High order approximation of Hodge Laplace problems with local coderivatives on cubical meshes
aa r X i v : . [ m a t h . NA ] O c t HIGH ORDER APPROXIMATION OF HODGE LAPLACEPROBLEMS WITH LOCAL CODERIVATIVES ON CUBICALMESHES
JEONGHUN J. LEE
Abstract.
In mixed finite element approximations of Hodge Laplace prob-lems associated with the de Rham complex, the exterior derivative operatorsare computed exactly, so the spatial locality is preserved. However, the nu-merical approximations of the associated coderivatives are nonlocal and it canbe regarded as an undesired effect of standard mixed methods. For numericalmethods with local coderivatives a perturbation of low order mixed methodsin the sense of variational crimes has been developed for simplicial and cubicalmeshes. In this paper we extend the low order method to all high orders oncubical meshes using a new family of finite element differential forms on cubi-cal meshes. The key theoretical contribution is a generalization of the lineardegree, in the construction of the serendipity family of differential forms, andthe generalization is essential in the unisolvency proof of the new family offinite element differential forms. Introduction
In this paper we consider finite element methods for the Hodge Laplace problemsof the de Rham complex where both the approximation of the exterior derivativeand the associated coderivative are spatially local operators. The locality of thecoderivative operator is not fulfilled in standard mixed methods for these problems(cf. [6, 7]). In fact, pursuing numerical methods with local coderivative is relatedto the development of various numerical methods for the Darcy flow problems.To discuss this local coderivative property in a more familiar context, let usconsider the mixed form of Darcy flow problems: Find ( σ, u ) ∈ H (div , Ω) × L (Ω)such that (cid:10) K − σ, τ (cid:11) − h u, div τ i = 0 , ∀ τ ∈ H (div , Ω) , h div σ, v i = h f, v i , ∀ v ∈ L (Ω) , (1.1)where the unknown functions σ and u are vector and scalar fields defined on abounded domain Ω in R n , and ∂ Ω is its boundary. The coefficient K is symmetric,matrix-valued, spatially varying, and uniformly positive definite. Note that u isthe scalar field of the pressure and σ is the fluid velocity given by the Darcy law σ = − K grad u . The standard mixed finite element method for this problem is: Date : July, 2019 .2000
Mathematics Subject Classification.
Primary: 65N30.
Key words and phrases. perturbed mixed methods, local constitutive laws.
Find ( σ h , u h ) ∈ Σ h × V h such that (cid:10) K − σ h , τ (cid:11) − h u h , div τ i = 0 , ∀ τ ∈ Σ h , h div σ h , v i = h f, v i , ∀ v ∈ V h , (1.2)where Σ h ⊂ H (div , Ω) and V h ⊂ L (Ω) are finite element spaces, and σ h is anapproximation of σ = − K grad u . Here the notation h· , ·i is used to denote the L inner product for both scalar fields and vector fields defined on Ω.The mixed method (1.2) has a local mass conservation property, and its stabil-ity conditions and error estimates are well-studied (cf. [10]). However, the mixedmethod (1.2) does not preserve the local property of the map u σ = − K grad u in the continuous problem. In other words, the map u h σ h , defined by the firstequation of (1.2), is not local because the inverse of the so–called “mass matrix”derived from the L inner product (cid:10) K − τ, τ ′ (cid:11) on Σ h is nonlocal. Since constitutivelaws are spatially local relations of quantities in many physical models, construc-tion of local numerical constitutive laws is one of key issues in the development ofnumerical methods following physical derivation of constitutive laws such as thefinite volume methods and the multi-point flux approximations.Here we give a brief overview on previous studies of numerical methods withlocal coderivatives mainly for the Darcy flow problems but we have to admit thatthis overview and the list of literature here are by no means complete.An early work for the locality property by perturbing the mixed method (1.2)was done in [8] on triangular meshes with the lowest order Raviart-Thomas space.The approach leads to a two-point flux method, which approximate flux acrossthe interface of two cells by two point values of pressure field in the two cells,but the two-point flux method is not consistent in general for anisotropic K , cf.[1, 2]. To circumvent this defect, various multi-point flux approximation schemeswere derived (cf. [1]) but the stability and error estimates of these schemes areusually nontrivial and are restricted to low order cases. It seems that the mostuseful approach for the stability and error estimates for these numerical schemesis to utilize connections between the schemes and perturbed mixed finite elementmethods, cf. [9, 16, 21, 22, 25, 27]. An alternative approach to perturbed mixedfinite element methods for the local coderivative property was proposed in [11, 27]independently for simplicial and quadrilateral meshes. The key to achieve localcoderivatives in this approach is a mass-lumping for vector-valued finite elements.Further extensions to hexahedral grids are studied in [20, 26]. Extensions to all highorder methods are studied in [3] with the development of a new family of H (div)finite elements on quadrilateral and hexahedral meshes, which is inspired by thenew family of low order finite elements in [23]. For the Maxwell equations, Cohenand Monk studied perturbed mixed methods based on anisotropic mass-lumpingof vector-valued finite elements but the methods may not be consistent when thematerial coefficients are not isotropic (cf. [14]). For the Hodge Laplace problemsthe discrete exterior calculus, proposed in [15, 19], cf. also [18], has a natural localcoderivative property by construction. However, a satisfactory convergence theoryseems to be limited (see [24] for 0-form case).The purpose of this paper is to extend the results in [3, 23] to the Hodge Laplaceproblems on cubical meshes. More precisely, we will construct high order perturbedmixed methods for the Hodge Laplace problems on cubical meshes which have thelocal numerical coderivatives. Since the Hodge Laplace problems are models of OCAL CODERIVATIVES 3 other specific problems, the numerical method in the present paper can be usedto develop numerical methods with the locality property for other problems. Forexample, the methods with the newly developed finite elements have potential ap-plications to high order mass-lumping for the time-dependent Maxwell equations.However, we will restrict our discussion only to steady problems in the paper be-cause developing the methods for steady problems is already quite involved.The paper is organized as follows. In the next section we will present a briefreview of exterior calculus, the de Rham complex with its discretizations, and theabstract analysis results in [23] for the analysis framework to be used in latersections. In Section 3 we develop a new family of finite element differential formson cubical meshes, say ˜ Q r Λ k , by defining the shape functions and the degrees offreedom, and proving the unisolvency. We also prove some properties of the newelements for construction of numerical methods with the local coderivative property.In Section 4, we construct numerical methods with local coderivatives using ˜ Q r Λ k and the framework in Section 2. Finally, we summarize our results with someconcluding remarks in Section 5.2. Preliminaries
Here we review the language of the finite element exterior calculus [6, 7] and alsointroduce new concepts of polynomial differential forms. We assume that Ω ⊂ R n is a bounded domain with a polyhedral boundary. We will consider finite elementapproximations of a differential equation which has a differential form defined on Ωas an unknown. Let Alt k ( R n ) be the space of alternating k –linear maps on R n . For1 ≤ k ≤ n let Σ k be the set of increasing injective maps from { , ..., k } to { , ..., n } .Then we can define an inner product on Alt k ( R n ) by h a, b i Alt = X σ ∈ Σ k a ( e σ , . . . , e σ k ) b ( e σ , . . . , e σ k ) , a, b ∈ Alt k ( R n ) , where σ i denotes σ ( i ) for 1 ≤ i ≤ k and { e , . . . , e n } is any orthonormal basis of R n .The differential k -forms on Ω are maps defined on Ω with values in Alt k ( R n ). If u is a differential k -form and t , . . . , t k are vectors in R n , then u x ( t , . . . , t k ) denotesthe value of u applied to the vectors t , . . . , t k at the point x ∈ Ω. The differentialform u is an element of the space L Λ k (Ω) if and only if the map x u x ( t , . . . , t k )is in L (Ω) for all tuples t , . . . , t k . In fact, L Λ k (Ω) is a Hilbert space with innerproduct given by h u, v i = Z Ω h u x , v x i Alt dx.
The exterior derivative of a k -form u is a ( k + 1)-form d u given byd u x ( t , . . . t k +1 ) = k +1 X j =1 ( − j +1 ∂ t j u x ( t , . . . , ˆ t j , . . . , t k +1 ) , where ˆ t j implies that t j is not included, and ∂ t j denotes the directional derivative.The Hilbert space H Λ k (Ω) is the corresponding space of k -forms u on Ω, which is JEONGHUN J. LEE in L Λ k (Ω), and where its exterior derivative, d u = d k u , is also in L Λ k +1 (Ω). The L version of the de Rham complex then takes the form H Λ (Ω) d −→ H Λ (Ω) d −→ · · · d n − −−−→ H Λ n (Ω) . (2.1)In the setting of k –forms, the Hodge Laplace problem takes the form(2.2) Lu = (d ∗ d + dd ∗ ) u = f, where d = d k is the exterior derivative mapping k –forms to ( k + 1)–forms, and thecoderivative d ∗ = d ∗ k can be seen as the formal adjoint of d k − . Hence, the HodgeLaplace operator L above is more precisely expressed as L = d ∗ k +1 d k + d k − d ∗ k . Atypical model problem studied in [6, 7] is of the form (2.2) and with appropriateboundary conditions. The mixed finite element methods are derived from a weakformulation, where σ = d ∗ u is introduced as an additional variable. It is of theform:Find ( σ, u ) ∈ H Λ k − (Ω) × H Λ k (Ω) such that h σ, τ i − (cid:10) u, d k − τ (cid:11) = 0 , τ ∈ H Λ k − (Ω) , (cid:10) d k − σ, v (cid:11) + (cid:10) d k u, d k v (cid:11) = h f, v i , v ∈ H Λ k (Ω) . (2.3)Here h· , ·i denotes the inner products of all the spaces of the form L Λ j (Ω) whichappears in the formulation, i.e., j = k − , k, k +1. We refer to Sections 2 and 7 of [6]for more details. We note that only the exterior derivate d is used explicitly in theweak formulation above, while the relation σ = d ∗ k u is formulated weakly in the firstequation. The formulation also contains the proper natural boundary conditions.The problem (2.3) with k = n − K is the identity matrix. The weakformulations (2.3) can be modified for variable by changing the L inner productsto the inner product with the variable coefficient, see [6, Section 7.3]. Throughoutthe discussion below we will restrict the discussion to the constant coefficient casebut the extension of the discussion to problems with variable coefficients which arepiecewise constants with respect to the mesh, is straightforward. We refer to [23,Section 6] for details.If the domain Ω is not homologically trivial, then there may exist nontrivialharmonic forms, i.e., nontrivial elements of the space H k (Ω) = { v ∈ H Λ k (Ω) : d v = 0 and h v, d τ i = 0 for all τ ∈ H Λ k − (Ω) } , and the solutions of the system (2.3) may not be unique. To obtain a system witha unique solution, an extra condition requiring orthogonality with respect to theharmonic forms:Find ( σ, u, p ) ∈ H Λ k − (Ω) × H Λ k (Ω) × H k (Ω) such that h σ, τ i − h d τ, u i = 0 , τ ∈ H Λ k − (Ω) , h d σ, v i + h d u, d v i + h p, v i = h f, v i , v ∈ H Λ k (Ω) , (2.4) h u, q i = 0 , q ∈ H k (Ω) . The basic construction in finite element exterior calculus is a correspondingsubcomplex of (2.1), V h d −→ V h d −→ · · · d −→ V nh , OCAL CODERIVATIVES 5 where the spaces V kh are finite dimensional subspaces of H Λ k (Ω). In particular, thediscrete spaces should have the property that d( V k − h ) ⊂ V kh . The finite elementmethods studied in [6, 7] are based on the weak formulation (2.3). These methodsare obtained by simply replacing the Sobolev spaces H Λ k − (Ω) and H Λ k (Ω) by thefinite element spaces V k − h and V kh . More precisely, we are searching for a triple(˜ σ h , ˜ u h , ˜ p h ) ∈ V k − h × V kh × H kh such that h ˜ σ h , τ i − h d τ, ˜ u h i = 0 , τ ∈ V k − h , h d˜ σ h , v i + h d˜ u h , d v i + h ˜ p h , v i = h f, v i , v ∈ V kh , (2.5) h ˜ u h , q i = 0 , q ∈ H kh , where the space H kh , approximating the harmonic forms, is given by H kh = { v ∈ V kh : d v = 0 and h v, d τ i = 0 for all τ ∈ V k − h } . Stability and error estimates for the numerical solution of (2.5) are discussed in [7,Theorem 3.9] with an error estimate of the form k ( σ, u, p ) − (˜ σ h , ˜ u h , ˜ p h ) k X . inf ( τ,v,q ) ∈X kh k ( σ, u, p ) − ( τ, v, q ) k X + E h ( u )(2.6)with k ( σ, u, p ) k X := (cid:0) k σ k + k d σ k + k u k + k d u k + k p k (cid:1) and E h ( u ) comes from the nonconformity of the space of discrete harmonic forms, H kh , to the space of continuous harmonic forms H k . E h ( u ) vanishes if there is nonontrivial harmonic forms, and will usually be of higher order than the other termson the right-hand side of (2.6). We refer to [6, Section 7.6] and [7, Section 3.4] formore details.In [23], abstract conditions are proposed to develop numerical methods satisfyingsuch properties and the lowest order methods were developed in simplicial andcubical meshes. In particular, the cubical mesh case required construction of newfinite element differential forms which were called S +1 Λ k spaces. In [3], the S +1 Λ k family is extended to all higher orders for the Darcy flow problems in the two andthree dimensions on cubical meshes, and as a consequence, higher order numericalmethods satisfying the local coderivative property in the two and three dimensionalDarcy flow problems on weakly distorted quadrilateral and hexahedral meshes areconstructed. The goal of this paper is developing high order numerical methods forthe Hodge Laplace equations satisfying the local coderivative property on cubicalmeshes. For the stability and error analysis we use the abstract framework in[23]. The main contributions of this paper are construction of a new family offinite element differential forms, a higher order extensions of S +1 Λ k , and the newtechniques for the development of the new elements. Although the new family isa higher order extension of S +1 Λ k , we will call it ˜ Q r Λ k family because it is closelyrelated to the Q r Λ k space rather than the S r Λ k family. One characteristic featureof ˜ Q r Λ k family is that the number of degrees of freedom is same as the one of Q r Λ k .Here we summarize the abstract conditions for stability and error estimatesestablished in [23]. (A) There is a symmetric bounded coercive bilinear form h· , ·i h on V k − h × V k − h such that the norm k τ k h := h τ, τ i / h is equivalent to k τ k for τ ∈ V k − h withconstants independent of h . JEONGHUN J. LEE (B)
There exist discrete subspaces W k − h ⊂ L Λ k − (Ω) and ˜ V k − h ⊂ V k − h suchthat h τ, τ i = h τ, τ i h , τ ∈ ˜ V k − h , τ ∈ W k − h , (2.7) and a linear map Π h : V k − h → ˜ V k − h such that dΠ h τ = d τ , k Π h τ k . k τ k ,and h Π h τ, τ i h = h τ, τ i h , τ ∈ W k − h . (2.8)If these assumptions are satisfied, then we find a solution ( σ h , u h , p h ) of theproblem h σ h , τ i h − h d τ, u h i = 0 , τ ∈ V k − h , h d σ h , v i + h d u h , d v i + h p h , v i = h f, v i , v ∈ V kh , (2.9) h u h , q i = 0 , q ∈ H kh . The following result is proved in [23].
Theorem 2.1.
Suppose that the assumptions (A) and (B) hold. Then the solutionof (2.9) , ( σ h , u h , p h ) , satisfies k ( σ h − ˜ σ h , u h − ˜ u h , p h − ˜ p h ) k X . k σ − P W h σ k + k σ − ˜ σ h k , (2.10) where P W h is the L -orthogonal projection into W k − h . Construction of ˜ Q r Λ k In this section we construct a new family of finite element differential forms˜ Q r Λ k ( T h ) on cubical meshes T h of Ω where the elements in T h are Cartesian productof intervals. If r = 1, then ˜ Q r Λ k is the S +1 Λ k space in [23]. For n = 2 , k = n −
1, ˜ Q r Λ k spaces are the H (div) finite element spaces which were discussedin [3]. The idea of these new elements construction in [23] and [3] is enriching theshape function space of the Q − Λ k and the Raviart-Thomas-Nedelec elements oncubical meshes with d-free and divergence-free shape functions in order to associatea basis of the shape function space to the degrees of freedom given by nodal pointevaluation. The most technical difficulty of new element construction in [3] is theunisolvency proof. For this, the authors in [3] used some features of the enrichedshape functions observed from their explicit expressions. However, it is highlynontrivial to use the same argument to ˜ Q r Λ k for general n and k because explicitforms of the enriched shape functions for general n and k are too complicatedto use conventional unisolvency arguments. To circumvent this difficulty we willintroduce new quantities of polynomial differential forms which allow us to extractuseful features of polynomial differential forms for unisolvency proof.In the paper the space of polynomial or polynomial differential forms withoutspecified domain will denote the space on R n . For example, Q r Λ k is the space ofpolynomial coefficient differential forms such that the polynomials coefficients are in Q r ( R n ). We use P Λ k to denote the space of all differential forms with polynomialcoefficients.For later discussion we introduce some additional notation for cubes in a hy-perspace in R n . For given I = { i , . . . , i m } ⊂ { , . . . , n } with i < i < . . . < i m ,consider an m -dimensional hyperspace F in R n determined by fixing all x j coor-dinates of j
6∈ I . If f is an m -dimensional cube in the m -dimensional hyperspace OCAL CODERIVATIVES 7 F , then Σ k ( f ) with k ≤ m is defined by the set of increasing injective map from { , . . . , k } to I . For σ ∈ Σ k ( f ) we will use J σ K to denote the range of σ , i.e., J σ K = { σ , σ , . . . , σ k } ⊂ I , and σ ∗ for σ ∈ Σ k ( f ) is the complementary sequence in Σ m − k ( f ) such that J σ K ∪ J σ ∗ K = I . For σ ∈ Σ k ( f ) with 1 ≤ k ≤ m we will use σ − i for i ∈ [[ σ ]] to denote the element τ ∈ Σ k − ( f ) such that [[ τ ]] = [[ σ ]] \ { i } . For σ ∈ Σ k ( f ) with 0 ≤ k ≤ m − σ + j is defined similarly for j ∈ [[ σ ∗ ]]. For i ∈ [[ σ ∗ ]] we let ǫ ( i, σ ) = ( − l where l = |{ j ∈ [[ σ ]] : j < i }| . For each σ ∈ Σ k ( f ) we define d x σ = d x σ ∧ · · · ∧ d x σ k andthe set { d x σ : σ ∈ Σ k ( f ) } is a basis of Alt k ( f ).Recall that Σ k is Σ k ( f ) with I = { , . . . , n } . A differential k -form u on Ω thenadmits the representation u = X σ ∈ Σ k u σ d x σ , where the coefficients u σ ’s are scalar functions on Ω. Furthermore, the exteriorderivative d u can be expressed asd u = X σ ∈ Σ k n X i =1 ∂ i u σ d x i ∧ d x σ , if ∂ i u σ is well-defined as a function on Ω. The Koszul operator κ : Alt k ( R n ) → Alt k − ( R n ) is defined by contraction with the vector x , i.e., ( κu ) x = x y u x . Asa consequence of the alternating property of Alt k ( R n ), it therefore follows that κ ◦ κ = 0. It also follows that κ (d x σ ) = κ (d x σ ∧ · · · ∧ d x σ k ) = k X i =1 ( − i +1 x σ i d x σ ∧ · · · ∧ d d x σ i ∧ · · · ∧ d x σ k , where d d x σ i means that the term d x σ i is omitted. This definition is extended to thespace of differential k -form on Ω by linearity, i.e., κu = κ X σ ∈ Σ k u σ d x σ = X σ ∈ Σ k u σ κ (d x σ ) . For future reference we note that κ d x σ = X i ∈ [[ σ ]] ǫ ( i, σ − i ) x i d x σ − i . (3.1)If f is an ( n − R n obtained by fixing one coordinate,for example, f = { x ∈ R n : x n = c } , then we can define the Koszul operator κ f for forms defined on f by ( κ f v ) x =( x − x f ) y v x , where x f = (0 , . . . , , c ). We note that the vector x − x f is in thetangent space of f for x ∈ f . Since tr f (( x − x f ) y u ) = tr f ( x − x f ) y u for x ∈ f and( κu ) x = ( x − x f ) y u x + x f y u x , we can conclude that(3.2) tr f κu = κ f tr f u + tr f ( x f y u ) . JEONGHUN J. LEE
For a multi-index α of n nonnegative integers, x α = x α · · · x α n n . If u is in Q r Λ k ,the space of polynomial k -forms with tensor product polynomials of order r , then u can be expressed as u = X σ ∈ Σ k u σ d x σ , u σ ∈ Q r . Denoting H r Λ k the space of differential k -forms with homogeneous polynomialcoefficients of degree r , we also have the identity( κ d + d κ ) u = ( r + k ) u, u ∈ H r Λ k , (3.3)cf. [6, Section 3]. Finally, throughout this paper we set ˆ T = [ − , n .3.1. The shape function space and the degrees of freedom of ˜ Q r Λ k . Inthis subsection we define the shape function space of ˜ Q r Λ k . The shape functions of˜ Q r Λ k will be obtained by enriching the shape functions of Q − r Λ k . There are otherfamilies of cubical finite element differential forms, cf. for example [4, 12, 13, 17],but these spaces are not involved in the construction of our new elements.If m is a k -form given by m = p d x σ , where σ ∈ Σ k and the coefficient polynomial p is a monomial, then we will call m form monomial. Before we define the shapefunctions of ˜ Q r Λ k let us introduce new quantities of form monomials. For a poly-nomial differential form u σ d x σ we call the indices in [[ σ ]] (in [[ σ ∗ ]], resp.) conformingindices ( nonconforming indices , resp.). For a form monomial m = c α x α d x σ = 0and α i = s for a conforming (nonconforming, resp.) index i of m , we call i a con-forming (nonconforming, resp.) index of degree s . We also define the conformingand nonconforming s -degrees of m = c α x α d x σ bycdeg s ( m ) = |{ i : i ∈ [[ σ ]] and α i = s }| , ncdeg s ( m ) = |{ i : i ∈ [[ σ ∗ ]] and α i = s }| . We can define cdeg s ( v ) and ncdeg s ( v ) if these quantities are same for any formmonomial of v . We remark that ncdeg ( m ) is same as the linear degree in [4] fora form monomial m . In contrast to the linear degree, we do not define cdeg s andncdeg s for general polynomial differential forms.The conforming degree gives a new characterization of the shape function spaceof Q − r Λ k by Q − r Λ k = span { x α d x σ ∈ Q r Λ k : cdeg r ( x α d x σ ) = 0 } . (3.4)Defining B r Λ k as B r Λ k = span { x α d x σ ∈ Q r Λ k : cdeg r ( x α d x σ ) > } , (3.5)it is easy to see that Q r Λ k = Q − r Λ k ⊕ B r Λ k . (3.6)We define the shape function space ˜ Q r Λ k as˜ Q r Λ k = Q − r Λ k + d κ B r Λ k . (3.7) Lemma 3.1.
Let m = x α d x σ ∈ P Λ k , ≤ k ≤ n − , be given with a multi-index α and a positive integer s . Assume that m ′ and m ′′ are given as m ′ = ∂ i x α d x i ∧ d x σ OCAL CODERIVATIVES 9 with i [[ σ ]] and m ′′ = x α x σ j d x σ ∧ · · · ∧ d d x σ j ∧ · · · ∧ d x σ k with j ∈ [[ σ ]] . Then thefollowing identities hold: ncdeg s ( m ′ ) = ncdeg s ( m ) − δ s,α i , (3.8) ncdeg s ( m ′′ ) = ncdeg s ( m ) + δ α σj ,s − , (3.9) ncdeg s +1 ( m ) + cdeg s ( m ) = ncdeg s +1 ( m ′ ) + cdeg s ( m ′ )(3.10) = ncdeg s +1 ( m ′′ ) + cdeg s ( m ′′ ) where δ i,j is the Kronecker delta. In particular, if m ∈ B r Λ k and m ′′ is a formmonomial of κm with j which is a nonconforming index of degree ( r + 1) , then j isa conforming index of degree r in m .Proof. We show (3.8) and the first identity in (3.10). If α i = s , then i is a non-conforming index of degree s of m but not of m ′ , and the other nonconformingindices of degree s of m and m ′ are same, so ncdeg s ( m ′ ) = ncdeg s ( m ) −
1. If s > m ′ with degree s − m with degree s − { i } . Therefore the first identityin (3.10) holds. If α i = s , then the sets of nonconforming indices of degree s ofboth m and m ′ are same, so ncdeg s ( m ′ ) = ncdeg s ( m ). If s >
1, then the sets ofconforming indices of degree s − m and m ′ are same as well, so the firstidentity in (3.10) holds.We show (3.9) and the second identity in (3.10). If α j = s −
1, then j is anonconforming index of degree s of m ′′ in addition to the nonconforming indices ofdegree s of m , so (3.9) holds. If α i = s −
1, then the sets of nonconforming indicesof degree s of m and m ′′ are same, so (3.9) again holds. The second identity in(3.10) can be verified in a way similar to the argument used for the first identity in(3.10).For the particular case, if m ∈ B r Λ k and m ′′ is a form monomial of κm whichhas a nonconforming index of degree ( r + 1), then the nonconforming index of m ′′ must be σ j and α σ j = r because α l ≤ r for 1 ≤ l ≤ n and α σ j + 1 = r + 1. Thiscompletes the proof. (cid:3) Corollary 3.2.
Let m be a form monomial. Then for any form monomial ˜ m in d m , cdeg s ( ˜ m ) ≥ cdeg s ( m ) for any s ≥ . Similarly, for any form monomial ˜ m in d m , ncdeg s ( ˜ m ) ≥ ncdeg s ( m ) if s ≥ .Proof. It is easy to check the assertions by Lemma 3.1. (cid:3)
Corollary 3.3.
The following inclusions hold: κ Q − r Λ k ⊂ Q − r Λ k − , (3.11) d B r Λ k ⊂ B r Λ k +1 ∩ d κ B r Λ k +1 , (3.12) d Q r Λ k ⊂ ˜ Q r Λ k +1 , (3.13) d κ Q r Λ k ⊂ ˜ Q r Λ k . (3.14) Proof.
The inclusion (3.11) and d B r Λ k ⊂ B r Λ k +1 can be easily checked by thecharacterizations of Q − r Λ k , B r Λ k in (3.4), (3.5), and by Lemma 3.1. To show (3.12),let u ∈ H s Λ k ∩ B r Λ k for a positive integer s . By (3.3), (d κ + κ d)d u = ( s + k )d u =d κ d u ∈ d κ B r Λ k +1 . As a consequence, d u ∈ B r Λ k +1 ∩ d κ B r Λ k +1 . (3.13) followsfrom (3.12) and d Q − r Λ k ⊂ Q − r Λ k +1 . Finally, (3.14) follows from (3.6), (3.11), thefact d Q − r Λ k ⊂ Q − r Λ k +1 , and (3.13). (cid:3) We now prove that ˜ Q r Λ k is invariant under dilation and translation. Lemma 3.4. If φ : R n → R n is a composition of dilation and translation, then φ ∗ ˜ Q r Λ k ⊂ ˜ Q r Λ k , where φ ∗ is the pullback of φ .Proof. Let φ ( x ) = Ax + b for a given invertible n × n diagonal matrix A and avector b ∈ R n . To show φ ∗ ˜ Q r Λ k ⊂ ˜ Q r Λ k , assume that u ∈ ˜ Q r Λ k is written as u = u − + d κu + with u − ∈ Q − r Λ k and u + ∈ B r Λ k . Then we have φ ∗ u = φ ∗ u − + φ ∗ d κu + = φ ∗ u − + d φ ∗ κu + = φ ∗ u − + d κφ ∗ u + + b y d( φ ∗ u + )where we used φ ∗ κu + = κφ ∗ u + + b y ( φ ∗ u + ) in the last equality (cf. [6, Section 3.2]).We can easily check φ ∗ u − ∈ Q − r Λ k from the definition of φ ∗ , and d κ Q − r Λ k ⊂ Q − r Λ k from (3.9) and (3.8). From (3.6) we haved κφ ∗ u + ∈ d κ Q r Λ k = d κ ( Q − r Λ k ⊕ B r Λ k ) ⊂ Q − r Λ k + d κ B r Λ k = ˜ Q r Λ k . It remains to show d( b y ( φ ∗ u + )) ∈ ˜ Q r Λ k . (3.15)To see this, note that b y ( φ ∗ u + ) ∈ Q r Λ k − . By (3.13) we have d( b y ( φ ∗ u + )) ∈ ˜ Q r Λ k ,so (3.15) is proved. (cid:3) Lemma 3.5.
The following hold: (a)
For a form monomial m = 0 in B r Λ k , κm generates at least one formmonomial whose nonconforming ( r + 1) -degree is . (b) The operator d κ is injective on B r Λ k .Proof. For m = x α d x σ ∈ B r Λ k there is at least one conforming index σ i suchthat α σ i = r . Then x α κ (d x σ ) has at least one form monomial such that σ i is anonconforming index and its polynomial coefficient has x r +1 σ i as a factor, so (a) isproved.For the injectivity of d κ on B r Λ k , it suffices to show that κ is injective on B r Λ k because d is injective on the image of κ . To show κ is injective on B r Λ k , we showthat form monomials with positive nonconforming ( r + 1)-degree generated by κm for m ∈ B := { x α d x σ : cdeg r ( x α d x σ ) > } are distinct. More precisely, if κm and κ ˜ m for m, ˜ m ∈ B have a same form monomial (up to ±
1) whose nonconforming( r + 1)-degree is 1, then m = ˜ m . To show it by contradiction, let m = x α d x σ and ˜ m = x ˜ α d x ˜ σ be two distinct elements in B and assume that κm and κ ˜ m havea common form monomial with nonconforming index of degree ( r + 1). From thedefinition of κ and the common form monomial assumption, there exist σ i ∈ [[ σ ]]and ˜ σ i ∈ [[˜ σ ]] such that x σ i x α d x σ ∧ · · · ∧ d d x σ i ∧ · · · ∧ d x σ k = ± x ˜ σ i x ˜ α d x ˜ σ ∧ · · · ∧ d d x ˜ σ i ∧ · · · ∧ d x ˜ σ k . (3.16)Since σ i and ˜ σ i are the only nonconforming indices of degree ( r + 1) by Lemma 3.1, σ i = ˜ σ i and therefore d x σ = d x ˜ σ . Moreover, comparison of x α and x ˜ α leads to α = ˜ α , so it contradicts to x α d x σ = x ˜ α d x ˜ σ . (cid:3) The following key result is a consequence of the above lemma.
Theorem 3.6.
For ≤ k ≤ n it holds that dim ˜ Q r Λ k = (cid:18) nk (cid:19) ( r + 1) n . OCAL CODERIVATIVES 11
Proof.
By Lemma 3.5 (b), the spaces d κ B r Λ k and B r Λ k have the same dimension,so it suffices to show that Q − r Λ k ∩ d κ B r Λ k = { } . Suppose that 0 = u ∈ B r Λ k and d κu ∈ Q − r Λ k . Every form monomial m of d κu satisfies cdeg r ( m ) ≥ r +1 ( m ) = 0. However, it is a contradiction to thecharacterization of Q − r Λ k in (3.4), so u = 0. (cid:3) Degrees of freedom and unisolvence of ˜ Q r Λ k . In this subsection we definethe degrees of freedom of ˜ Q r Λ k and prove the unisolvency for the degrees of freedom.For the degrees of freedom we define two polynomial spaces for σ ∈ Σ k ( f ) for acube f included in an m -dimensional hyperspace by Q r,σ ( f ) = O i ∈ [[ σ ]] P r ( x i ) , Q r,σ ∗ ( f ) = O i ∈ [[ σ ∗ ]] P r ( x i )where P r ( x i ) is the space of polynomials of x i with degree less than or equal to r .The degrees of freedom of ˜ Q r Λ k ( ˆ T ) is u Z f tr f u ∧ v, f ∈ ∆ l ( ˆ T ) , k ≤ l ≤ n, v ∈ X τ ∈ Σ l − k ( f ) ( Q r − ,τ ⊗ Q r,τ ∗ )( f )d x τ (3.17)where ( Q r − ,τ ⊗ Q r,τ ∗ )( f ) = Q r − ,τ ( f ) ⊗ Q r,τ ∗ ( f ). Theorem 3.7.
The number of degrees of freedom given by (3.17) is (cid:18) nk (cid:19) ( r + 1) n . Proof.
For τ ∈ Σ l − k ( f ) with f ∈ ∆ l ( ˆ T ), l ≥ k ,dim Q r − ,τ ( f ) = ( r − l − k , dim Q r,τ ∗ ( f ) = ( r + 1) k . We can easily check that | ∆ l ( ˆ T ) | = (cid:18) nn − l (cid:19) n − l , | Σ l − k ( f ) | = (cid:18) ll − k (cid:19) for f ∈ ∆ l ( ˆ T ) . Therefore the number of degrees of freedom given by (3.17) is X k ≤ l ≤ n (cid:18) nn − l (cid:19) n − l (cid:18) ll − k (cid:19) ( r − l − k ( r + 1) k = n ! ( r + 1) k k ! ( n − k )! X k ≤ l ≤ n ( n − k )!( n − l )! ( l − k )! 2 n − l ( r − l − k = (cid:18) nk (cid:19) ( r + 1) k X ≤ i ≤ n − k ( n − k )!( n − k − i )! i ! 2 n − k − i ( r − i = (cid:18) nk (cid:19) ( r + 1) n , so the proof is complete. (cid:3) The following result will be useful to reduce unisolvency proof.
Theorem 3.8 (trace property) . Let f be a hyperspace in R n determined by fixingone coordinate x i . Then tr f ˜ Q r Λ k ⊂ ˜ Q r Λ k ( f ) . Proof.
Without loss of generality we assume that f = { x ∈ R n : x n = c } forsome constant c . It is easy to check by definition that tr f Q − r Λ k ⊂ Q − r Λ k ( f ), soit is enough to show that tr f d κ B r Λ k ⊂ ˜ Q r Λ k ( f ). We will show tr f d κ ( x α d x σ ) ∈ ˜ Q r Λ k ( f ) for all x α d x σ ∈ B r Λ k below. By (3.2) and the commutativity of tr f andd, tr f d κ ( x α d x σ ) = d κ f tr f ( x α d x σ ) + d tr f ( x f y ( x α d x σ ))where x f is the vector field (0 , . . . , , c ) on R n .If n ∈ [[ σ ]], then tr f ( x α d x σ ) = 0 and tr f ( x f y ( x α d x σ )) = ( cx α | x n = c )d x ˜ σ ∈Q r Λ k − ( f ) where d x ˜ σ is defined by d x σ = d x ˜ σ ∧ d x n . From (3.13) we can con-clude that tr f d κ ( x α d x σ ) ∈ ˜ Q r Λ k ( f ). If n [[ σ ]], then x f y ( x α d x σ ) = 0. Sincetr f ( x α d x σ ) = x α | x n = c d x σ ∈ Q r Λ k ( f ), the conclusion follows from (3.14). (cid:3) Before we start the unisolvence proof we need a lemma and auxiliary definitions.
Lemma 3.9.
Suppose that σ, ˜ σ ∈ Σ k , ≤ k ≤ n − , σ = ˜ σ satisfy σ + i = ˜ σ + ˜ i for some i ∈ [[˜ σ ]] , ˜ i ∈ [[ σ ]] . Then ǫ ( i, σ ) ǫ ( i, ˜ σ − i ) − ǫ (˜ i, ˜ σ ) ǫ (˜ i, σ − ˜ i ) = 0 . Proof.
We first prove it under the assumption i > ˜ i . Let a = |{ l : l < ˜ i, l ∈ [[ σ ]] ∩ [[˜ σ ]] }| ,b = |{ l : ˜ i < l < i, l ∈ [[ σ ]] ∩ [[˜ σ ]] }| ,c = |{ l : i < l, l ∈ [[ σ ]] ∩ [[˜ σ ]] }| . Then one can check ǫ ( i, σ ) = ( − a + b +1 , ǫ (˜ i, ˜ σ ) = ( − a ,ǫ ( i, ˜ σ − i ) = ( − a + b , ǫ (˜ i, σ − ˜ i ) = ( − a , so the assertion follows. If i < ˜ i , then we set a = |{ l : l < i, l ∈ [[ σ ]] ∩ [[˜ σ ]] }| ,b = |{ l : i < l < ˜ i, l ∈ [[ σ ]] ∩ [[˜ σ ]] }| ,c = |{ l : ˜ i < l, l ∈ [[ σ ]] ∩ [[˜ σ ]] }| , and one can check that ǫ ( i, σ ) = ( − a , ǫ (˜ i, ˜ σ ) = ( − a + b +1 ,ǫ ( i, ˜ σ − i ) = ( − a , ǫ (˜ i, σ − ˜ i ) = ( − a + b . The proof is complete. (cid:3)
We define D r,l Λ k as D r,l := { u ∈ P Λ k : ncdeg r +1 ( m ) + cdeg r ( m ) = l for every form monomial m in u } . By Lemma 3.1 d κ maps D r,l Λ k into itself. Considering the decomposition of B r Λ k B r Λ k = M ≤ l ≤ k M r ≤ s ≤ nr H s Λ k ∩ D r,l Λ k ∩ B r Λ k OCAL CODERIVATIVES 13 we have a decomposition of d κ B r Λ k d κ B r Λ k = M ≤ l ≤ k M r ≤ s ≤ nr H s Λ k ∩ D r,l Λ k ∩ d κ B r Λ k . (3.18)Let us recall the degrees of freedom of Q − r Λ k ( ˆ T ) with vanishing trace in [5]. If u ∈ Q − r Λ k ( ˆ T ) and tr f u = 0 for all f ∈ ∆ n − ( ˆ T ), then Z ˆ T u ∧ v = 0 , ∀ v ∈ Q − r − Λ n − k ( ˆ T )(3.19)implies that u = 0.We are now ready to prove unisolvency of ˜ Q r Λ k ( ˆ T ), r ≥
1, with the degrees offreedom (3.17).
Proposition 3.10 (unisolvence with vanishing trace assumption) . Suppose that u ∈ ˜ Q r Λ k ( ˆ T ) , r ≥ , and tr f u = 0 for all f ∈ ∆ n − ( ˆ T ) . If Z ˆ T u ∧ v = 0 ∀ v ∈ X τ ∈ Σ n − k ( ˆ T ) ( Q r − ,τ ⊗ Q r,τ ∗ )( ˆ T )d x τ , (3.20) then u = 0 . Here we accept Q − = ∅ for convention.Proof. If k = 0 or k = n , then ˜ Q r Λ k ( ˆ T ) = Q r Λ k ( ˆ T ) and (3.20) is a standard set ofdegrees of freedom for the shape functions with vanishing trace, so there is nothingto prove.Assume that 0 < k < n , and let u = P σ ∈ Σ k ( ˆ T ) u σ d x σ ∈ ˜ Q r Λ k ( ˆ T ) be a shapefunction with vanishing trace. From the vanishing trace assumption tr f u = 0 forall f ∈ ∆ n − ( ˆ T ), u σ vanishes on all faces f ∈ ∆ n − ( ˆ T ) determined by x i = ± i ∈ [[ σ ]]. Therefore u σ = b σ ∗ ˜ u σ with b σ ∗ := Q l ∈ [[ σ ∗ ]] (1 − x l ) for all σ ∈ Σ k ( ˆ T ).In the degree properties (3.8), (3.9), (3.10), the coefficients of all form monomialsin ˜ Q r Λ k ( ˆ T ) have at most one variable of degree ( r + 1). From this observation and(3.20), u has a form u = X σ ∈ Σ k ( ˆ T ) b σ ∗ X i ∈ [[ σ ∗ ]] L wr − ( x i ) p σ,i d x σ (3.21)where L ws ( t ) is the monic Legendre polynomial of degree s on [ − ,
1] with weight(1 − t ), and p σ,i is a polynomial in ( Q r,σ ⊗ Q r − ,σ ∗ )( ˆ T ) which is independent of x i . Recall the decomposition (3.18) and let u = u + X l,s u l,s ∈ Q − r Λ k ( ˆ T ) ⊕ M ≤ l ≤ k M r ≤ s ≤ nr H s Λ k ( ˆ T ) ∩ D r,l Λ k ( ˆ T ) ∩ d κ B r Λ k ( ˆ T ) . Let l be the largest index l such that u l,s = 0 for some s , and let s be the largestindex s such that u l ,s = 0. In the proof below, we will show u l ,s = 0. Byinduction, this implies that u = u ∈ Q − r Λ k ( ˆ T ), and therefore u = 0 by (3.20),(3.19), and the inclusion Q − r − Λ n − k ( ˆ T ) ⊂ X τ ∈ Σ n − k ( ˆ T ) ( Q r − ,τ ⊗ Q r,τ ∗ )( ˆ T )d x τ . We now start the proof of u l ,s = 0. In the form (3.21), if p is a monomialof p σ,i for fixed σ , then cdeg r ( p d x σ ) < l because any form monomial m con-taining the factors x r +1 i and p in its polynomial coefficient, which comes from theexpansion of the coefficient of d x σ in (3.21), satisfies both of ncdeg r +1 ( m ) = 1 andncdeg r +1 ( m ) + cdeg r ( m ) ≤ l . Note that it also implies that u l ,s does not haveany form monomials with conforming r -degree l .Let q σ,i be the homogeneous polynomial of p σ,i for each σ and i ∈ [[ σ ∗ ]] whichcontribute to consist u l ,s , i.e., u l ,s = X σ ∈ Σ k ( ˆ T ) Y l ∈ [[ σ ∗ ]] x l X i ∈ [[ σ ∗ ]] x r − i q σ,i d x σ (3.22)and cdeg r ( q σ,i d x σ ) = l − u l ,s ∈ d κ B r Λ k ( ˆ T )by definition, d u l ,s = 0.We now show u l ,s = 0 for l = 1. If l = 1, then q σ,i ∈ ( Q r − ,σ ⊗ Q r − ,σ ∗ )( ˆ T )because cdeg r ( q σ,i d x σ ) < l . We claim that q σ,i = 0 for all σ and i . To prove itwe show that the form monomials in d u l ,s with conforming r -degree 1, are alldistinct. Note that such form monomials are from( r + 1) Y l ∈ [[ σ ∗ ]] x l x r − i q σ,i d x i ∧ d x σ (3.23)for i ∈ [[ σ ∗ ]]. For fixed σ the form monomials in this formula are all distinct fordifferent i ’s. Further, if we assume that there is ˜ σ ∈ Σ k ( ˆ T ) with ˜ σ = σ and ˜ i ∈ [[˜ σ ∗ ]]such that the form monomials in( r + 1) Y l ∈ [[˜ σ ∗ ]] x l x r − i q ˜ σ, ˜ i d x ˜ i ∧ d x ˜ σ are not linearly independent with the ones in (3.23), then it leads to a contradictionbecause i and ˜ i are the only conforming indices of degree r in these form monomials,and therefore i = ˜ i , σ = ˜ σ . Thus, all form monomials in d u l ,s with conforming r -degree 1 are distinct and have the form (3.23). From d u l ,s = 0, q σ,i = 0 for all σ ∈ Σ k ( ˆ T ) and i ∈ [[ σ ∗ ]], and therefore u l ,s = 0 by (3.22).If l >
1, then we consider the expression of q σ,i as a sum of homogeneouspolynomials q σ,i = X τ ⊂ [[ σ ]] , | τ | = l − Y j ∈ τ x rj q σ,i,τ (3.24)in which q σ,i,τ ∈ ( Q r − ,σ ⊗ Q r − ,σ ∗ )( ˆ T ) is independent of the variables x i and x j ’sfor j ∈ τ . Here q σ,i,τ can be vanishing for some τ ⊂ [[ σ ]]. In the discussion below, wewill make a system of equations with all possibly involving q σ,i,τ ’s as its unknown.The equations are obtained from κu l ,s and d u l ,s , and the right-hand sides arezero. We then show that the system is well-posed, so all q σ,i,τ ’s vanish.To derive equations from κu l ,s , note that κu l ,s ∈ κ d κ B r Λ k ( ˆ T ) = κ B r Λ k ( ˆ T ),so the nonconforming ( r + 1)-degree of form monomials in κu l ,s is at most 1 byLemma 3.1. In the formal expression of κu l ,s using (3.22) and (3.24), for fixed σ ,the form monomials which have nonconforming ( r + 1)-degree 2 are obtained by Y l ∈ [[ σ ∗ ]] x l x r − i Y l ∈ τ x rl x j ǫ ( j, σ − j ) q σ,i,τ d x σ − j (3.25) OCAL CODERIVATIVES 15 for σ ∈ Σ k ( ˆ T ), i ∈ [[ σ ∗ ]], τ ⊂ [[ σ ]] with | τ | = l −
1, and j ∈ τ . We claim that all ofthese terms are linearly independent for fixed σ ∈ Σ k ( ˆ T ). To see it, assume that theform (3.25) with ( i, τ, j ) and ( i ′ , τ ′ , j ′ ) are linearly dependent for i ′ ∈ [[ σ ∗ ]], τ ′ ⊂ [[ σ ]], j ′ ∈ τ ′ . Comparison of the nonconforming indices with degree ( r + 1) gives i = i ′ , j = j ′ or i = j ′ , j = i ′ . If i = i ′ and j = j ′ , then τ = τ ′ from the comparisonof conforming indices of degree r . If i = j ′ and j = i ′ , then d x σ − j = d x σ − j ′ , sothey cannot be linearly dependent. Therefore the terms of the form (3.25) are alldistinct for fixed σ .We now assume that (3.25) with ( σ, i, τ, j ) and (˜ σ, ˜ i, ˜ τ , ˜ j ) are linearly dependentfor ˜ σ = σ . Comparing the nonconforming indices of degree ( r + 1), i = ˜ i , j = ˜ j or i = ˜ j , j = ˜ i . However, if i = ˜ i and j = ˜ j , then d x σ − j = d x ˜ σ − ˜ j implies σ = ˜ σ which is a contradiction, so i = ˜ j and j = ˜ i . Regarding this and comparing theconforming indices of degree r , linear dependence occurs only when i = ˜ j, j = ˜ i, τ ∪ { i } = ˜ τ ∪ { ˜ i } , σ − ˜ i = ˜ σ − i. (3.26)As a consequence, for fixed ( σ, i, τ, j ) with i ∈ [[ σ ]] and j ∈ τ , there is a uniquequadruple (˜ σ, ˜ i, ˜ τ , ˜ j ) determined by (3.26) which may generate a linearly dependentpolynomial differential form in the form of (3.25). Since all form monomials whichhave nonconforming ( r + 1)-degree 2 are vanishing, ǫ (˜ i, σ − ˜ i ) q σ,i,τ + ǫ ( i, ˜ σ − i ) q ˜ σ, ˜ i, ˜ τ = 0(3.27)for the quadruples ( σ, i, τ, j ) and (˜ σ, ˜ i, ˜ τ , ˜ j ) satisfying (3.26).We now consider the expressions of form monomials with conforming r -degree l in d u l ,s . From (3.22) and (3.24), they have a form( r + 1) x r − i Y l ∈ [[ σ ∗ ]] x l Y l ∈ τ x rl ǫ ( i, σ ) q σ,i,τ d x σ + i . (3.28)By checking the differential form component and the conforming indices of degree r one can check that all of these terms are distinct over i ∈ [[ σ ∗ ]] and τ ⊂ [[ σ ]] with | τ | = l − σ .Considering ( σ, i, τ ) and (˜ σ, ˜ i, ˜ τ ) generating linearly dependent differential formsof the form (3.28), a direct comparison gives σ + i = ˜ σ + ˜ i, τ ∪ { i } = ˜ τ ∪ { ˜ i } which is exactly same as (3.26). The identity d u l ,s = 0 with (3.25) gives a newequation ǫ ( i, σ ) q σ,i,τ + ǫ (˜ i, ˜ σ ) q ˜ σ, ˜ i, ˜ τ = 0 . (3.29)By Lemma 3.9, (3.27), and (3.29), it follows that q σ,i,τ = q ˜ σ, ˜ i, ˜ τ = 0. Since this istrue for any σ , i , τ , u l ,s = 0 and this completes the proof. (cid:3) Theorem 3.11 (unisolvence) . Let u ∈ ˜ Q r Λ k ( ˆ T ) . If Z f tr f u ∧ v = 0 ∀ v ∈ X τ ∈ Σ l − k ( f ) ( Q r − ,τ ⊗ Q r,τ ∗ )( f )d x τ , (3.30) for all f ∈ ∆ l ( ˆ T ) , l ≥ k , then u = 0 . Proof. If l = k , then tr f u ∈ ˜ Q r Λ k ( f ) = Q r Λ k ( f ) for f ∈ ∆ k ( ˆ T ) by Theorem 3.8,and (3.30) gives a standard set of degrees of freedom for Q r Λ k ( f ), so tr f u = 0.Suppose that tr f u = 0 for all f ∈ ∆ l ( ˆ T ) for some l ≥ k . For any given ˜ f ∈ ∆ l +1 ( ˆ T ),tr ˜ f u ∈ ˜ Q r Λ k ( ˜ f ) and tr f tr ˜ f u = tr f u = 0 for all f ∈ ∆ l ( ˜ f ) by the assumption. ByProposition 3.10 and the degrees of freedom (3.30), tr ˜ f u = 0. By induction, wecan show that tr f u = 0 for all f ∈ ∆ n − ( ˆ T ), so u = 0 by Proposition 3.10. (cid:3) Nodal tensor product degrees of freedom.
In this subsection we showthat ˜ Q r Λ k ( ˆ T ) has a set of degrees of freedom given by evaluating nodal values at aset of points in ˆ T . This alternative set of degrees of freedom will be used to definenumerical methods with local coderivatives in the next section.The Gauss–Lobatto quadrature rule with r + 1( r ≥
1) points uses r − I = [ − ,
1] with positive weights and gives exactintegration of polynomials of order 2 r −
1. Let { ξ j } rj =0 be the quadrature pointsof the Gauss-Lobatto quadrature on I and { λ j } rj =0 with λ j > T by taking tensor productsof the Gauss-Lobatto quadrature nodes and weights, and we discuss the formalexpressions for the tensor product quadrature rules below.For σ ∈ Σ k ( ˆ T ) let N σ,r be the set of points in N ≤ i ≤ k I ( x σ i ) defined by N σ,r = { ( x σ , . . . , x σ k ) = ( ξ j , . . . , ξ j k ) : 0 ≤ j l ≤ r, ≤ l ≤ k } .N σ ∗ ,r is defined similarly as N σ ∗ ,r = { ( x σ ∗ , . . . , x σ ∗ n − k ) = ( ξ j , . . . , ξ j n − k ) : 0 ≤ j l ≤ r, ≤ l ≤ n − k } , the set of points in the ( n − k )-dimensional cube N ≤ i ≤ n − k I ( x σ ∗ i ). Therefore thetensor product N r := N σ,r ⊗ N σ ∗ ,r is the set given by the tensor product of Gauss-Lobatto quadrature points on ˆ T . Letting N be the set of nonnegative integers, wewill use ξ i σ and ξ j σ ∗ with multi-indices i ∈ N k and j ∈ N n − k to denote the points in N σ,r and N σ ∗ ,r , respectively. We also use λ j σ and λ j σ ∗ to denote the correspondingtensor-product weights of the Gauss-Lobatto quadrature.For a continuous function v on ˆ T , we define R r ( v ), E r,ξ i σ ( v ), E r ( v ) as R r ( v ) = (cid:16) v ( ξ i σ ⊗ ξ j σ ∗ ) (cid:17) ξ i σ ⊗ ξ j σ ∗ ∈ N r ∈ R ( r +1) n ,E r,ξ i σ ( v ) = X ξ j σ ∗ ∈ N r,σ ∗ λ j σ ∗ v ( ξ i σ ⊗ ξ j σ ∗ ) ,E r ( v ) = X ξ i σ ∈ N r,σ λ i σ E r,ξ i σ ( v ) . Similarly, for polynomial differential forms u = u σ d x σ ∈ P Λ k ( ˆ T ) we define R r ( u )as the element in R M ⊗ Λ k with M = (cid:18) nk (cid:19) ( r + 1) n , which consists of R r ( u σ ) ⊗ d x σ for σ ∈ Σ k ( ˆ T ). OCAL CODERIVATIVES 17
Lemma 3.12.
For v = P σ ∈ Σ k ( ˆ T ) v σ d x σ suppose that a polynomial v σ has a formwith v σ = b σ ∗ ˜ v σ , b σ ∗ := Y l ∈ [[ σ ∗ ]] (1 − x l ) , ˜ v σ ∈ ( Q r − ,σ ∗ ⊗ Q r,σ )( ˆ T ) . Suppose also that nonconforming ( r + 1) -degree of every form monomial in v σ d x σ is at most 1 for all σ ∈ Σ k ( ˆ T ) . If R r ( v σ ) = 0 all σ ∈ Σ k ( ˆ T ) , then v = 0 .Proof. If we rewrite ˜ v σ with the weighted Legendre polynomial L wj ’s, then theassumption on nonconforming ( r + 1)-degree leads us to have˜ v σ = X i ∈ [[ σ ∗ ]] L wr − ( x i ) p σ,i + ˜ v σ, (3.31)with p σ,i ∈ ( Q r,σ ⊗ Q r − ,σ ∗ )( ˆ T ) independent of x i , in which ˜ v σ, is a sum of poly-nomials of the form Y i ∈ [[ σ ∗ ]] L wd i ( x i ) q d ( x σ )(3.32)where d = ( d σ ∗ , . . . , d σ ∗ n − k ) ∈ N n − k with max i ∈ [[ σ ∗ ]] { d i } ≤ r − q d ∈ Q r,σ ( ˆ T ).We first to show that ˜ v σ, = 0. Let 0 = ψ = Q i ∈ [[ σ ∗ ]] L wd ′ i ( x i ) ∈ Q r − ,σ ∗ ( ˆ T ) with d ′ i ’s satisfying 0 ≤ d ′ i ≤ r − i ∈ [[ σ ∗ ]]. We claim that E r,ξ i σ (cid:0) b σ ∗ L wr − ( x i ) p σ,i ψ (cid:1) = 0for all i ∈ [[ σ ∗ ]] and ξ i σ ∈ N r,σ . To see it, note that the quadrature along x i coordinate can be replaced by integration with x i variable on [ − ,
1] because thedegree of x i variable is r + 1 + d ′ i ≤ r − L wr − ( x i ) is orthogonal to L wd ′ i ( x i )with (1 − x i ) weight. Therefore we have E r,ξ i σ ( vφ ) = E r,ξ i σ ( b σ ∗ ˜ v σ, ψ ) . If we consider the expression of ˜ v σ, in (3.32), a completely analogous argumentusing orthogonality of Legendre polynomials gives E r,ξ i σ ( vφ ) = E r,ξ i σ (cid:0) b σ ∗ ψ ( x σ ∗ ) q d ′ ( x σ ) (cid:1) . where d ′ = ( d ′ σ ∗ , . . . , d ′ σ ∗ n − k ). Note that this result is obtained without using theassumption R r ( v σ ) = 0. Since R r ( v σ ) = 0, the above quantity vanishes. Bythe definition of E r,ξ i σ , we have either b σ ∗ ( ξ j σ ∗ ) ψ ( ξ j σ ∗ ) = 0 for all ξ j σ ∗ ∈ N r,σ ∗ or q d ′ ( ξ i σ ) = 0. However, the first case implies that b σ ∗ ψ = 0 because the nodal valueevaluations at the points in N r,σ ∗ is already a set of degrees of freedom for Q r,σ ∗ ( ˆ T ),and it leads to a contradiction to ψ = 0. Therefore q d vanishes at ξ i σ , and we canshow that q d vanishes at any point in N r,σ with the same argument. Recall that q d ′ ∈ Q r,σ ( ˆ T ), so these vanishing conditions of q d ′ implies that q d ′ = 0. Finally,this holds for any d ′ , and therefore ˜ v σ, = 0.To show v = 0, we notice that v σ with (3.31) is exactly the form of u σ in (3.21),so the same argument in the unisolvency proof can be used to show v = 0. (cid:3) Theorem 3.13.
Suppose that v ∈ ˜ Q r Λ k ( ˆ T ) and R r ( v ) = 0 holds. Then v = 0 . Proof.
Note that tr f v ∈ ˜ Q r Λ k ( f ) = Q r Λ k ( f ) for f ∈ ∆ k ( ˆ T ). Since the restrictionof R r on f becomes a set of quadrature degrees of freedom of Q r Λ k ( f ), tr f v = 0for all f ∈ ∆ k ( ˆ T ) holds. For tr g v with g ∈ ∆ k +1 ( ˆ T ), all traces of tr g v on k -dimensional subcubes are vanishing, so the assumption of Lemma 3.12 is satisfiedfor tr g v . Applying Lemma 3.12 with the restriction of R r on g , one can concludethat tr g v = 0 for any g ∈ ∆ k +1 ( ˆ T ). We can continue this argument for any g ∈ ∆ l ( ˆ T ), l ≥ k + 1, by induction, so the conclusion follows. (cid:3) Numerical methods with local coderivatives
We construct numerical methods with local coderivatives using ˜ Q r Λ k ( T h ). Forthis we need a modified bilinear form h· , ·i h in (A) , and the auxiliary spaces ˜ V k − h , W k − h and the map Π h in (B) . The conditions (A) , (B) are stated with index k − V kh , ˜ V kh , W kh whichwe choose as V kh = ˜ Q r Λ k ( T h ) , ˜ V kh = Q − r Λ k ( T h ) , W kh = Q dr − Λ k ( T h )(4.1)where Q dr − Λ k ( T h ) is the space of discontinuous polynomial differential forms. Wewill show that the finite elements ˜ Q r Λ k ( T h ) and Q − r Λ k ( T h ) satisfy the conditions (A) , (B) . In addition to the conditions in (A) and (B) , for local coderivatives, wealso need to show that h· , ·i h on ˜ Q r Λ k ( T h ) can give a block diagonal matrix withappropriate choice of global DOFs.We have shown that ˜ Q r Λ k ( ˆ T ) has a set of DOFs determined by evaluations atnodal points. For T ∈ T h we can define the evaluation operator E Tr for continuousfunctions on T with the scaled Gauss-Lobatto quadrature rules on T .For u, v ∈ ˜ Q r Λ k ( T ) with expressions u = P σ ∈ Σ k ( T ) u σ d x σ , v = P σ ∈ Σ k ( T ) v σ d x σ ,we define h u, v i h,T by(4.2) h u, v i h,T = | T | X σ ∈ Σ k ( T ) E Tr ( u σ v σ ) . It is easy to see that h· , ·i h,T is an inner product on ˜ Q r Λ k ( T ) and the norm definedby this inner product is equivalent to the L norm with constants independent onthe scaling of T . We can define the inner product h· , ·i h on ˜ Q r Λ k ( T h ) by h u, v i h = X T ∈T h h u | T , v | T i h,T (4.3)for u, v ∈ ˜ Q r Λ k ( T h ), and the norm k·k h defined by this inner product is equivalentto the L norm with constants independent of h . Therefore (A) is satisfied.We now verify (B) , but with k -forms instead of ( k − V kh and W kh in (4.1), so verification of (B) consists of the following three steps:Step 1. Prove (2.7)Step 2. Define Π h satisfying the conditions in (B) Step 3. Prove (2.8)For Step 1, it is enough to show the identity in (2.7) for the restrictions ofpolynomial differential forms on any T ∈ T h . Moreover, by scaling argument, it OCAL CODERIVATIVES 19 suffices to show the equality on the reference element ˆ T , i.e., h u, v i h, ˆ T = h u, v i , u ∈ Q − r Λ k ( ˆ T ) , v ∈ Q r − Λ k ( ˆ T ) . In fact, this equality is true because the Gauss-Lobatto quadrature rule with ( r + 1)points give exact integration for polynomials of degree 2 r −
1. This completes theproof of Step 1.The following result gives a proof of Step 2.
Theorem 4.1.
Let Π h : ˜ Q r Λ k ( T h ) → Q − r Λ k ( T h ) be the interpolation operatordefined by the canonical degrees of freedom of Q − r Λ k ( T h ) , i.e., Π h u for u ∈ ˜ Q r Λ k ( T ) is characterized by Z f tr f Π h u ∧ v = Z f tr f u ∧ v, v ∈ Q − r − Λ l − k ( f ) , f ∈ ∆ l ( T h ) , l ≥ k. (4.4) Then Π h is bounded in L Λ k (Ω) with a norm independent of h , and d( u − Π h u ) = 0 for u ∈ ˜ Q r Λ k ( T h ) .Proof. Note the identity Q − r Λ k = M σ ∈ Σ k Q σ,r − ⊗ Q σ ∗ ,r d x σ from the characterization of Q − r Λ k . By the definitions of the degrees of freedom of Q − r Λ k and ˜ Q r Λ k , the degrees of freedom of Q − r Λ k ( T h ) is a subset of the degrees offreedom of ˜ Q r Λ k ( T h ). Therefore the uniform L boundedness of Π h is a consequenceof equivalence of the L norm and a discerete norm defined by the degrees of freedomon each of these spaces.To show d( u − Π h u ) = 0, without loss of generality, we consider u defined onlyon one element T . Since d( u − Π h u ) ∈ Q − r Λ k +1 ( T ), it is sufficient to show that Z f tr f d( u − Π h u ) ∧ v = 0 ∀ v ∈ Q − r − Λ l − k − ( f ) , f ∈ ∆ l ( T ) , l ≥ k + 1(4.5)by the canonical degrees of freedom of Q − r Λ k +1 ( T ). These vanishing identitiesfollow from the commutativity of d and tr f , and Stokes’ theorem by Z f tr f d( u − Π h u ) ∧ v = Z f d tr f ( u − Π h u ) ∧ v = Z ∂f tr ∂f tr f ( u − Π h u ) ∧ tr ∂f v + Z f tr f ( u − Π h u ) ∧ d v = 0where the last equality follows from tr ∂f tr f = tr ∂f , the hierarchical trace propertyof Q − r Λ k spaces, the inclusion d v ∈ Q − r − Λ l − k ( f ), and (4.4). (cid:3) For Step 3, we can reduce (2.8) to the corresponding identity on ˆ T , i.e., it isenough to show h u − Π h u, v i h, ˆ T = 0for u ∈ ˜ Q r Λ k ( ˆ T ) and v ∈ Q r − Λ k ( ˆ T ). Before we start its proof, recall that the quadrature nodes of the Gauss-Lobattorule with ( r + 1) points are the zeros of ddt L r ( t ) on [ − , r + 1)( L r +1 ( t ) − L r − ( t )) = (2 r + 1)( tL r ( t ) − L r − ( t )) = (2 r + 1) t − r ddt L r ( t ) , so L r +1 ( ξ j ) − L r − ( ξ j ) = 0 0 ≤ j ≤ r, (4.6)i.e., the evaluation of L r +1 ( t ) − L r − ( t ) at the quadrature nodes of the Gauss-Lobatto rule with ( r + 1) points vanishes.As Π h , we define ˆΠ h : ˜ Q r Λ k ( ˆ T ) → Q − r Λ k ( ˆ T ) as Z f u ∧ v = Z f ˆΠ h u ∧ v v ∈ Q − r − Λ l − k ( f ) , f ∈ ∆ l ( ˆ T ) , l ≥ k. Theorem 4.2.
Let v = u − ˆΠ h u for u ∈ ˜ Q r Λ k ( ˆ T ) . In v = P σ ∈ Σ k ( ˆ T ) v σ d x σ every v σ has a form v σ = X i ∈ [[ σ ∗ ]] ( L r +1 ( x i ) − L r − ( x i )) p σ,i + v σ, (4.7) where p σ,i ∈ Q σ ∗ ,r ( ˆ T ) ⊗Q σ,r − ( ˆ T ) , and every term in v σ, written with the Legendrepolynomials has a factor of L r ( x j ) for some j ∈ [[ σ ]] .Proof. Let v = u − ˆΠ h u = P σ ∈ Σ k ( ˆ T ) v σ d x σ be given. To prove the assertionby induction we need to show the following two results. First, for f ∈ ∆ k ( ˆ T )every polynomial coefficient of tr f v ∈ ˜ Q r Λ k ( f ) satisfies (4.7). Second, if everypolynomial coefficient of tr f v satisfies (4.7) for all f ∈ ∆ l ( ˆ T ) with given l ≥ k ,then tr g v satisfies the same property corresponding to (4.7) for all g ∈ ∆ l +1 ( ˆ T ).We prove the first claim. Recalling tr f v ∈ ˜ Q r Λ k ( f ) = Q r Λ k ( f ) for f ∈ ∆ k ( T )and the definition of ˆΠ h , the polynomial coefficient of tr f v is a polynomial in Q r ( f )which is orthogonal to all Q r − ( f ) in the L inner product. Then it has at leastone factor of L r ( x j ) for j ∈ [[ σ ]], therefore it is a form of (4.7).For the second claim, from the trace property and the definition of ˆΠ h commut-ing with the trace operator tr, we can reduce the claim to the case l = n − f v satisfies (4.7)for all f ∈ ∆ n − ( ˆ T ).By (4.4) and the definition of shape function space of ˜ Q r Λ k ( ˆ T ), v σ has a form X l ∈ [[ σ ∗ ]] L r +1 ( x l ) p σ,l + v σ, + v σ, where p σ,i ∈ ( Q σ ∗ ,r ⊗ Q σ,r − )( ˆ T ) but p σ,i is independent of x i , v σ, is as in theassertion, and v σ, is a polynomial such that every term in its expression with theLegendre polynomials, has a factor of L r ( x l ) or L r − ( x l ) for some l ∈ [[ σ ∗ ]]. Weremark that v σ, may have polynomial terms of degree greater than r . Let usrewrite v σ as v σ = X l ∈ [[ σ ∗ ]] ( L r +1 ( x l ) − L r − ( x l )) p σ,l + v σ, + q σ where q σ ∈ ( Q σ ∗ ,r ⊗ Q σ,r − )( ˆ T ). Note that q σ is a polynomial such that every termin its Legendre polynomial expression has L r ( x l ) factor for some l ∈ [[ σ ∗ ]] or is in OCAL CODERIVATIVES 21 Q r − ( ˆ T ). Let f be an ( n − x i = 1 forsome i ∈ [[ σ ]]. Then the polynomial coefficient of the differential form d x σ in tr f v is X l ∈ [[ σ ∗ ]] ,l = i ( L r +1 ( x l ) − L r − ( x l )) p σ,l | x i =1 + v σ, | x i =1 + q σ | x i =1 . Since tr f v can be written in the form of (4.7), the comparison of the above andexpressions in the form of (4.7) on f leads to q σ | x i =1 = 0. Since it holds for any i ∈ [[ σ ∗ ]] with x i = ± q σ = ˜ q σ Y i ∈ [[ σ ∗ ]] (1 − x i ) , ˜ q σ ∈ ( Q σ ∗ ,r − ⊗ Q σ,r − )( ˆ T ) . If we set η = ˜ q σ d x σ ∗ , then R ˆ T v ∧ η = 0 due to the fact v = u − ˆΠ h u and thedefinition of ˆΠ h . However, Z ˆ T v ∧ η = ± Z ˆ T v σ ˜ q σ vol ˆ T = ± Z ˆ T Y i ∈ [[ σ ∗ ]] (1 − x i )˜ q σ vol ˆ T , therefore q σ = 0. As a consequence v σ has a form (4.7). (cid:3) Corollary 4.3.
For u ∈ ˜ Q r Λ k ( T h ) and v ∈ Q dr − Λ k ( T h ) , h u − Π h u, v i h = 0 . Proof.
It suffices to show the assertion for u ∈ ˜ Q r Λ k ( T ) and v ∈ Q dr − Λ k ( T ) forany T ∈ T h . We define ˆ u ∈ ˜ Q r Λ k ( ˆ T ) and ˆ v ∈ Q dr − Λ k ( ˆ T ) as the pullbacks φ ∗ u and φ ∗ v with the dilation map φ : ˆ T → T . From the definition of Π h and the property ofpullback on differential forms, φ ∗ ( u − Π h u ) = ˆ u − ˆΠ h ˆ u holds. Since h u − Π h u, v i h,T is a constant multiple of D ˆ u − ˆΠ h ˆ u, ˆ v E h, ˆ T with the constant depending on scaling.By the characterization of ˆ u − ˆΠ h ˆ u in Theorem 4.2, the exactness of the Gauss-Lobatto quadrature rule for polynomials of degree 2 r −
1, the orthogonality ofLegendre polynomials, and (4.6), D ˆ u − ˆΠ h ˆ u, ˆ v E h, ˆ T = 0. (cid:3) Finally, we check that the proposed method give local coderivatives with anargument completely analogous to the one in [23].To see more details, let the set of nodal degrees of freedom and the basis of˜ Q r Λ k ( T h ) associated to the nodal degrees of freedom be { φ ( T,z ) : z ∈ N r ( T ) , T ∈ T h } , { ψ ( T,z ) : z ∈ N r ( T ) , T ∈ T h } , such that φ ( T,z ) ( ψ ( T ′ ,z ′ ) ) = δ T T ′ δ zz ′ with the Kronecker delta. For each z ∈ N r ( T )for some T ∈ T h and we can consider the set of all basis functions ψ ( T ′ ,z ) with thecommon point z . We denote this set of basis functions by Ψ z . Then the quadraturebilinear form h· , ·i h with { ψ ( T,z ) } gives a block diagonal matrix such that each blockis associated to Ψ z for some z . The influence of the inverse of this block matrixassociated to Ψ z is only on the local domain consists of { T ′ } such that ψ ( T ′ ,z ) ∈ Ψ z ,so it is a spatially local operator and then gives a local coderivative. Concluding remarks
In this paper we develop high order numerical methods with finite elements forthe Hodge Laplace problems on cubical meshes that admit local approximations ofthe coderivatives. The main task of the development is construction of a new familyof finite element differential forms on cubical meshes which satisfy the propertiesfor the stability analysis framework presented in Section 2.In our study we have only considered Hodge-Laplace problems with the identitycoefficients. However, the discussion can be easily extended to a matrix-valuedcoefficient K − which is symmetric positive definite, and piecewise constant on T h .We point out that this is an important difference between our approach and thework in [14] for the Maxwell equations. More precisely, the quadrature rules in[14] are combinations of Gauss and Gauss-Lobatto rules for different components,so the methods are not robust for matrix-valued coefficients because the matrix-valued coefficients in bilinear forms mix the components with different quadraturerules. We refer the interested readers to [23, Section 6] for details of the analysiswith K − = I .Finally, the extension of the methods to curvilinear meshes is not easy becausethe conditions in (B) strongly rely on the properties of quadrature rules on cubicalmeshes which do not hold on distorted cubical (or called curvilinear) meshes. In [3],this extension was done for weakly distorted quadrilateral and hexahedral meshes bymeasuring changes of bilinear forms and related quantities under mesh distortion.However, this is possible in the cases because explicit forms of the bilinear andtrilinear maps and their derivatives are available for the specific setting k = n − n = 2 ,
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