New degrees of freedom for differential forms on cubical meshes

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ New degrees of freedom for differential forms on cubical meshes © The Author(s) 2023 Published version Lohi, Jonni Lohi, J. (2023). New degrees of freedom for differential forms on cubical meshes. Advances in Computational Mathematics, 49(3), Article 42. https://doi.org/10.1007/s10444-023-10047-x 2023 Advances in Computational Mathematics (2023) 49:42 https://doi.org/10.1007/s10444-023-10047-x New degrees of freedom for differential forms on cubical meshes Jonni Lohi1 Received: 10 October 2022 / Accepted: 4 May 2023 © The Author(s) 2023 Abstract We consider new degrees of freedom for higher order differential forms on cubical meshes. The approach is inspired by the idea of Rapetti and Bossavit to define higher order Whitney forms and their degrees of freedom using small simplices. We show that higher order differential forms on cubical meshes can be defined analogously using small cubes and prove that these small cubes yield unisolvent degrees of freedom. Importantly, this approach is compatible with discrete exterior calculus and expands the framework to cover higher order methods on cubical meshes, complementing the earlier strategy based on simplices. Keywords Cochains ·Cubical mesh ·Degrees of freedom ·Differential forms · Discrete exterior calculus Mathematics Subject Classification (2010) Primary 65N30; Secondary 58A10 · 65D05 ·41A10 1 Introduction Finite element exterior calculus [4] highlights the importance of suitable finite element spaces in discretisations of partial differential equations. The principal finite elements for differential forms are presented in the periodic table of finite elements [1]. Along with the shape functions, the table provides degrees of freedom (dofs), defined as weighted moments, and together they specify the finite element space on a given mesh. Although these traditional dofs suit the finite element method excellently, for cochain-based methods it is desirable to obtain dofs for p-forms through integration Communicated by: Francesca Rapetti BJonni Lohi [email protected] 1Faculty of Information Technology, University of Jyväskylä, PO Box 35, FI-40014 Jyväskylä, Finland 0123456789().: V,-vol 123 42 Page 2 of 14 J. Lohi on p-chains of the mesh. For example, in the case of (lowest order) Whitney forms (i.e. the space P− 1p), the basis p-forms are in correspondence with p-cochains of the mesh, and hence they can be used as a tool in methods that are based on discrete exterior calculus. With higher order Whitney forms (P− kpfor k>1) this is no longer the case, and the traditional dofs lack physical interpretation. Rapetti and Bossavit [10] addressed this issue by introducing an approach based on small simplices, which are images of the mesh simplices through homothetic transformations. The idea is to define the shape functions and their dofs using these: to each small p-simplex of order kcorresponds a Whitney p-form of order k, and the dofs are obtained through integration over kth order small p-simplices. Although the approach generalises the lowest order case (in that k=1 yields the standard Whitney forms on the initial simplices), the higher order case is not equally simple. In particular, the small simplices do not pave the initial mesh, and the spanning forms corresponding to small simplices are not linearly independent. Despite these downsides, the approach can be reconciled with discrete exterior calculus and has been adopted for use [6–8]. In this work, we provide an analogous approach for the space Q− kp, the (tensor product) finite element space of differential forms on cubical meshes, to which we hereafter refer as “cubical forms” for short. The approach uses small cubes, which are similar to small simplices but defined on cubical meshes. We first give a definition of the small cubes and use them to define cubical forms similarly as higher order Whitney forms are defined using small simplices. The new degrees of freedom resulting from integration over small cubes are considered next: we provide an explicit formula for integrating basis functions and prove that the dofs are unisolvent. Finally, we conclude with the properties of the resulting interpolation operator. Two improvements over the analogous strategy based on small simplices are that the small cubes completely pave the initial mesh and the spanning cubical forms are linearly independent. The approach is hence readily compatible with discrete exterior calculus and enables higher order methods on cubical meshes. 2 Small cubes and cubical forms We first define the small cubes and the cubical forms in the unit n-cube n=[0,1]n. Cubical meshes are considered in Section4. Definition 2.1 (Small cubes). Let J(n,k−1)denote the set of multi-indices k= (k1,...,kn)with ncomponents ki≤k−1. For the unit n-cube n=[0,1]n, each multi-index k∈J(n,k−1)defines a map kk−1:n→nby kk−1(x1,...,xn)=(k1+x1,...,kn+xn) k. For k≥1, the set of kth order small p-cubes of nis Sp k(n)={kk−1(τ) |k∈J(n,k−1)and τis a p-face of n}. 123 New degrees of freedom for differential forms on cubical meshes Page 3 of 14 42 Remark 2.2 Since J(n,k−1)⊂J(n,k),themapkk−1is not defined by the components of kalone. The subscript specifies the set of multi-indices whose element k is considered. Examples of small cubes are shown in Fig. 1. Cubical forms can be seen as counterparts of Whitney forms for cubes. These are the shape functions of the Q− kpfamily in finite element exterior calculus, and they can be obtained using a tensor product construction [3]. We define cubical forms using small cubes similarly as higher order Whitney forms are defined using small simplices. Henceforth, we say that two p-cells (or hyperplanes) are parallel if one of them can be moved to the hyperplane of the other by translation. Definition 2.3 (Lowest order cubical forms) Let σbe a p-face of n.Letxi1,...,xip be the coordinates whose plane is parallel to σand xip+1,...,xinthe other coodinates, whose values yip+1,...,yinare either 0 or 1 on σ. The lowest order cubical form Wσ corresponding to σis Wσ=n  j=p+1 xyij ij(1−xij)1−yijdxi1∧...∧dxip. Fig. 1 Small cubes of orders 1–4 in three dimensions 123 42 Page 4 of 14 J. Lohi Definition 2.4 (Higher order cubical forms) Let k∈J(n,k−1)and τbe a p-face of n.Thekth order cubical p-form corresponding to the small cube kk−1(τ) is w(kk−1(τ)) =n  i=1 xki i(1−xi)k−1−kiWτ. The space of kth order cubical p-forms is Qp k(n)=span w(kk−1(τ)) |k∈J(n,k−1)and τis a p-face of n. The forms given in Definition 2.4 yield exactly the shape functions of the family Q− kp. To prove this claim, recall from [3] that Q− kp(n)is the span of p-forms of the form fdxi1∧...∧dxip, where the coefficient function fis at most kth order polynomial in all variables and at most (k−1)th order polynomial in the variables xi1,...,xip. Proposition 2.5 In the unit n-cube n, we have Q p k(n)=Q− kp(n). Proof That Qp k(n)⊂Q− kp(n)follows directly from Definitions 2.3 and 2.4. It remains to prove Q− kp(n)⊂Qp k(n), and for this it is sufficent to show that Qp k(n)contains all p-forms of the form xy1 1·...·xyn ndxi1∧...∧dxip, where the yiare integers such that 0 ≤yi≤kfor all iand yi≤k−1ifi∈{i1,...,ip}. Let ω=xy1 1·...·xyn ndxi1∧...∧dxipfor such integers yi. We choose zi=yi+1 if i∈{i1,...,ip},zi=yiif i/∈{i1,...,ip}, and write xyi i=xyi i(xi+(1−xi))k−zi=xyi i k−zi  j=0k−zi jxj i(1−xi)k−zi−j. Expanding ωin this way, we get a linear combination of terms of the form n  i=1 xai i(1−xi)bidxi1∧...∧dxip, where ai+bi=k−1ifi∈{i1,...,ip}and ai+bi=kotherwise. From Definitions 2.3 and 2.4, we see that such terms are in Qp k(n). From existing results for Q− kp(see [3]), we know that the exterior derivative d satisfies d(Qp k(n)) ⊂Qp+1 k(n)and the dimension of the space Qp k(n)is n pkp(k+1)n−p. It is easy to see that this is also the number of distinct kth order small p-cubes of n. The spanning forms given in Definition 2.4 are hence linearly independent, which is an improvement over the analogous approach based on small simplices and higher order Whitney forms. 123 New degrees of freedom for differential forms on cubical meshes Page 5 of 14 42 3 New degrees of freedom Since p-forms can be integrated over small p-cubes, we can take the integrals over kth order small p-cubes as degrees of freedom for kth order cubical p-forms. Note that each dof can be associated with a specific face of n— the one that contains the small simplex but has no faces of lower dimension that also contain it. Hence the basic requirement for degrees of freedom is fulfilled: the values of dofs associated with a face only depend on the trace of the differential form on that face. 3.1 Integrating basis functions over small simplices In this subsection we provide a formula for computing the values of the new dofs for basis functions. The following lemmas play a key role. Lemma 3.1 For integers m,n≥0and for y,z∈R, 1 0 (z+x)n(y+1−x)mdx = m  i=0 n  j=0m in jym−izn−ji!j! (i+j+1)!. Proof 1 0 (z+x)n(y+1−x)mdx=1 0n  j=0n jzn−j·xj m  i=0m iym−i·(1−x)idx = m  i=0 n  j=0m in jym−izn−j1 0 (1−x)ixjdx = m  i=0 n  j=0m in jym−izn−ji!j! (i+j+1)!, where we used a well-known integration rule for products of barycentric functions [11] in the last step.  Lemma 3.2 Let τbe a p-face of n. Let xi1,...,xipbe the coordinates whose plane is parallel to τand xip+1,...,xinthe other coodinates, whose values yip+1,...,yin are either 0 or 1 on τ. Let k∈J(n,k),k∈J(n,k), and υ=k k(τ). The average of n i=1xki i(1−xi)k−kiover the small p-cube υis 1 |υ|υ n  i=1 xki i(1−xi)k−ki =1 (k+1)nk n  j=p+1 (k ij+yij)kij(k−k ij+1−yij)k−kij 123 42 Page 6 of 14 J. Lohi ·p  j=11 0 (k ij+x)kij(k−k ij+1−x)k−kijdx. Proof Recall that k kmaps (x1,...,xn)to (k 1+x1,...,k n+xn)/(k+1).For j>p, xkij ij(1−xij)k−kijhas the constant value k ij+yij k+1kij1− k ij+yij k+1k−kij =1 (k+1)k(k ij+yij)kij(k−k ij+1−yij)k−kij on υand hence 1 |υ|υ n  i=1 xki i(1−xi)k−ki =1 (k+1)(n−p)k·n  j=p+1 (k ij+yij)kij(k−k ij+1−yij)k−kij ·1 |υ|υ p  j=1 xkij ij(1−xij)k−kij. Since 1/(k+1)pis the Jacobian determinant of k kregarded as a map from τonto υ and 1 |υ|=(k+1)p, we can write 1 |υ|υ p  j=1 xkij ij(1−xij)k−kij=τ p  j=1k ij+xij k+1kij1− k ij+xij k+1k−kij =1 (k+1)pk τ p  j=1 (k ij+xij)kij(k−k ij+1−xij)k−kij. The result follows, since the integral above is τ p  j=1 (k ij+xij)kij(k−k ij+1−xij)k−kij =[0,1]pp  j=1 (k ij+xij)kij(k−k ij+1−xij)k−kijdxi1...dxip = p  j=11 0 (k ij+x)kij(k−k ij+1−x)k−kijdx.  123 New degrees of freedom for differential forms on cubical meshes Page 7 of 14 42 The integral of any kth order spanning p-form given in Definition 2.4 over any kth order small p-cube can now be computed by combining Lemmas 3.1 and 3.2 with the following proposition. Proposition 3.3 Let σbe a p-face of the unit n-cube n, and let ωbe a smooth 0-form. For any small p-cube υ, we have υ ωWσ=1 |υ|υ ωWσ(x), vect(υ), where 1 |υ|υωis the average of ωover υ, x is any point in υ, and vect(υ) is the p-vector of υ. Proof Let xi1,...,xipbe the coordinates whose plane is parallel to σ.Ifυis not parallel to σ, then both sides become zero because some of the coordinates is constant and hence d xi1∧...∧dxipvanishes on υ.Butifυis parallel to σ,Wσis constant in υand hence υ ωWσ=υω(x)Wσ(x), vect(υ) |υ|dx =1 |υ|υ ωWσ(x), vect(υ).  3.2 Proof of unisolvence Let us next show that these new degrees of freedom are unisolvent. Note that since the number of small p-cubes is equal to the number of (linearly independent) spanning p-forms, it is sufficient to prove that ω∈Qp k(n)has zero integral over all kth order small p-cubes only if ω=0. This is shown in Theorem 3.6, whose proof uses the following two lemmas. Lemma 3.4 For each i ∈{1,...,n},letk i≥0be an integer and Kia set of ki+1 distinct real numbers. Suppose that f :Rn→Ris a polynomial of order kiat most in the variable xi, for all i =1,...,n. If f (x)=0for all x ∈K1×...×Kn, then f=0. Proof A well-known result for univariate polynomials states that a polynomial of order k≥1 can have at most kroots. Hence the case n=1 is clear. Suppose as an induction hypothesis that the statement holds for n=m−1, with m≥2, and consider the case n=m.If f(x)=0 for all x∈K1×...×Km, then for each yj∈Kmthe function gj:Rm−1→Rdefined by gj(x)=f(x,yj)is zero by the induction hypothesis. Hence for any (x1,...,xm−1), the function y→ f(x1,...,xm−1,y)vanishes in Km and hence has km+1 roots. Since it is an univariate polynomial of order kmat most, it must be zero. Hence the statement holds for n=m. Lemma 3.5 Suppose that f :Rn→Ris a nonzero polynomial. For any h1,...,hn> 0, there exist , M1,...,Mn>0such that |f(x)|≥for all x in [M1,M1+h1]× ...×[Mn,Mn+hn]. 123 42 Page 8 of 14 J. Lohi Proof We can write f= k1  i1=0 k2  i2=0 ... kn  in=0 a(i1,i2,...,in)xi1 1xi2 2·...·xin n = k1  i1=0 xi1 1 k2  i2=0 xi2 2... kn  in=0 a(i1,i2,...,in)xin n, where kiis the order of fin the variable xiand each coefficient a(i1,i2,...,in)is constant. For each j∈{1,...,n}and i1,...,in−jsuch that 0 ≤il≤klfor all l∈{1,...,n−j}, let us define a function gj[i1,...,in−j]:Rj→Rby gj[i1,...,in−j](xn−j+1,...,xn) = kn−j+1  in−j+1=0 xin−j+1 n−j+1 kn−j+2  in−j+2=0 xin−j+2 n−j+2... kn  in=0 a(i1,i2,...,in)xin n. In other words, we have g1[i1,...,in−1](xn)= kn  in=0 a(i1,i2,...,in)xin n, gj[i1,...,in−j](xn−j+1,...,xn)= kn−j+1  in−j+1=0 gj−1[i1,...,in−j+1](xn−j+2,...,xn)xin−j+1 n−j+1, and gn(x)=f(x). We proceed as follows. At step 1, we can find n,Mn>0 such that each g1[i1,...,in−1]is either identically zero or satisfies |g1[i1,...,in−1](xn)|≥n for all xn∈[Mn,Mn+hn].Atstep j(for 2 ≤j≤n), suppose we have found n−j+2,Mn−j+2,...,Mn>0 such that each gj−1[i1,...,in−j+1]is either identically zero or satisfies |gj−1[i1,...,in−j+1](xn−j+2,...,xn)|≥n−j+2 for all (xn−j+2,...,xn)∈[Mn−j+2,Mn−j+2+hn−j+2]×...×[Mn,Mn+hn]. Then we can find n−j+1,Mn−j+1>0 such that each gj[i1,...,in−j]is either identically zero or satisfies |gj[i1,...,in−j](xn−j+1,...,xn)|≥n−j+1 for all (xn−j+1,...,xn)∈[Mn−j+1,Mn−j+1+hn−j+1]×...×[Mn,Mn+hn].The proof is completed at step n, since gn=f, which is nonzero by assumption.  Theorem 3.6 Let ω∈Qp k(n).Ifυω=0for all small p-cubes υ∈Sp k(n), then ω=0. 123