Part I, Chapter 3 Simplicial finite elements This chapter deals with finite elements (K,P,Σ) where K is a simplex in ♥♥ Rd with d 2, the degrees of freedom Σ are either nodal values in K or ≥ integrals over the faces (or edges) of K, and P is the space Pk,d composed of multivariate polynomials of total degree at most k 0. We focus our attention on scalar-valued finite elements, but most of the results≥ in this chapter can be extended to the vector-valued case by reasoning componentwise. Examples of vector-valued elements where the normal and the tangential components at the boundary of K play specific roles are studied in Chapters 8 and 9. 3.1 Simplices and barycentric coordinates Simplices in Rd, d 1, are ubiquitous objects in the finite element theory. We review important properties≥ of simplices in this section. 3.1.1 Simplices Definition 3.1 (Simplex, vertices). Let d 1. Let zi i 0: d be a set of d ≥ { } ∈{ } points in R such that the vectors z1 z0,..., zd z0 are linearly inde- pendent. Then, the convex hull of these{ − points is called− a}simplex in Rd, say K = conv(z0,..., zd). By definition, K is a closed set. If d = 1, a simplex is an interval [xl,xr] with real numbers xl < xr; if d = 2, a simplex is a triangle; and if d = 3, a simplex is a tetrahedron. The points zi i 0: d are the vertices of K. { } ∈{ } Example 3.2 (Unit simplex). The unit simplex in Rd corresponds to the d choice x R 0 xi 1, i 1:d , i 1: d xi 1 . This set has { ∈ | ≤ ≤ ∀ ∈ { } ∈{ } ≤ } d-dimensional measure equal to 1 . d! P Definition 3.3 (Faces, normals). For all i 0:d , the convex hull of the set z ,..., z z is denoted F and is called∈ { the face} of K opposite to the { 0 d} \ { i} i 30 Chapter 3. Simplicial finite elements d vertex zi. By construction, Fi is a subset of an affine hyperplane in R , and the unit normal to Fi pointing outward K is denoted nK,i. The unit outward normal vector on ∂K is denoted nK . Definition 3.4 (l-faces). For all l 0:d 1 , an l-face of K is the convex ∈ { − } hull of a subset of zi i 0: d of cardinality (l + 1). By definition the l-faces are closed. The 0-faces{ } ∈{ of K} are the vertices of K. The 1-faces of K are called edges. In dimension d = 2, the notions of edge and face coincide. In dimension d = 1, the notions of vertex, edge, and face coincide. Example 3.5 (Number of faces and edges). The number of l-faces in a d d+1 simplex in R is l+1 since an l-face is defined by selecting a subset of vertices with cardinality (l +1). In particular, there are (d +1) faces and vertices, and d(d+1) there are 2 edges. 3.1.2 Barycentric coordinates in a simplex d d Let K be a simplex in R with vertices zi i 0: d . For all x R , we denote { } ∈{ } ∈ by λi(x), for all i 1:d , the components of the vector x z0 in the basis (z z ,..., z ∈z {), i.e.,} − 1 − 0 d − 0 x z = λ (x)(z z ). (3.1) − 0 i i − 0 i 1: d ∈{X } Differentiating twice (3.1), we infer that the second Fr´echet derivative of λi 2 d vanishes identically, i.e., D λi(x)(h1, h2) = 0 for all h1, h2 R . This implies ∈ d that λi is an affine function of x, i.e., there exist γi R and gi R such d ∈ ∈ that λi(x)= γi + gi x for all x R . Note that Dλi is independent of x and · d ∈ Dλi(h)= gi h for all h R ; in other words, λi = gi. To allow· all the vertices∈ of K to play a symmetric∇ role, we introduce the additional function λ0(x) := 1 i 1: d λi(x), so that the following holds − ∈{ } for all x Rd: ∈ P λi(x)=1 and x = λi(x)zi. (3.2) i 0: d i 0: d ∈{X } ∈{X } A consequence of the above definitions is that λ (z )= δ , i,j 0:d , (3.3) i j ij ∀ ∈ { } This implies that the functions λi i 0: d are linearly independent: if the { } ∈{ } linear combination i 0: d βiλi(x) vanishes identically, evaluating it at a vertex z yields β = 0∈{ for all} j 0:d . Moreover, since K is the convex hull j j P ∈ { } of zi i 0: d , we infer that { } ∈{ } 0 λ (x) 1, x K, i 0:d . (3.4) ≤ i ≤ ∀ ∈ ∀ ∈ { } Part I. Getting Started 31 Definition 3.6 (Barycentric coordinates). The functions λi i 0: d are called the barycentric coordinates in K. { } ∈{ } Proposition 3.7 (Geometric expression). The barycentric coordinates are such that F λ (x) = 1 | i| n (x z ), (3.5) i − d K K,i· − i | | Fi for all x K and all i 0:d , whence λi = d| K| nK,i. ∈ ∈ { } ∇ − | | Proof. Since λi is constant on Fi, its gradient is collinear to nK,i. Since λi is affine, we can write λi(x) = 1 cinK,i (x zi). We obtain ci by using that K = 1 F (n (z z )) for− all j = i·. − | | d | i| K,i· j − i 6 ⊓⊔ Remark 3.8 (Geometric identities). The following holds: F n = 0, F n (c c )= K I , (3.6) | i| K,i | i| K,i ⊗ Fi − K | | d i 0: d i 0: d ∈{X } ∈{X } I where cFi is the barycenter of Fi, cK that of K, and d the identity matrix in d d R × . These identities hold in any polyhedron, see Exercise 3.1 for proofs. ⊓⊔ (♦) 3.1.3 Geometric map TA The following construction is important for finite elements having degrees of freedom over edges or faces, see 3.4 and Chapters 8 and 9 for examples. Let A be an affine subspace of Rd of§ dimension l 1:d 1 . We can build an l ∈ { − } A affine geometric map TA : R A as follows: Fix a simplex S in A with A → l l vertices zi i 0: l and fix a simplex S in R with vertices zi i 0: l and ∈{ } ∈{ } { } l { } l barycentric coordinates λi i 0: l . Typically, S is the unit simplex in R . { } ∈{ } A b l b l A Then, set TA(y) = i 0: l λi(y)zi for all y R . Note that TA(S ) = S ∈{ } ∈ and T (z ) = zA for all ib 0:l ; we often considerb the restriction of T to A i i P A l ∈ {b } b S and abuse the notation by still writing TA instead of TA Sl . b | b Propositionb 3.9 (Diffeomorphism). The map TA is a smooth diffeomor- phism. Proof. Since the barycentric coordinates λi i 0: l are affine functions, the { } ∈{ } map TA is of class C∞ and DTA(y) is independent of y. Thus, we only need l l to verify that the linear map DTA : R Rb is invertible, which will follow by → l verifying that DTA is injective. Let h R be such that DTA(h) = 0. This A A ∈ yields i 1: l Dλi(h)(zi z0 ) = 0, so that Dλi(h) = 0 for all i 0:l ∈{ } − ∈ { } since the vectors (zA zA,..., zA zA) are linearly independent. We conclude P 1 0 l 0 using Exercise 3.2(iv).b − − b ⊓⊔ 32 Chapter 3. Simplicial finite elements 3.2 The polynomial space Pk,d The real vector space Pk,d is composed of d-variate polynomial functions p : Rd R of total degree at most k. Thus, → P = span xα1 ...xαd , 0 α ,...,α k, α + ... + α k . (3.7) k,d { 1 d ≤ 1 d ≤ 1 d ≤ } The importance of the polynomial space Pk,d is rooted in the fact that the Taylor expansion of order k of a d-variate function belongs to Pk,d. Another important fact is that, for any smooth function v : Rd R, → [ v P ] [ Dk+1v(x) = 0, x Rd ]. (3.8) ∈ k,d ⇐⇒ ∀ ∈ The vector space Pk,d has dimension k +1 if d = 1, k + d 1 dim Pk,d = = 2 (k + 1)(k +2) if d = 2, (3.9) d 1 6 (k + 1)(k + 2)(k +3) if d = 3. See Exercise 3.3 for the proof. We omit the subscript d and simply write Pk when the context is unambiguous. d An element α = (α1,...,αd) of N is called a multi-index, and its length is defined to be α = α1 + ... + αd. A generic polynomial function p Pk,d can be written in| the| form ∈ α α α1 αd p(x)= aαx , x := x1 ...xd , (3.10) α ∈AXk,d with real coefficients a and multi-index set = α Nd α k . Note α Ak,d { ∈ || |≤ } that card( ) = dim(P )= k+d . Ak,d k,d d d Lemma 3.10 (Trace). Let Hbe an affine hyperplane in R and let TH : d 1 R − H be an affine bijective map. Then, p TH Pk,d 1 for all p Pk,d. → ◦ ∈ − ∈ k+1 d 1 Proof. Observe first that D (p TH )(y) =0 for all y R − by using the chain rule and the fact that T is◦ affine, then apply (3.8).∈ H ⊓⊔ 3.3 Lagrange (nodal) finite elements We begin with a simple example, see Table 3.1 for k = 1.