26.1 Definition and examples

Transcription

1 Part V, Chapter 26 Quadratures Evaluating integrals over cells and faces are frequent tasks when implementing a solution method based on finite elements. Such integrals are often approximately evaluated by so-called quadratures. In this chapter, we define the notion of quadratures and review one- and multi-dimensional quadratures frequently used in finite element codes. Then, as an example of application, we analyze the impact of the quadrature error in the finite element approximation of a model second-order elliptic problem (see Chapter 25). 26. Definition and examples Let φ be a smooth function and let T h be a mesh of a domain D in R d that covers D exactly. Suppose we want to evaluate the integral Dφ(x)dx. Since φ(x)dx = φ(x) dx, D K T h this problem reduces to evaluating the integral over each mesh cell. An effective way of doing this approximately is by means of quadratures. Definition 26. (Quadrature nodes and weights). Let K be a compact, connected, Lipschitz subset of R d with non-empty interior. Let l q be an integer. A quadrature in K with l q nodes is specified through a set of l q points {ξ l } l {:lq} in K, called Gaussnodes and a set of l q real numbers {ω l } l {:lq}, called quadrature weights. The quadrature then consists of the approximation φ(x)dx ω l φ(ξ l ). (26.) K K l {:l q} The largest integer k such that (26.) is an equality for any polynomial in P k,d is called the quadrature order and is denoted k q.

2 374 Chapter 26. Quadratures Given a quadrature on the reference element K and a mesh T h, a quadrature in any cell K T h can be generated by using the geometric map T K : K K. Let J K the Jacobian matrix of T K. Proposition 26.2 (Quadrature generation). Consider a quadrature in K with nodes { ξ l } l {:lq} and weights { ω l } l {:lq}. Then, setting ξ lk := T K ( ξ l ) and ω lk := ω l det ( J K ( ξ l ) ), (26.2) for all l {:l q }, generates a quadrature on K. If the quadrature on K is of order k q and the geometric map T K is affine, then the quadrature on K is also of order k q. Proof. Since T K is a C -diffeomorphism, the change of variables x = T K ( x) yields K φ(x)dx = φ ( T K K ( x) ) det ( J K ( x) ) d x, and we can apply the quadrature over K to the right-hand side. The statement on the quadrature order is immediate to verify; indeed, whenever T K is affine, J K is constant and φ T K is in P k,d if φ P k,d. Remark 26.3 (Surface quadrature). When generating a surface quadrature from a quadrature on a reference surface, Lemma 9.4 must be used to account for the transformation of the surface measure; see Exercise A large amount of literature is devoted to quadratures; see Abramowitz and Stegun [, Chap. 25], Hammer and Stroud [287], Stroud [467], Davis and Rabinowitz [90], Brass and Petras [96]. we refer the reader to 6..2 for onedimensional quadratures. Example 26.4 (Cuboids). Quadratures on cuboids can be deduced from one-dimensional quadratures by taking the Gauss nodes in tensor-product form. Note though that tensor-product formulas are not optimal in the sense of using the fewest function evaluations for a given order. Although no general formula for non-tensor-product quadratures for the cube is known, many quasi-optimal quadratures are available in the literature, see e.g., Cools [7, 2.3.] and Cools and Rabinowitz [72, 4.]. Example 26.5 (Quadratures on simplices). Table 26. lists some quadratures on triangles in two dimensions. In this table, we call multiplicity the number of permutations that must be performed on the barycentric coordinates to obtain the list of all the Gauss nodes of the quadrature. For instance, the first-order formula in the second line has three Gauss nodes; the corresponding barycentric coordinates and weights are {,0,0; 3 S}, {0,,0; 3 S}, {0,0,; 3 S}, where S denotes the surface of the triangle. Table 26.2 lists some quadratures on a tetrahedron. As in two dimensions, the multiplicity is the number of permutations to perform on the barycentric coordinates to obtain all the Gauss

3 Part V. Elliptic PDEs 375 nodes of the quadrature. For instance, the third-order formula has five Gauss nodes which are the node ( 4, 4, 4, 4 ) with the weight 4 5V and the four nodes ( 6, 6, 6, 2 ), ( 6, 6, 2, 6 ), ( 6, 2, 6, 6 ), ( 2, 6, 6, 6 ) with the weight 9 20V; here, V denotesthe volumeofthetetrahedron.aformulavalidonanysimplexin R d is K λα0 0...λα d d dx = K α 0!...α d!d! (α α d +d)!, where {λ 0,...,λ d } are the barycentric coordinates in K and α 0,...,α d are natural numbers. This formula is useful to numerically verify the order of quadratures. k q l q Barycentric coord. Multiplicity Weights ω l (,, 3 3 3) S 3 S 3 (,0,0) (,, ) 3 3 ( 2 3,,0) ( 3 4,, 3 3 3) 9 S 6 (,, ) 3 48 ( 3 7,, ) 20 (,,0) (,0,0) 3 S 5 S (a i,a i, 2a i) for i =,2 3 ω i for i =,2 5 7 a = ω = S a 2 = ω 2 = S (,, 3 3 3) (a i,a i, 2a i) for i =,2 3 a = a 2 = (a i,a i, 2a i) for i =, S 55 5 S S 200 a = S a 2 = S (a,b, a b) 6 a = S b = Table 26.. Nodes and weights for quadratures on a triangle of area S.

4 376 Chapter 26. Quadratures k q l q Barycentric coord. Multiplicity Weights ω l (,,, ) V 4 (,0,0,0) (a,a,a, 3a) a = (,,0,0) V 4 V 5 V (,0,0,0) 4 V 20 (,,, ) 4 V 5 (,,, ) 4 V 20 (,,, ) (a i,a i,a i, 3a i) for i =,2 4 a = a 2 = (a,a, a, a) a = V V V V Table Nodes and weights for quadratures on a tetrahedron of volume V Discrete problem with quadratures We now study how quadratures impact the well-posedness of a discrete model problem and the convergence properties of its finite element approximation. This is a crucial question since quadratures cannot be avoided in practice. For simplicity, we focus on the purely diffusive, homogeneous Dirichlet problem. As in 25.,this problem is approximatedusing H -conformingfinite elements.the discretespacev h isbuilt usinganaffinemeshfromashape-regular mesh sequence and a reference finite element ( K, P, Σ) of degree k. The noveltyisthat we nowtakeinto accountthe useofquadraturesto evaluatethe bilinear form a(v,w) = D v wdx and the linear form l(w) = D fwdx Quadrature error For all K T h, we generate a quadrature in each mesh cell K T h with nodes {ξ lk } l {:lq} and weights {ω lk } l {:lq} from a reference quadrature of order k q. For any function φ that is smooth enough to have point values, we define the quadrature error as follows: E K (φ) := K φ(x)dx l q l= ω lk φ(ξ lk ). (26.3)

5 Part V. Elliptic PDEs 377 Lemma 26.6 (Quadrature error with polynomial factor). Let m, n N, r [, ] and assume that k q n+m. Assume that the mesh sequence (T h ) h>0 is affine. Then, there is c, uniform, such that E K (φp) ch m K φ W m, (K) p L (K), (26.4) for all φ W m, (K), all p such that p T K P P n,d, and all K T h. Proof. Let K be the reference element. Let p such that p := p T K P n,d. Let us also denote φ = φ T K. Since T h is affine, after making a change of variable we obtain E K (φp) = det(j K ) Ê( φ p) (with obvious notation for Ê). The linear form φ Ê( φ p) is bounded on W m, ( K) since Ê( φ p) c φ C0 ( K) p C 0 ( K) c φ W m, ( K) p L ( K), where we have used (.8) and norm equivalence in P for p (since dim( P) < )). Moreover, Ê( φ p) = Ê(( φ ĝ) p) for all ĝ P m,d, since p P n,d and k q n+m. Owing to the Bramble Hilbert Lemma 0.8, we infer that Ê( φ p) c inf ĝ P n,d φ ĝ W m, ( K) p L ( K). c φ W m, ( K) p L ( K). We conclude by using Lemma Discrete problem and well-posedness For all K T h, we assume that the diffusion coefficients ij and the source term f are continuous in K, and we define a Q (v h,w h ) = ω lk v h (ξ lk ) (ξ lk ) w h (ξ lk ), l Q (w h ) = K T h l {:l q} K T h l {:l q} ω lk f(ξ lk )w h (ξ lk ), for all (v h,w h ) V h V h. Observe that it is not possible in general to extend a Q and l Q to H0 (D), since functions in H 0 (D) are not necessarily continuous. The discrete problem with quadratures is formulated as follows: { Find uh V h such that (26.5) a Q (u h,w h ) = l Q (w h ), w h V h. Lemma 26.7 (Well-posedness). Assume that P Pl,d for some integer l. Assume that k q 2l 2 and that W, (T h ;R d d ). Then, the discrete problem (26.5) is well-posed for h small enough.

6 378 Chapter 26. Quadratures Proof. We need to prove that there is α Q > 0 such that α Q v h H (D) sup wh V h a Q(v h,w h ) w h L 2 (D) for all v h V h. Since a Q (v h,w h ) a(v h,w h ) (a a Q )(v h,w h ) sup sup sup, w h V h w h L 2 (D) w h V h w h L 2 (D) w h V h w h L 2 (D) and since α v h L 2 (D) sup wh V h a(v h,w h ) w h L 2 (D) with α > 0 owing to the coercivity of a on V h, the assertion follows if we prove that (a a Q )(v h,w h ) ch v h L 2 (D) w h L 2 (D). We observe that (a a Q )(v h,w h ) = K T h E K ( v h w h ). For all i,j {:d}, we use Lemma 26.6 with φ = ij, p = i v h j w h, m =, and n = 2l 2 (so that n + m k q ). Note that indeed p T K P 2l 2,d, because the mesh is affine, i v h = d i = JT K,ii ( i v h) T K, i v h P l,d and a similar argument holds for j w h. Since i v h j w h L (K) i v h L 2 (K) j w h L 2 (K), we infer that E K ( v h w h ) ch K W, (K;R d d ) v h L 2 (K) w h L 2 (K). The desired bound follows by a discrete Cauchy Schwarz inequality (the constant c depends on W, (T h ;R )). d d Error analysis Theorem 26.8 (Error estimate). Assume that P k,d P P l,d for some integers l k. Assume that k q l + k 2. Assume that W k, (T h ;R d d ) and f W k, (T h ). Assume that (26.5) is well-posed and that u H k+ (D). Then, u u h H (D) ch k ( u H k+ (D)(+C Q ( ))+C Q (f)), (26.6) with C Q ( ) = W k, (T h ;R d d ), C Q (f) = max(c P,D l D f W k, (T h ), f W k, (T h )), and C P,D from the Poincaré inequality (2.0). Proof. We use Strang s First Lemma with the quasi-interpolation operator I g,av h0 introduced in Using the H -seminorm, (2.4) becomes (u u h ) L2 (D) C (u I g,av h0 u) L 2 (D) + g,av (a a Q )(Ih0 sup u,w h) (l l Q )(w h ), α Q w h V h w h L 2 (D) where α Q results from the well-posedness of the discrete problem. The supremum over w h V h represents the quadrature error that we now bound by estimating each of the two terms separately.

7 Part V. Elliptic PDEs 379 () Bound on (a a Q ). Letting v h = I g,av h0 u, we use Lemma 26.6 with φ = ij i v h,p = j w h,m = k,andn = l (sothat n+m = l+k 2 k q ) for all i,j {:d} to infer that E K ( ij i v h j w h ) ch k K ij i v h W k, (K) j w h L (K). Combining the Leibniz product rule with the inverse inequality (5.2) (with p = and r = 2) leads to This leads to ij i v h W k, (K) c k ij W m, (K) i v h W k m, (K) m=0 c K 2 ij W k, (K) i v h H k (K). E K ( v h w h ) ch k K W k, (K;R d d ) v h Hk (K) w h L 2 (K). As a result, (a a Q )(Ih0 av sup u,w h) w h V h w h L 2 (D) ch k (max K T h W k, (K;R d d )) u H k+ (D), where we have used the estimate I g,av h0 u H k+ (D) c u H k+ (D) which follows from Theorem 8.2. (2) Bound on (f f Q ). We have (l Q l)(w h ) = K T h E K (fw h ). We cannot apply Lemma 26.6 with φ = f, p = w h, m = k, and n = l since n+m = l+k may be larger than k q. Fortunately, we can use instead Exercise 26.2 with µ = k and ν = l (which satisfy ν +µ 2 = l+k 2 k q ) to infer that E K (fw h ) ch k K( f W k, (K) w h L (K) + f W k, (K) w h L (K)). We conclude using inverse inequalities for w h, a discrete Cauchy Schwarz inequality and the global Poincaré inequality. Remark 26.9 (Literature). The above analysis is inspired from Ciarlet [52, 4.], Ciarlet and Raviart [54], Dautray and Lions [88, XII.5]. It is possible to refine the analysis by assuming f W k,q (T h ) with q > d k and q 2, and that a surface quadrature of order k+l is used to approximate boundary integrals resulting from Neumann conditions; see [88, XII.5] Implementation This section addresses practical aspects of quadrature implementation in view of matrix assembling.

8 380 Chapter 26. Quadratures Nodesand weights. Letm {:N c }andletk m bethecorrespondingmesh cell. The nodes and weights of the quadrature are defined in Proposition Recall from Definition?? that the geometric map T Km is built from reference shape functions { ψ n } n {:ngeo}. This leads us to define the double-entry array psi(:n geo,:l q ) such that psi(n,l) = ψ n ( ξ l ). The k-th Cartesian component of the Gauss node ξ lkm = T Km ( ξ l ) is given by n geo (ξ lkm ) k = coord(k, j(n, m)) psi(n, l). n= We also need the the triple-entry arraydpsi dhatk(:d,:n geo,:l q ) providing the derivatives of the geometric shape functions at the Gauss nodes, i.e., dpsi dhatk(k,n,l) = ψ n x k ( ξ l ). Then,theentriesoftheJacobianmatrixJ Km at ξ l canbecomputedasfollows: ( ) n geo J Km ( ξ l ) = coord(k,j(n,m))dpsi dhatk(k 2,n,l), k,k 2 n= for all k,k 2 {:d}. Since the determinant of J Km ( ξ l ) is always multiplied by the weight ω l in the quadratures, it may be useful to store this product once and for all in a double-entry array of weights weight K({:l q },{:N c }): weight K(l,m) = ω l det ( J Km ( ξ l ) ). Notice that when the mesh is affine, the partial derivatives of the shape functions ψ n areconstant in K, so that the size of the arraydpsi dhatkis reduced to d n geo ; moreover, memory space is saved by storing separately the reference quadrature weights ω l and the determinants det(j Km ). Shape functions. Let { θ n } n N be the referenceshape functions. Since these functions and their derivatives generally need to be evaluated many times at the Gauss nodes in K, it can be useful to compute these values once and for all and store them in the double-entry array theta({:n sh },{:l q }) such that theta(n,l) = θ n ( ξ l ), and in the triple-entry array dtheta dhatk({:d},{:n sh },{:l q }) such that dtheta dhatk(k,n,l) = θ n x k ( ξ l ).

9 Part V. Elliptic PDEs 38 Let us assume for simplicity that the linear bijective map used to generate the local shape functions is the pullback by the geometric map. As a result, the values of the local shape functions at the Gauss nodes in the mesh cell K m are given by θ n (ξ lkm ) = θ n ( ξ l ) = theta(n,l), for all n {:n sh }, all l {:l q }, and all m {:N c } (notice that the value of θ n (ξ lkm ) is independent of K m ). Let us now consider the first-order derivatives of the local shape functions at the Gauss nodes. Using the chain rule yields, for all k {:d}, θ n x k (ξ lkm ) = = d k 2= d k 2= θ n ( ξ l ) (T K m ) k2 (ξ lkm ) x k2 x k θ ( ) n ( ξ l ) J K x m ( ξ l ). k2 k 2,k One can evaluate these derivatives once and for all and store them in an array dtheta dk({:d},{:n sh },{:l q },{:N c }) such that dtheta dk(k,n,l,m) = θ n x k (ξ lkm ). Notice that the size of this array, d n sh l q N c, can be extremely large. If the mesh is affine, an alternative strategy is possible once observing that the Jacobian matrix J Km and its inverse do not depend on the Gauss nodes. In this case, one may store the inverse of the Jacobian matrix in a triple-entry array inv jac K({:d},{:d},{:N c }) such that ( [JKm ] ) inv jac K(k,k 2,m) = k,k 2. Each time the quantity θn x k (ξ lkm ) is needed, the following operations must be performed: θ n x k (ξ lkm ) = d dtheta dhatk(k 2,n,l)inv jac K(k 2,k,m). k 2= The array inv jac K has d d N c entries; this number is much smaller than the number of entries of dtheta dk if d n sh l q. In this situation, storing inv jac K will save memory space at the prize of some additional computations. Assembling. Let us first consider the assembling of the stiffness matrix. We consider a slightly more general problem than in 26.2, namely a diffusionadvection-reaction problem for which

10 382 Chapter 26. Quadratures a Q (v h,w h ) = ω lkm A(ξ lkm,v h Km,w h Km ), m {:N c} l {:l q} with A(ξ,ϕ,ψ) = k,k 2 {:d} +ψ(ξ) ϕ x k (ξ) k k 2 (ξ) ψ x k2 (ξ) k {:d} β k (ξ) ϕ x k (ξ)+ϕ(ξ)µ(ξ)ψ(ξ). A general assembling procedure for a stiffness matrix (stored in the CSR format) has been outlined in Algorithm Our goal is now to detail the evaluation of the array tmp used in this algorithm by means of quadratures. We assume for simplicity that the coefficients ( k k 2 ) k,k 2 {:d}, (β k ) k {:d}, and µ are known analytically; see Exercise 26.5 for discrete data. The assembling procedure is shown in Algorithm 26.. Notice that we preliminarily evaluate and store the coordinates of the Gauss nodes ξ lkm since we need to evaluate the values of the coefficients at these nodes. Algorithm 26.. Assembling of A for analytic data. A = 0 for m {:N c} do for l {:l q} do; tmp = 0 for k {:d} do n geo xi l(k) = coord(k,j(n,m)) psi(n,l) n= for ni {:n sh } do for nj {:n sh } do d x = dtheta dk(k,nj,l,m) k k 2 (xi l) dtheta dk(k 2,ni,l,m) k,k 2 = x 2 = theta(ni,l) d β k (xi l) dtheta dk(k,nj,l,m) k = x 3 = theta(ni,l) µ(xi l) theta(nj,l) tmp(ni,nj) = tmp(ni,nj)+[x +x 2 +x 3] weight K(l,m) Accumulate tmp in A as in Algorithm 23.2 The assembling of the right-hand side vector can be performed similarly. Let us consider that l Q (w h ) = m {:N c} l {:l ω q} lk m F(ξ lkm,w h Km )

11 Part V. Elliptic PDEs 383 with F h (ξ,ψ) = f(ξ)ψ(ξ). The assembling procedure is presented in Algorithm Algorithm Assembling of B for analytic data. B = 0 for m {:N c} do for l {:l q} do; tmp = 0 for k {:d} do n geo xi l(k ) = coord(k,j(n,m))psi(n,l) n= for ni {:n sh } do tmp(ni) = tmp(ni)+f(xi l) theta(ni,l) weight K(l,m) for ni {:n sh } do; i = gdof(ni,m) B(i) = B(i)+tmp(ni) Exercises Exercise 26. (Quadrature error). Consider a quadrature oforderk q. Let p [, ] and let m N be such that k q + m > d p. Prove that there is c, uniform with respect to h, such that E K (φ) ch m+d( p ) K φ W m,p (K), for all φ W m,p (K) and all K T h. (Hint: use Lemma??.) Prove that D φ(x)dx K T h l {:l q} ω lk φ(ξ lk ) ch m φ W m,p (D) for all φ W m,p (D) and all h > 0. (Hint: use a discrete Hölder s inequality.) Exercise 26.2 (Quadrature error with polynomial). Let ν, µ and assume that k q ν +µ 2. The goal is to prove that E K (φp) ch µ K ( φ W µ, p L (K) + φ W µ, p L (K)), (26.7) for all φ W µ, (K), all p P ν,d, all K T h, and all h > 0. (Note that φ W µ, p L (K) is not needed if k q ν +µ ; see Lemma 26.6,) (i) Prove that E K (φp K ) ch µ K φ W µ, p L (K), where p K is the meanvalue of p over K.

12 384 Chapter 26. Quadratures (ii) Prove (26.7) for µ 2. (Hint: Apply Lemma 26.6 with m = µ.) (iii) Prove (26.7) for µ =. (Hint: prove first that Ê( φ( p p K)) c φ( p p K) W, ( K) ; note that p K = p K.) Exercise 26.3 (Asssembling). Let D = (0,) 2. Consider the problem u+u = in D and u D = 0. Approximate its solution with P H - conforming finite elements on the two meshes shown on the right. 0 0 Evaluate the discrete solution in both cases. (Hint: there is only one degree of freedom in both cases; see Exercise 25.5 for computing the gradient part of the stiffness coefficient and use a quadrature from Table 26. for the zeroorder term.) For a fine mesh composed of 800 elements, u h ( 2, 2 ) Comment. Exercise 26.4 (Surface quadrature). Let F be a face of a three-dimensional element. Let F R 2 be the reference face and let T F : F F be a geometric transformation for F. Let t (ŝ),t 2 (ŝ) be the two column vectors of the Jacobian matrix of T F (ŝ), say J F (ŝ) = [t (ŝ),t 2 (ŝ)]. (i) Compute the metric tensor F := J T F J F in terms of the dot products t i t j (ŝ), i,j 2. (ii) Show that ds = t (ŝ) t 2 (ŝ) l 2 (R 3 )dŝ (Hint: use Lagrange s identity). (iii) Let {(ŝ l,ŵ l } l {:l q } be a quadrature on F. What is the corresponding quadrature on F? Exercise 26.5 (Assembling with discrete data). Assume that the coefficients ( k k 2 ) k,k 2 {:d}, (β k ) k {:d}, and µ are in the discrete space V h. Let dif, beta, and mu be the corresponding coordinate vectors. Adapt Algorithm 26. to this situation. (Hint: µ(ξ lkm ) = n {:n sh } mu(gdof(n,m)) theta(n,l).) Exercise 26.6 (Assembling of RHS). Write the assembling algorithm for the right-hand side vector in the case where F h (ξ,w h ) = f(ξ)w h (ξ) + d k = β k (ξ) w h x k (ξ) with analytically-known data.