For j {0,..., k(n+)}, consider the linear form γ j : C 0 (Ω) v γ j (v) = v(a j ). (.6) Proposition.9. {ϕ 0,..., ϕ k(n+) } is a basis for P k h, and {γ 0,..., γ k(n+) } is a basis for L(P k h ; R). Proof. Similar to that of Proposition. since γ j (ϕ j ) = δ jj k(n + ). for 0 j, j The global degrees of freedom in Ph k are chosen to be the (k(n+)+) linear forms defined in (.6); hence, the global shape functions in Ph k are the functions {ϕ 0,..., ϕ k(n+) }. The main advantage of using high-degree polynomials is that smooth functions can be interpolated to high-order accuracy. Define the interpolation op- PSfrag replacements erator Ih k to be ϕ 2i ϕ 2j+ a x i ξ i, x i ξ i, x i+ x j ξ j, x j+ b Fig..5. Global shape functions in the approximation space P 2 h.

8 0 Chapter. Finite Element Interpolation I k h : C 0 (Ω) v k(n+) j=0 γ j (v)ϕ j P k h. (.7) Ih kv is called the Lagrange interpolant of v of degree k. Clearly, I h k is a linear operator, and Ih kv is the unique function in P h k that takes the same value as v at all the mesh nodes. The approximation space Ph k is the codomain of Ik h. Lemma.0. Ph k H (Ω). Proof. Similar to that of Lemma.3. To investigate the properties of Ih k, it is convenient to introduce a family of local interpolation operators. On I i = [x i, x i+ ] T h, choose the local degrees of freedom to be the (k + ) linear forms {σ i,0,..., σ i,k } defined as follows: σ i,m : P k p σ i,m (p) = p(ξ i,m ), 0 m k. (.8) The triplet {I i, P k, Σ i } is called a (one-dimensional) P k Lagrange finite element, and the points {ξ i,0,..., ξ i,k } are called the nodes of the finite element. Clearly, the local shape functions {θ i,0,..., θ i,k } are the (k + ) Lagrange polynomials associated with the nodes {ξ i,0,..., ξ i,k }, i.e., θ i,m = L k i,m for 0 m k. Finally, introduce the family {I k I i } Ii T h of local interpolation operators such that, for i {0,..., N}, I k I i : C 0 (I i ) v k σ i,m (v)θ i,m, (.9) m=0 i.e., for all 0 i N and v C 0 (Ω), (Ih kv) I i = II k i (v Ii ). Let us show that the family {II k i } Ii T h can be generated from a single reference interpolation operator. Let = [0, ] be the unit interval, henceforth referred to as the reference interval. Set P = P k, and define the (k + ) linear forms { σ 0,..., σ k } as follows: σ m : P k p σ m ( p) = p( ξ m ), 0 m k, (.20) where ξ m = m k. Let { L k 0,..., L k k } be the Lagrange polynomials associated with the nodes { ξ 0,..., ξ k }; see Figure.3. Set θ m = L k m, 0 m k, so that σ m ( θ n ) = δ mn for 0 m, n k. Then, {, P, Σ} is a P k Lagrange finite element, and the corresponding interpolation operator is I k : C 0 ( ) v k σ m ( v) θ m. m=0 {, P, Σ} is called the reference finite element and I k the reference interpolation operator. For i {0,..., N}, consider the affine transformations

9 .. One-Dimensional Interpolation T i : t x = xi + th i I i. (.2) Since T i ( ) = I i, the mesh T h can be constructed by applying the affine transformations T i to the reference interval. Moreover, owing to the fact that T i ( ξ m ) = ξ i,m for 0 m k, it is clear that θ i,m T i = θ m and σ i,m (v) = σ m (v T i ) for all v C 0 (I i ). Hence, using I k I i (v)(t i ( x)) = = k k σ i,m (v)θ i,m (T i ( x)) = σ i,m (v) θ m ( x) = m=0 m=0 k σ m (v T i ) θ m ( x) = I k (v T i )( x), m=0 we infer v C 0 (I i ), I k I i (v) T i = I k (v T i ). (.22) In other words, the family {II k i } Ii T h is entirely generated by the transformations {T i } Ii T h and the reference interpolation operator I k. The property (.22) plays a key role when estimating the interpolation error; see the proof of Proposition.2 below. Proposition.. I k h is a linear continuous mapping from H (Ω) to H (Ω), and I k h L(H (Ω);H (Ω)) is uniformly bounded with respect to h. Proof. () To prove that Ih k maps H (Ω) to H (Ω), use the argument of step in the proof of Proposition.4. (2) Let v H (Ω) and I i T h. Since k m=0 θ i,m = 0, (I k I i v) = k [v(ξ i,m ) v(x i )]θ i,m. m=0 Inequality (.8) yields v(ξ i,m ) v(x i ) h 2 i v,ii for 0 m k. Furthermore, changing variables in the integral, it is clear that θ i,m,ii = h 2 i θ m,. Set c k = max 0 m k θ m, and observe that this quantity is mesh-independent. A straightforward calculation yields I k I i v,ii (k + )c k v,ii, showing that I k h v,ω is controlled by v,ω uniformly with respect to h. In addition, since k m=0 θ i,m =, I k I i v v(x i ) = k [v(ξ i,m ) v(x i )]θ i,m, m=0

10 2 Chapter. Finite Element Interpolation implying, for x I i, II k i v(x) v L (Ω) + (k + )d k h 2 i v,ii with the meshindependent constant d k = max 0 m k θ m L ( ). Then, using(.9) yields Ih kv L (Ω) is controlled by v,ω uniformly with respect to h. To conclude, use the fact that Ih kv 0,Ω b a 2 Ih kv L (Ω). we Proposition.2. Let 0 l k. Then, there exists c such that, for all h and v H l+ (Ω), and for l, v I k hv 0,Ω + h v I k hv,ω c h l+ v l+,ω, (.23) l+ m=2 h m ( N i=0 v I k hv 2 m,i i ) 2 c h l+ v l+,ω. (.24) Proof. Let 0 l k and 0 m l +. Let v H l+ (Ω). () Consider a mesh interval I i. Set v = v T i. Then, use (.22) and change variables in the integral to obtain Similarly, v l+, = h l+ 2 i v l+,ii. (2) Consider the linear mapping v II k i v m,ii = h m+ 2 i v I k v m,. F : H l+ ( ) v v I k v H m ( ). Note that I k v is meaningful since in one dimension, v H l+ ( ) with l 0 implies v C 0 ( ). Moreover, F is continuous from H l+ ( ) to H m ( ). Indeed, one can easily adapt the proof of Proposition. to prove that I k is continuous from H ( ) to H s ( ) for all s. Furthermore, it is clear that P k is invariant under F since, for all p P k with p = k n=0 α n θ n, I p = k m,n=0 α n σ m ( θ n ) θ m = k m,n=0 α n δ mn θm = (3) Since l k, P l is invariant under F. As a result, k α n θn = p. n=0 v I k v m, = F( v) m, = inf p P l F( v + p) m, F L(H l+ ( );H m ( )) inf p P l v + p l+, c inf p P l v + p l+, c v l+,,

11 .. One-Dimensional Interpolation 3 the last estimate resulting from the Deny Lions Lemma; see Lemma B.67. The identities derived in step yield v II k i v m,ii = h m+ 2 i v I k v m, c h m+ 2 i v l+, c h l+ m i v l+,ii. (4) To derive the estimates (.23) and (.24), sum over the mesh intervals. When m = 0 or, global norms over Ω can be used since Ph k H (Ω) owing to Lemma.0. Remark.3. (i) The proof of Proposition.2 shows that the interpolation properties of Ih k are local. (ii) If the function to be interpolated is smooth enough, say v H k+ (Ω), the interpolation error is of optimal order. In particular, (.23) yields h, v H k+ (Ω), v I k hv 0,Ω + h v I k hv,ω c h k+ v k+,ω. However, one should bear in mind that the order of the interpolation error may not be optimal if the function to be interpolated is not smooth. For instance, if v H s (Ω) and v H s+ (Ω) with s 2, considering polynomials of degree larger than s does not improve the interpolation error. (iii) If the function to be interpolated is in H (Ω) only, one can still prove lim h 0 v I k h v,ω = 0. To this end, use the density of H 2 (Ω) in H (Ω) and (.23); details are left as an exercise...4 Interpolation by discontinuous functions Let P k d,h = {v h L (Ω); i {0,..., N}, v h Ii P k }. Since the restriction of a function v h Pd,h k to an interval I i can be chosen independently of its restriction to the other intervals, Pd,h k is a vector space of dimension (k + ) (N + ). However, instead of taking the Lagrange polynomials as local shape functions, it is often more convenient to consider the Legendre polynomials or modifications thereof based on the concept of hierarchical bases; see..5. Let = [0, ] be the reference interval. Definition.4 (Legendre polynomials). The Legendre polynomials on the reference interval [0, ] are defined to be Êk(t) = d k k! (t 2 t) k for k 0. dt k The Legendre polynomial Êk is of degree k, Êk(0) = ( ) k, Êk() =, and its k roots are in. The roots of the Legendre polynomials are called Gauß Legendre points and play an important role in the design of quadratures; see 8.. The first four Legendre polynomials are (see Figure.6)

12 4 Chapter. Finite Element Interpolation Fig..6. Legendre polynomials of degree at most 3 on the reference interval [0, ]. Ê 0 (t) =, Ê (t) = 2t, Ê 2 (t) = 6t 2 6t +, Ê 3 (t) = 20t 3 30t 2 + 2t. In the literature, the Legendre polynomials are sometimes defined using the reference interval [, +]. Up to rescaling, both definitions are equivalent. In the context of finite elements, an important property of Legendre polynomials is that Ê m (t)ên(t) dt = 2m+ δ mn. (.25) 0 Introduce the functions {ϕ i,m } 0 i N,0 m k such that ϕ i,m Ij = δ ij Ê m T i where the geometric transformation T i is defined in (.2). Clearly, {ϕ i,m } 0 i N,0 m k is a basis for Pd,h k. The corresponding degrees of freedom are the linear forms γ i,m, 0 i N and 0 m k, such that γ i,m : L (Ω) v γ i,m (v) = 2m+ h i v(x) Êm T i (x) dx, I i since, for 0 i, i N and 0 m, m k, γ i,m (ϕ i,m ) 2m+ = h i ϕ i,m (x) Êm T i (x) dx I i = (2m + )δ ii δ mm Ê m (t) 2 dt = δ ii δ mm. Define the interpolation operator I k d,h by I k d,h : L (Ω) v N k γ i,m (v)ϕ i,m Pd,h. k (.26) i=0 m=0

13 .. One-Dimensional Interpolation 5 For instance, Id,h 0 v is the unique piecewise constant function that takes the same mean value as v over the mesh intervals. Let I i = [x i, x i+ ] T h and choose for the local degrees of freedom in P k the set Σ i = {γ i,m } 0 m k. The triplet {I i, P k, Σ i } is often called a modal finite element; see..5 for further insight. The local shape functions are θ i,m = Ê m T i. Introduce the family {Id,I k i } Ii T h of local interpolation operators such that, for 0 i N, I k d,i i : L (I i ) v k σ i,m (v)θ i,m. (.27) m=0 Then, it is clear that, for all v L (Ω), (Id,h k v) I i = Id,I k i (v Ii ). Using the family {Id,I k i } Ii T h, one easily verifies the following results: Proposition.5. I k d,h is a linear continuous mapping from L (Ω) to L (Ω), and I k d,h L(L (Ω);L (Ω)) is uniformly bounded with respect to h. Proposition.6. Let k 0 and let 0 l k. Then, there exists c such that, for all h and v H l+ (Ω), v I k d,hv 0,Ω + l+ m= h m ( N i=0 v I k d,hv 2 m,i i ) 2 c h l+ v l+,ω. Proof. Use steps, 2, and 3 in the proof of Proposition.2. Example.7. Taking k = l = 0 in Proposition.6 yields, for all h and v H (Ω), v Id,h 0 v 0,Ω c h v,ω...5 Hierarchical polynomial bases Although the emphasis in this book is set on h-type finite element methods for which convergence is achieved by refining the mesh, it is also possible to consider p-type finite element methods for which convergence is achieved by increasing the polynomial degree of the interpolation in every element. The hp-type finite element method is a combination of these two strategies. The idea that the p version of the finite element method can be as efficient as the h version is rooted in a series of papers by Babuška et al. [BaS8, BaD8]. When working with high-degree polynomials, it is important to select carefully the polynomial basis. The material presented herein is set at an introductory level; see, e.g., [as99 b, pp. 3 59]. The following definition plays an important role in the construction of polynomial bases: Definition.8 (Hierarchical modal basis). A family {B k } k 0, where B k is a set of polynomials, is said to be a hierarchical modal basis if, for all k 0: (i) B k is a basis for P k.

14 6 Chapter. Finite Element Interpolation (ii) B k B k+. Example.9. The simplest example of hierarchical modal basis is B k = {, x,..., x k }. So far, the local shape functions { θ 0,..., θ k } we have used are the Lagrange polynomials { L k 0,..., L k k } or the Legendre polynomials {Ê0,..., Êk}. Clearly, the Legendre polynomial basis is a hierarchical modal basis. This is not the case for the Lagrange polynomial basis, which instead has the remarkable property that L k l ( ξ l ) = δ ll at the associated nodes { ξ 0,..., ξ k }. Because of this property, the Lagrange polynomial basis is said to be a nodal basis. A first important criterion to select a high-degree polynomial basis is that the basis is orthogonal or nearly orthogonal with respect to an appropriate inner product. Let = [0, ] be the reference interval and define the matrix M of order k + with entries m, n {0,..., k}, M,mn = θ m (t) θ n (t) dt. (.28) The matrix M is symmetric positive definite and is called the elemental mass matrix. The high-degree polynomial basis can be constructed so that M is diagonal or almost diagonal. Define the condition number of M to be the ratio between its largest and smallest eigenvalue; see 9.. Instead of diagonality, an alternative criterion to select a polynomial basis can be that the condition number of M does not increase too much as k grows; see Remark.20(i). A second important criterion is that interface conditions between adjacent mesh elements can be imposed easily. For instance, imposing continuity at the interfaces ensures that the codomain of the global interpolation operator is in H (Ω); see, e.g., Lemmas.3 and.0. Remark.20. (i) The conditioning of the elemental mass matrix has important consequences on computational efficiency. For instance, in time-dependent problems discretized with explicit time-marching algorithms, this matrix has to be inverted at each time step; see, e.g., (6.27). Furthermore, for time-dependent advection problems, explicit time step restrictions are less severe when the elemental mass matrix is well-conditioned; see [as99 b, p. 87] and also Exercises 6.7 and 6.9. (ii) Instead of the elemental mass matrix, one can also consider the elemental stiffness matrix A defined by d m, n {0,..., k}, A,mn = θ dt m (t) d θ dt n (t) dt. This matrix, which is symmetric and positive, arises when approximating the Laplace equation; see 3.. The high-degree polynomial basis can then be constructed so that A remains relatively well-conditioned.

15 .. One-Dimensional Interpolation 7 The Legendre polynomial basis satisfies the first criterion above. Owing to (.25), the mass matrix is diagonal and its condition number is (2k + ). However, Legendre polynomials do not vanish at the boundary of, making it cumbersome to enforce C 0 -continuity between adjacent mesh intervals. On the other hand, the Lagrange polynomial basis satisfies the C 0 -continuity criterion provided the nodal points contain the interval endpoints, but the mass matrix is dense and its condition number explodes exponentially with k; see [OlD95] for a proof and [as99 b, p. 44] for an illustration. We now discuss appropriate modifications of the above bases designed to better fulfill the above criteria. Modal (C 0 -continuous) basis. We first define the Jacobi polynomials. Definition.2 (Jacobi polynomials). Let α > and β >. The Jacobi polynomials {J α,β k } k 0 are defined by J α,β k (t) = ( )k k! 2 α β ( t) α ( β dk t dt ( t) α+k t β+k). (.29) k The Jacobi polynomials satisfy the important orthogonality property ( t) α t β Jm α,β (t)jn α,β (t) dt = c m,α,β δ mn, (.30) with constant c m,α,β = 2m+α+β+ Γ (m+α+)γ (m+β+) m!γ (m+α+β+). The first Jacobi polynomials for α = β = are J, 0 (t) =, J, (t) = 4t 2, and J, 2 (t) = 5t 2 5t + 3. Note that the Legendre polynomials introduced in Definition.4 are Jacobi polynomials with parameters α = β = 0. For more details on Jacobi polynomials, see [AbS72, Chap. 22] and [as99 b, p. 350]. The modal (C 0 -continuous) basis is the set of functions { θ 0,..., θ k } such that t if l = 0, θ l (t) = ( t)t J, l (t) if 0 < l < k, t if l = k. This basis possesses several attractive features: (.3) (i) It is a hierarchical modal basis according to Definition.8. (ii) C 0 -continuity at element endpoints can be easily enforced since only the first and last basis functions do not vanish at the endpoints. (iii) Owing to the use of Jacobi polynomials with parameters α = β =, the elemental mass matrix M is such that M,mn = 0 for m n > 2 and 0 m, n k, unless m = k and n 2 or n = k and m 2. Furthermore, this matrix remains relatively well-conditioned. A precise result in arbitrary dimension d using tensor products of modal hierarchical bases is that the condition number of the elemental mass matrix (resp., stiffness matrix) is equivalent to 4 kd (resp., 4 k(d ) ) uniformly in k; see [HuG98].

16 8 Chapter. Finite Element Interpolation Fig..7. Left: Modal (C 0 -continuous) basis functions of degree at most 4 on the reference interval [0, ]. Right: Nodal (C 0 -continuous) basis functions of degree at most 3 on the same interval. The modal (C 0 -continuous) basis functions are shown in the left panel of Figure.7 for k = 5. Remark.22. Note that in the present case the degrees of freedom have no evident definition. It is more natural to define directly the local shape functions without resorting to the notion of degrees of freedom. Nodal (C 0 -continuous) basis. Nodal basis functions are interesting in the context of quadratures; see 8. for an introduction to these techniques. The principle of quadratures is to approximate the integral of a function over by a linear combination of the values it takes at (k + ) points in, say { ξ 0,..., ξ k }, in the form f(t) dt k ω l f( ξ l ). (.32) The points { ξ 0,..., ξ k } are called the quadrature nodes and the numbers { ω 0,..., ω k } the quadrature weights. For k 2, the Gauß Lobatto quadrature nodes are defined to be the two endpoints of and the (k ) roots of Ê k. The resulting quadrature rule is exact for polynomials up to degree 2k. Define the local degrees of freedom { σ 0,..., σ k } such that, for 0 i k, l=0 σ i : P k p σ i ( p) = p( ξ i ) R. Then, the local shape functions { θ 0,..., θ k } are the Lagrange polynomials associated with the nodes { ξ 0,..., ξ k }. Using standard induction relations on the Legendre polynomials, it is possible to show that the local shape functions { θ 0,..., θ k } are given by

17 .2. Finite Elements: Definitions and Examples 9 m {0..., k}, θm (t) = (t )tê k (t) k(k + )Êk( ξ m )(t ξ m ). (.33) These functions are shown in the right panel of Figure.7 for k = 4. Although these nodal basis functions are not hierarchical, they present attractive features in the context of spectral element methods; see [as99 b, p. 5] and [Pat84] for more details. If the quadrature (.32) is used to evaluate M in (.28), the elemental mass matrix becomes diagonal, and each diagonal entry is equal to the row-wise sum of the entries of the exact elemental matrix. Summing rowwise the entries of the mass matrix and using the result as diagonal entries is often referred to as lumping..2 Finite Elements: Definitions and Examples The purpose of this section is to give a general definition of finite elements and local interpolation operators. Numerous two- and three-dimensional examples are listed..2. Main definitions Following Ciarlet, a finite element is defined as a triplet {, P, Σ}; see, e.g., [Cia9, p. 93]. Definition.23. A finite element consists of a triplet {, P, Σ} where: (i) is a compact, connected, Lipschitz subset of R d with non-empty interior. (ii) P is a vector space of functions p : R m for some positive integer m (typically m = or d). (iii) Σ is a set of n sh linear forms {σ,..., σ nsh } acting on the elements of P, and such that the linear mapping P p ( σ (p),..., σ nsh (p) ) R n sh, (.34) is bijective, i.e., Σ is a basis for L(P ; R). The linear forms {σ,..., σ nsh } are called the local degrees of freedom. Proposition.24. There exists a basis {θ,..., θ nsh } in P such that σ i (θ j ) = δ ij, i, j n sh. Proof. Direct consequence of the bijectivity of the mapping (.34). Definition.25. {θ,..., θ nsh } are called the local shape functions.

18 20 Chapter. Finite Element Interpolation Remark.26. Condition (iii) in Definition.23 amounts to proving that (α,..., α nsh ) R n sh,!p P, σ i (p) = α i for i n sh, which, in turn, is equivalent to { dim P = card Σ = nsh, p P, (σ i (p) = 0, i n sh ) = (p = 0). This property is usually referred to as unisolvence. In the literature, the bijectivity of the mapping (.34) is sometimes not included in the definition and, if this property holds, the finite element is said to be unisolvent. Definition.27 (Lagrange finite element). Let {, P, Σ} be a finite element. If there is a set of points {a,..., a nsh } in such that, for all p P, σ i (p) = p(a i ), i n sh, {, P, Σ} is called a Lagrange finite element. The points {a,..., a nsh } are called the nodes of the finite element, and the local shape functions {θ,..., θ nsh } (which are such that θ i (a j ) = δ ij for i, j n sh ) are called the nodal basis of P. Example.28. See..2 and..3 for one-dimensional examples of Lagrange finite elements. Remark.29. In the literature, Lagrange finite elements as defined above are also called nodal finite elements..2.2 Local interpolation operator Let {, P, Σ} be a finite element. Assume that there exists a normed vector space V () of functions v : R m, such that: (i) P V (). (ii) The linear forms {σ,..., σ nsh } can be extended to V (). Then, the local interpolation operator I can be defined as follows: I : V () v n sh σ i (v)θ i P. (.35) i= V () is the domain of I and P is its codomain. Note that the term interpolation is used in a broad sense since I v is not necessarily defined by matching point values of v. Proposition.30. P is invariant under I, i.e., p P, I p = p. Proof. Letting p = n sh j= α jθ j yields I p = n sh i,j= α jσ i (θ j )θ i = p.

19 .2. Finite Elements: Definitions and Examples 2 Example.3. (i) For Lagrange finite elements, one may choose V () = [C 0 ()] m or V () = [H s ()] m with s > d 2. The local Lagrange interpolation operator is defined as follows: n sh I : V () v I v = v(a i )θ i, (.36) i.e., the Lagrange interpolant is constructed by matching the point values at the Lagrange nodes. (ii) For the modal finite elements discussed in..4, an admissible choice is V () = L (). Remark.32. It may seem more appropriate to define a finite element as a quadruplet {, P, Σ, V ()}, where the triplet {, P, Σ} complies with Definition.23 and V () satisfies properties (i) (ii). However, for the sake of simplicity, we hereafter employ the well-established triplet-based definition, and always implicitly assume that there exists a normed vector space V () satisfying properties (i) (ii). In many textbooks, V () is implicitly assumed to be of the form C s () for some integer s 0; see, e.g., [Cia9, p. 96] or [BrS94, p. 79]. i=.2.3 Simplicial Lagrange finite elements Simplices and barycentric coordinates. Let {a 0,..., a d } be a family a points in R d, d. Assume that the vectors {a a 0,..., a d a 0 } are linearly independent. Then, the convex hull of {a 0,..., a d } is called a simplex, and the points {a 0,..., a d } are called the vertices of the simplex. The unit simplex of R d is the set { } d x R d ; x i 0, i d, and x i. A simplex can be equivalently defined to be the image of the unit simplex by a bijective affine transformation. For 0 i d, define F i to be the face of opposite to a i, and define n i to be the outward normal to F i. Note that in dimension 2 a face is also called an edge, but this distinction will not be made unless necessary. Given a simplex in R d, it is often convenient to consider the associated barycentric coordinates {λ 0,..., λ d } defined as follows: For 0 i d, λ i : R d x λ i (x) = (x a i) n i (a j a i ) n i R, (.37) where a j is an arbitrary vertex in F i (the definition of λ i is clearly independent of the choice of the vertex in F i ). The barycentric coordinate λ i is an affine i=

20 22 Chapter. Finite Element Interpolation function; it is equal to at a i and vanishes at F i. Furthermore, its level-sets are hyperplanes parallel to F i. Note that the barycenter G of has barycentric coordinates ( d+,..., d+ ). The barycentric coordinates satisfy the following properties: For all x, 0 λ i (x), and for all x R d, d+ λ i (x) = i= and d+ λ i (x)(x a i ) = 0. See Exercise.4 for further properties in dimension 2 and 3. Example.33. In the unit simplex, λ 0 = x x 2, λ = x, and λ 2 = x 2 in dimension 2, and λ 0 = x x 2 x 3, λ = x, λ 2 = x 2, λ 3 = x 3 in dimension 3. The polynomial space P k. Let x = (x,..., x d ) and let P k be the space of polynomials in the variables x,..., x d, with real coefficients and of global degree at most k, P k = p(x) = 0 i,...,i d k i +...+i d k i= α i...i d x i... xi d d ; α i...i d R. One readily verifies that P k is a vector space of dimension ( ) k + if d =, d + k dim P k = = k 2 (k + )(k + 2) if d = 2, (k + )(k + 2)(k + 3) if d = 3. 6 Proposition.34. Let be a simplex in R d. Let k, let P = P k, and let n sh = dim P k. Consider the set of nodes {a i } i nsh with barycentric coordinates ( i0 k,..., i ) d k, 0 i0,..., i d k, i i d = k. Let Σ = {σ,..., σ nsh } be the linear forms such that σ i (p) = p(a i ), i n sh. Then, {, P, Σ} is a Lagrange finite element. Proof. See Exercise.3. Table. presents examples for k =, 2, and 3 in dimension 2 and 3. For k =, the (d + ) local shape functions are the barycentric coordinates θ i = λ i, 0 i d. For k = 2, the local shape functions are { λi (2λ i ), 0 i d, 4λ i λ j, 0 i < j d,

21 .2. Finite Elements: Definitions and Examples 23 P P 2 P 3 Table.. Two- and three-dimensional P, P 2, and P 3 Lagrange finite elements; in three dimensions, only visible degrees of freedom are shown. and for k = 3, 2 λ i(3λ i )(3λ i 2), 0 i d, 9 2 λ i(3λ i )λ j, 0 i, j d, i j, 27λ i λ j λ k, 0 i < j < k d..2.4 Tensor product Lagrange finite elements Cuboids. Given a set of d intervals {[c i, d i ]} i d, all with non-zero measure, the set = d i= [c i, d i ] is called a cuboid. For x, there exists a unique vector (t,..., t d ) [0, ] d such that, for all i d, x i = c i + t i (d i c i ). The vector (t,..., t d ) is called the local coordinate vector of x in. The polynomial space Q k. Let Q k be the polynomial space in the variables x,..., x d, with real coefficients and of degree at most k in each variable. In dimension, Q k = P k ; in dimension d 2, Q k = q(x) = 0 i,...,i d k α i...i d x i... xi d d ; α i...i d R. One readily verifies that Q k is a vector space of dimension

22 24 Chapter. Finite Element Interpolation Q Q 2 Q 3 Table.2. Two- and three-dimensional Q, Q 2, and Q 3 Lagrange finite elements; in three dimensions, only visible degrees of freedom are shown. dim Q k = (k + ) d = Note the inclusions P k Q k P kd. { (k + ) 2 if d = 2, (k + ) 3 if d = 3. Proposition.35. Let be a cuboid in R d. Let k, let P = Q k, and let n sh = dim Q k. Consider the set of nodes {a i } i nsh with local coordinates ( ik,..., i ) d k, 0 i,..., i d k. Let Σ = {σ,..., σ nsh } be the linear forms such that σ i (p) = p(a i ), i n sh. Then, {, P, Σ} is a Lagrange finite element. Table.2 presents examples for k =, 2, and 3 in dimension 2 and 3. For i d, set ξ i,l = c i + l k (d i c i ), 0 l k, and let {L k i,0,..., Lk i,k } be the Lagrange polynomials in the variable x i associated with the nodes {ξ i,0,..., ξ i,k }; see Definition.7. Then, the local shape functions are θ i...i d (x) = L k,i (x )... L k d,i d (x d ), 0 i,..., i d k..2.5 Prismatic Lagrange finite elements Prisms. For x R d, set x = (x,..., x d ). Let be a simplex in R d and let [a, b] be an interval with non-zero measure. Then, the set = {x R d ; x ; x d [a, b]} is called a prism. Let (λ 0,..., λ d ) be the barycentric coordinates of x in and let t [0, ] be such that x d = a + t(b a). Then, the prismatic coordinates of x are defined to be (λ 0,..., λ d ; t).

23 .2. Finite Elements: Definitions and Examples 25 PR PR 2 PR 3 Table.3. Prismatic Lagrange finite elements of degree, 2, and 3; only visible degrees of freedom are shown. Prismatic polynomials. Let P k [x ] (resp., P k [x d ]) be the set of polynomials with real coefficients in the variable x (resp., x d ) of global degree at most k. Set PR k = {p(x) = p (x ) p 2 (x d ); p P k [x ], p 2 P k [x d ]}. Clearly, P k PR k and dim PR k = 2 (k + )2 (k + 2) in dimension 3. Proposition.36. Let be a prism in R d. Let k, let P = PR k, and let n sh = dim PR k. Consider the set of nodes {a i } i nsh with prismatic coordinates ( ) i0 k,..., i d k ; i d k, 0 i 0,..., i d, i d k, i i d = k. Let Σ = {σ,..., σ nsh } be the linear forms such that σ i (p) = p(a i ), i n sh. Then, {, P, Σ} is a Lagrange finite element. Table.3 presents examples for k =, 2, and 3. The local shape functions can be expressed in tensor product form using the local shape functions on the simplex and the Lagrange polynomials in x d..2.6 The Crouzeix Raviart finite element Let be a simplex in R d, set P = P, and take for the local degrees of freedom the mean-value over the (d + ) faces of, i.e., for 0 i d, σ i (p) = p. meas(f i ) F i Proposition.37. Let Σ = {σ i } 0 i d. Then, {, P, Σ} is a finite element. Using the barycentric coordinates {λ 0,..., λ d } defined in (.37), the local shape functions are ) θ i (x) = d( d λ i(x), 0 i d. (.38)

24 26 Chapter. Finite Element Interpolation Fig..8. Crouzeix Raviart finite element in two (left) and three (right) dimensions; in three dimensions, only visible degrees of freedom are shown. Indeed, θ i P and σ j (θ i ) = δ ij for 0 i, j d. Note that θ i Fi = and θ i (a i ) = d. A conventional representation of the Crouzeix Raviart finite element is shown in Figure.8. The dot means that the mean-value is taken over the corresponding face. This finite element has been introduced by Crouzeix and Raviart; see [CrR73] and also [BrF9 b, pp ]. An admissible choice for the domain of the local interpolation operator is V () = W, (). Indeed, owing to the Trace Theorem B.52 applied with p =, the trace of a function in W, () is in L ( ). The local Crouzeix Raviart interpolation operator is then defined as follows: I CR : V () v I CR v = d ( i=0 meas F i F i v ) θ i P. (.39) Remark.38. (i) Since a polynomial in P is linear, its mean-value over a face is equal to the value it takes at the barycenter. Therefore, another possible choice for the degrees of freedom is to take the value at the face barycenters. The resulting finite element is a Lagrange finite element according to Definition.27. The only difference with the Crouzeix Raviart finite element is that it is no longer possible to take W, () for the domain of the local interpolation operator; an admissible choice is, for instance, V () = C 0 (). (ii) Another choice for the local degrees of freedom is σ i (p) = F i p for ( d 0 i d; then, the local shape functions are θ i = meas(f i) d λ ) i..2.7 The Raviart Thomas finite element Let be a simplex in R d. Consider the vector space of R d -valued polynomials RT 0 = [P 0 ] d x P 0. (.40) Clearly, the dimension of RT 0 is d +. For p RT 0, the local degrees of freedom are chosen to be the value of the flux of the normal component of p across the faces of, i.e., for 0 i d,

25 .2. Finite Elements: Definitions and Examples 27 Fig..9. Raviart Thomas finite element in two (left) and three (right) dimensions; in three dimensions, only visible degrees of freedom are shown. σ i (p) = p n i. F i Proposition.39. Let Σ = {σ i } 0 i d. Then, {, RT 0, Σ} is a finite element. The local shape functions are θ i (x) = d meas() (x a i), 0 i d. (.4) Indeed, θ i RT 0 and σ j (θ i ) = δ ij for 0 i, j d. Note that the normal component of a local shape function is constant on the face with which it is associated and is zero on the other faces. A conventional representation of the degrees of freedom of the Raviart Thomas finite element is shown in Figure.9. An arrow means that the flux of the normal component is taken over the corresponding face. This finite element has been introduced by Raviart and Thomas and is often referred to as the RT 0 finite element [RaT77]. It is used, for instance, in applications related to fluid mechanics where the functions to be interpolated are velocities. The domain of the local interpolation operator can be taken to be V div () = {v [L p ()] d ; v L s ()} for p > 2, s q, q = p + d. 2d d+2 Note that V div () = W,t () with t > is also an admissible choice. Indeed, one can show that for v V div () and for a face F i of, the quantity F i v n i is meaningful. The local Raviart Thomas interpolation operator is then defined as follows: I RT : V div () v I RT v = d ( i=0 F i v n i ) θ i RT 0. (.42) Remark.40. (i) See Exercise.5 for the proofs of the above results and for an alternative expression of the local shape functions in terms of barycentric coordinates. Further results can be found in [BrF9 b, p. 3] and [QuV97, p. 82]. (ii) In the spirit of Remark.38, the Raviart Thomas finite element can

26 28 Chapter. Finite Element Interpolation be defined as a Lagrange finite element. Another choice for the degrees of freedom is σ i (p) = meas(f i) F i p n i ; then, the local shape functions are θ i = meas (F i) d meas() (x a i). Lemma.4. Let I RT be defined in (.42). Let π0 be the orthogonal projection from L 2 () to P 0. The following diagram commutes: V div () L 2 () Proof. Left as an exercise. I RT RT 0 π0 P0.2.8 The Nédélec (or edge) finite element Let be a simplex in R d, d = 2 or 3. Define the polynomial space of dimension 2 d(d + ), N0 = [P0] d R, R = {p [P] d ; x p = 0}. (.43) Introducing the mapping R : R 2 R 2 such that R(x, x 2 ) = (x 2, x ), the following equivalent definition of N 0 holds in dimension 2: N 0 = [P 0 ] 2 (R(x)P 0 ). (.44) In dimension 3, the following equivalent definition of N 0 holds: N 0 = [P 0 ] 3 (x [P 0 ] 3 ). (.45) For p N 0, the local degrees of freedom are chosen to be the integral of the tangential component of p along the three (resp., six) edges of in two (resp., three) dimensions. Set n e = 3 if d = 2 and n e = 6 if d = 3. Denote by {e i } i ne the set of edges of and, for each edge e i, let t i be one of the two unit vectors parallel to e i. For i n e, the local degrees of freedom are σ i (p) = p t i. e i Proposition.42. Let Σ = {σ i } i ne. Then, {, N 0, Σ} is a finite element. In two dimensions, the local shape function associated with the edge e i, i 3, is R(x a i ) θ i (x) = t i [R( ai +ai 2 2 a i ) ] meas(e i ), i, i 2 i. (.46)

27 .2. Finite Elements: Definitions and Examples 29 Fig..0. Edge finite element in dimension 2 (left) and 3 (right); in three dimensions, only visible degrees of freedom are shown. In three dimensions, define the mapping j : {,..., 6} {,..., 6} such that j(i) is the index of the edge opposite to e i, i.e., e i does not intersect e j(i). Note that j = j. Let m i be the midpoint of e i. Then, the local shape function associated with the edge e i, i 6, is θ i (x) = (x m j(i) ) t j(i) t i [(m i m j(i) ) t j(i) ] meas(ei ). (.47) In both cases, the tangential component of a local shape function is constant along the edge with which it is associated and vanishes along the other edges. A conventional representation of the edge finite element is shown in Figure.0. An arrow means that the integral of the component parallel to this direction is taken over the corresponding edge. This finite element has been introduced by Nédélec [Néd80, Néd86]; see also [Whi57]. It is used, for instance, in electromagnetism and in magneto-hydrodynamics; see [Bos93, Chap. 3]. In two dimensions, the domain of the local interpolation operator can be taken to be V curl () = {v = (v, v 2 ) [L p ()] 2 ; 2 v v 2 L p ()} for p > 2. Indeed, one can show that for v V curl () and for an edge e i of, the quantity e i v t i is meaningful. In three dimensions, a suitable choice is V curl () = {v [H s ()] 3 ; v [L p ()] 3 } for s > 2 and p > 2; see, e.g., [AmB98]. The local Nédélec interpolation operator is then defined as follows: n e ( ) I N : V curl () v Iv N = v t i θ i N 0. (.48) e i Remark.43. (i) See Exercise.6 for the proofs of the above results and for an alternative expression of the local shape functions in terms of barycentric coordinates. (ii) In the spirit of Remark.38, the Nédélec finite element can be defined as a Lagrange finite element. Another choice for the degrees of freedom is σ i (p) = meas(e i) e i p t i for i n e ; the local shape functions are then readily derived from (.46) and (.47). Lemma.44. Assume d = 3. Let I RT and IN be defined in (.42) and (.48), respectively. The following diagram commutes: i=

28 30 Chapter. Finite Element Interpolation V curl () V div () I N I RT N 0 RT0 Proof. Let v V curl (). It is clear that I Nv [P 0] 3 RT 0. Let F be a face of and n F be the corresponding outward normal. Then, ( (Iv)) n N F = Iv t N e = v t e F = e F F e ( v) n F = e F F e (I RT ( v)) n F, where t e is a unit vector parallel to the edge e so that the edge integrals are taken anti-clockwise along F. The above equality implies I RT( v) = (I Nv), since these two functions are in RT 0 and their fluxes across the faces of are identical. Lemma.45. Assume d = 2 or 3. Let I be the interpolation operator associated with the P Lagrange finite element and let V () = H s () be its domain, s > d 2. The following diagram commutes: V () V curl () P I IN N0 Proof. Le v V (). Let e be an edge of and denote by a, a 2 the two vertices of e. Set t = to obtain e a2 a a 2 a d (Iv) t = Iv(a 2 ) Iv(a ) = v(a 2 ) v(a ) = ( v) t = I( v) t. N e e Conclude using the fact that both I N ( v) and (I v) belong to N High-order finite elements As in the one-dimensional case, basis functions must be selected carefully when working with high-degree polynomials.

29 .3. Meshes: Basic Concepts 3 Nodal finite elements. When is a simplex in R d and P = P k with k large, it is possible to define sets of quadrature points with near optimal interpolation properties: the so-called Fekete points; see [ChB95, TaW00]. Then, these points can be used as Lagrange nodes to define nodal bases. Finite element methods using the Fekete points as Lagrange nodes when k is large are often referred to as spectral element methods; see, e.g., [as99 b ]. When is a cuboid in R d and P = Q k, one can use the tensor product of Gauß Lobatto nodes instead of equi-distributing the Lagrange nodes in each space direction. Then, the local shape functions are θ i,...,i d (x,..., x d ) = θ i (x )... θ id (x d ), 0 i,..., i d k, (.49) where the functions {θ i } 0 i k are the images by suitable mappings of the nodal (C 0 -continuous) basis functions defined in (.33). An interesting property of the Gauß Lobatto points is that they are the Fekete points for the d-dimensional cuboid, i.e., these points have near optimal interpolation properties; see [BoT0]. Modal finite elements. When is a cuboid, hierarchical modal bases can be constructed using tensor products of one-dimensional hierarchical modal bases. For instance, we can consider the basis functions defined in (.49), where the functions {θ i } 0 i k are now the images by suitable mappings of the modal (C 0 -continuous) basis functions defined in (.3). When is a simplex or a prism, the construction of hierarchical bases is more technical. The idea is to introduce a nonlinear transformation mapping to a square or a cube and to use tensor products of one-dimensional bases. See [as99 b, pp ] for a detailed presentation..3 Meshes: Basic Concepts This section presents the general principles governing the construction of a mesh. Implementation aspects are investigated in Chapter Domains and meshes Throughout this book, we shall use the following: Definition.46 (Domain). In dimension, a domain is an open, bounded interval. In dimension d 2, a domain is an open, bounded, connected set in R d such that its boundary Ω satisfies the following property: There are α > 0, β > 0, a finite number R of local coordinate systems x r = (x r, x r d ), r R, where x r R d and x r d R, and R local maps φr that are Lipschitz on their definition domain {x r R d ; x r < α} and such that