Nonstandard finite element de Rham complexes on cubical meshes
aa r X i v : . [ m a t h . NA ] A p r Nonstandard finite element de Rham complexeson cubical meshes
Andrew Gillette ∗∗ Kaibo Hu †† Shuo Zhang ‡‡ Abstract
We propose two general operations on finite element differential complexes on cubical meshesthat can be used to construct and analyze sequences of “nonstandard” convergent methods. Thefirst operation, called DoF-transfer, moves edge degrees of freedom to vertices in a way that reducesglobal degrees of freedom while increasing continuity order at vertices. The second operation, calledserendipity, eliminates interior bubble functions and degrees of freedom locally on each elementwithout affecting edge degrees of freedom. These operations can be used independently or intandem to create nonstandard complexes that incorporate Hermite, Adini and “trimmed-Adini”elements. We show that the resulting elements provide convergent, non-conforming methods forproblems requiring stronger regularity and satisfy a discrete Korn inequality. We discuss potentialbenefits of applying these elements to Stokes, biharmonic and elasticity problems.
To obtain compatible and stable numerical schemes in the framework of discrete differential formsand finite element exterior calculus [5, 6, 25], physical variables are discretized in discrete differentialcomplexes. There have been many successful finite element de Rham complexes, consisting of theLagrange, 1st and 2nd N´ed´elec, Raviart-Thomas, Brezzi-Douglas-Marini (BDM) and discontinuouselements. On simplicial meshes, the construction of these elements is classical [33, 32, 37, 13]. Oncubical meshes, the analogue of the Raviart-Thomas family has a tensor product nature [9], while theanalogue of the BDM families, denoted as S r Λ k , are developed in [2] as a resolution of the serendipityelement (c.f. [3]). These results are summarized in the Periodic Table of the Finite Elements [7].The recently defined trimmed serendipity family, denoted S − r Λ k , provides a distinct resolution of theserendipity element as subcomplex of the de Rham sequence with some computational advantages [23].While the de Rham complex describes regularity spaces appropriate for Hodge-Laplacian problems,it is not suited for flow problems that involve the same differential operators but require strongerregularities. A prominent example is the Stokes problem and the corresponding “Stokes complex”, ∗∗ Department of Mathematics, University of Arizona, 617 N. Santa Rita Ave., Tucson, Arizona, USA. email: [email protected] †† Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO 0316 Oslo, Norway. email: [email protected] ‡‡ LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematicsand System Sciences, and National Centre for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences,Beijing 100190, People’s Republic of China email: [email protected] ✲ R ✲ H (Ω) curl ✲ (cid:2) H (Ω) (cid:3) ✲ L ✲ , in two space dimensions. The importance of precisely preserving mass conservation (also called thedivergence-free condition) has been shown in [29]. Accordingly, finite dimensional subcomplexes ofthe Stokes complexes can be used to construct suitable elements. Falk and Neilan [21] construct asequence starting with the C Argyris element on 2D triangular meshes. Neilan and Sap [35] give ananalogous construction on cubical meshes, which is a resolution of the Bogner-Fox-Schimt (BFS) C element. Tensor product constructions based on splines in the context of the isogeometric analysisare proposed in [14]. We also refer to [34, 35] and the reference therein for relevant work in 3D andrefer to [43, 8, 16] for constructions on macroelements. To obtain these finite element complexes withhigher regularity, one usually needs a high polynomial degree and extra derivative degrees of freedom,or certain mesh conditions.An attractive approach to constructing Stokes elements in a simple fashion while ensuring massconservation is to use non-conforming elements. For instance, the discrete Stokes complex0 ✲ R ✲ Morley curl h ✲ [Crouzeix − Raviart] h ✲ DG ✲ , ( . )consists of the Morley and the vector Crouzeix-Raviart elements and piecewise derivatives curl h anddiv h . Although the Morley element is an elegant non-conforming choice for the biharmonic problem, itis not convergent for the Poisson equation (c.f. [41, 36]). Correspondingly, the vector Crouzeix-Raviartelement is not desirable as an H (div) element for the Hodge Laplacian problem. For the Darcy-Stokes-Brinkman problem, which is a perturbation of the flow problem with an H (div) Hodge Laplacian, thenumerical experiments in [31] demonstrate that the convergence with the Crouzeix-Raviart elementdeteriorates as the coefficient in front of the Laplace operator tends to zero. As argued in [31, 39], apossible cure for this problem is to seek elements that are conforming in H (div) and non-conformingconvergent in (cid:2) H (cid:3) n , then further construct discrete sequences that are conforming to the de Rhamcomplex and non-conforming convergent as a Stokes complex. A similar complex is constructed onquadrilateral grids [44].On a given mesh, any approximation space can be characterized by local functions on each cell andthe continuity across the boundaries of the cells, which we shall call the “inter-element continuity”in subsequent discussions. Finite elements are a special kind of approximation in which inter-elementcontinuity can be ensured via constraints local to each element. Precisely, in Ciarlet’s classical definitionof finite elements, a finite element is denoted as a triple ( P, G, D ) where P is the approximation space(in this case, a space of polynomial differential forms), G is the geometry (in this case, an n -cube)and D is the set of functionals on P that are associated to portions of G (e.g. evaluation at a specificvertex of G , or integration against a specific test function along an edge of G ). The inter-elementcontinuity is imposed by requiring that these DoFs are single valued on neighboring elements. This isan extra locality condition and in general, approximation spaces, such as some spline spaces, may notfulfill this condition. In the theory of finite element systems (FES) [17], another framework of finiteelements developed by Christiansen and collaborators, no degrees of freedom are explicitly used. Theapproximation space is viewed as a collection of spaces living on all subcells (of all dimensions) of themesh. The inter-element continuity is imposed by requiring that all the traces of functions on higherdimensional cells coincide with the functions on lower dimensional cells (therefore the global functions2re single valued). The original theory of the FES is tailored for finite element differential forms [17],and is generalized to spaces with higher continuity in [16].In general, there are two ways to reduce a piecewise-defined approximation space: reduce the size oflocal shape function spaces or increase the inter-element continuity. These two operations can always bedone. However, from the perspective of finite elements, we require the first operation (local reduction)to preserve certain local approximation properties (e.g. containment of polynomials of a certain degree)and require the second (global reduction) to retain the locality of the approximation (e.g. finiteelements defined in Ciarlet’s sense or in the FES sense). From a viewpoint of approximation theories,the locality of the approximation spaces plus the local approximation properties guarantees the globalapproximation properties. In light of the finite element exterior calculus, one further demands thatthese two operations also preserve the exactness of a sequence of finite element spaces.In this paper, we provide a canonical procedure for modifying conforming deRham subcomplexesinto non-conforming convergent sequences of spaces of elements that can be used in problems requiringgreater regularity. In 2D, our construction leads to simple elements for the Darcy-Stokes-Brinkmanproblem while in 3D our construction yields the Adini plate element [1] and an H (curl) element thatis conforming for Poisson/Stokes type problems. Instead of enriching an existing space with bubblefunctions as in, e.g. [31, 39], we reduce the local and global degrees of freedom of standard de Rhamfinite element complexes to sequences that provide the desired properties. From the perspective of non-conforming methods, consistency is preserved in the space reduction. More precisely, if the consistencyerror in the Strang lemma converges to zero as h → V h , then the consistency error forany subspace of V h also tends to zero.The ideas presented here are related to recent investigations into “nonstandard” vector elements.Stenberg introduced a family of “nonstandard elements” for the H (div) space on triangles and tetrahe-dra in [38]. These elements are subspaces of the classical BDM spaces that are continuous at vertices ofthe mesh. Enforcing this continuity constraint reduces the number of global degrees of freedom (DoFs).The canonical interpolations defined by the DoFs, however, do not commute with the differential op-erations; instead, Stenberg uses macroelement techniques to show the inf-sup condition holds [38].In [15], Christiansen, Hu, and Hu derive complexes of nonstandard elements from a different perspec-tive. The elements are constructed as variations of copies of Lagrange and Hermite elements, and as aresult, canonical nodal bases are available in some cases. The work is partly motivated by an elementfor linear elasticity [28, 26]. The extended periodic table in [15] includes the triangular and tetrahedralStenberg elements as a special case for the H (div) space and several new H (curl) finite elements.The results presented in this paper can be regarded as a continuation of the discussions in [15]. Inboth the simplicial and the cubical cases, the number of global DoFs is reduced. However, there aresome prominent differences. The triangular Hermite element does not lead to convergent schemes forbiharmonic problems, although modifications on the shape functions are possible to make it convergent[42]. In contrast, cubical elements with vertex derivative DoFs usually do lead to convergent schemesfor the biharmonic equation due to the geometric symmetry. The Adini plate element serves as anilluminating example: it is globally C with derivative DoFs at vertices and comparable to the Hermiteelement on triangular meshes (c.f. [27] and the references therein). In this paper, we will constructthe Adini element by a new procedure, and place it within an entire finite element complex on squaresor cubes.Our approach to element construction is to provide two operations that can be applied to known3nite element complexes: a “DoF-transfer” operation and a “serendipity” operation. The DoF-transferoperation moves some edge DoFs to vertex DoFs. The local shape function spaces and bubbles donot change and the DoF-transfer preserves unisolvence and exactness. The serendipity operation elim-inates some interior bubbles and DoFs at the same time in a way that preserves unisolvence andexactness. The idea of the serendipity operation bears some similarity to the serendipity reductionprocess in virtual element methods as described in [18, 19], however, virtual element techniques arechiefly concerned with applicability to many-sided polygonal and generic polyhedral element geome-tries. Combining these two operations, we reduce the local and global spaces of various known discretecomplexes on cubical meshes [2, 4, 23] to obtain Hermite, Adini and trimmed-Adini type families.The H (div) element in the 2D Hermite complex coincides with the nonstandard element introducedin Stenberg [38].The rest of this paper will be organized as follows. In Section 2, we introduce notation andbackground material. In Section 3, we define the DoF-transfer and serendipity operations. In Section4, we define the Hermite, Adini, and trimmed-Adini elements and complexes. Section 5 is devoted tothe approximation and convergence properties of the new finite elements, including the discrete Korninequality and the convergence as non-conforming elements. In Section 6, we further use the idea ofthe serendipity operation to construct a complex of minimal elements that are conforming as a deRham complex and non-conforming as a Stokes complex. We conclude in Section 7 with some remarksand future directions. Let Ω ⊂ R n be a polyhedral domain and T h be a mesh on Ω consisting of n -cubes. Sometimes we use T h and T h to denote 2D and 3D meshes respectively. A single square element is denoted by (cid:3) and asingle cube element by (cid:3) . Given a mesh, we use V , E , F and K to denote the sets of vertices, edges,faces and three dimensional cells (zero to three dimensional cells) respectively. We use ν f and τ f todenote the unit normal and tangential vectors of a cell f , respectively, when applicable.We use P r,s ( K ), or simply P r,s , to denote the polynomial space of degree r in x and degree s in y on a domain K , i.e. P r,s := P r ( x ) × P s ( y ) . Similarly, in three dimensions, we define P r,s,t := P r ( x ) × P s ( y ) × P t ( z ) , When the orders are equal, we get the tensor product polynomial spaces, denoted Q r := P r,r , and Q r := P r,r,r . Following [2], we define the superlinear degree of a monomial to be the total degree of factors thatenter with quadratic or higher degrees. For example, the superlinear degree of x y z is 5, where wecount the degrees of x and y , but not z . We use S r to denote the polynomials with superlinear degreeless than or equal to r . There is a simple nesting of spaces given by P r ⊂ S r ⊂ Q r .We review some basic facts from homological algebra; further details can be found, for instance, in[10, 20, 5]. A differential complex is a sequence of spaces V i and operators d i such that0 ✲ V d ✲ V d ✲ · · · d n − ✲ V n d n ✲ , ( . )4atisfying d i +1 d i = 0 for i = 1 , , · · · , n −
1; typically, this condition is denoted as d = 0. Let ker( d i )be the kernel space of the operator d i in V i , and Im( d i ) be the image of the operator d i in V i +1 .Due to the complex property d = 0, we have ker( d i ) ⊂ Im( d i − ) for each i ≥
2. Furthermore, ifker( d i ) = Im( d i − ), we say that the complex ( . ) is exact at V i . At the two ends of the sequence, thecomplex is exact at V if d is injective (with trivial kernel), and is exact at V n if d n is surjective (withtrivial cokernel). The complex ( . ) is called exact if it is exact at all the spaces V i . If each space in( . ) has finite dimension, then a necessary (but not sufficient) condition for the exactness of ( . ) isthe following dimension condition: n X i =1 ( − i dim( V i ) = 0 . For finite element spaces, the exactness of the local shape function space on each mesh elementis called local exactness , while that of the whole space, with the inter-element continuity taken intoaccount, is called global exactness . In subsequent discussions, the local exactness follows from classicalresults. Therefore we mean global exactness as a default.Another complex0 ✲ W d ✲ W d ✲ · · · d n − ✲ W n ✲ subcomplex of ( . ) if for each i = 1 , , · · · , n , W i is a subspace of V i and the differentialoperators are the same.Define H ( d, Ω) := (cid:8) u ∈ L (Ω) : du ∈ L (Ω) (cid:9) , where d is a differential operator. We use k u k s, Ω to denote the H s norm of u on Ω. When there isno possible confusion, we also use k · k s instead of k · k s, Ω and use k · k instead of the L norm k · k .There are two de Rham sequences with the above type of Sobolev regularity in 2D:0 −−−−→ R −−−−→ H (curl , Ω) curl −−−−→ H (div , Ω) div −−−−→ L (Ω) −−−−→ , ( . )and 0 −−−−→ R −−−−→ H (grad , Ω) grad −−−−→ H (rot , Ω) rot −−−−→ L (Ω) −−−−→ . ( . )Here, the 2D curl operator maps a scalar function to a vector valued function, defined bycurl u := ( − ∂ u, ∂ u ) T . ( . )One can obtain ( . ) by rotating the spaces in ( . ) by π/
2. Accordingly, in the remainder of thispaper, we only consider ( . ) in 2D. There is only one 3D version of the sequence, which reads:0 −−−−→ R −−−−→ H (grad , Ω) grad −−−−→ H (curl , Ω) curl −−−−→ H (div , Ω) div −−−−→ L (Ω) −−−−→ . ( . )Here, the 3D curl has its usual meaning, mapping a vector valued function to a vector valued function:curl u := ( ∂ u − ∂ u , ∂ u − ∂ u , ∂ u − ∂ u ) T . ( . )When there is no possible confusion, we will use curl to denote both ( . ) and ( . ), as determined bythe spatial dimension. The sequences ( . ), ( . ) and ( . ) are all exact on contractible domains. We5ill work with three known finite element subcomplexes of the de Rham complex. In 2D these are:0 ✲ R ✲ Q − r Λ ( T h ) curl ✲ Q − r Λ ( T h ) div ✲ Q − r Λ ( T h ) ✲ , ✲ R ✲ S r +2 Λ ( T h ) curl ✲ S r +1 Λ ( T h ) div ✲ S r Λ ( T h ) ✲ , ✲ R ✲ S − r Λ ( T h ) curl ✲ S − r Λ ( T h ) div ✲ S − r Λ ( T h ) ✲ , ( . )namely, the 2D Raviart-Thomas (tensor product) type finite element complex [4], the 2D BDM typecomplex [2], and the 2D trimmed serendipity complex [23], respectively, for r ≥
1. The 3D familiescan be similarly denoted by0 ✲ R ✲ Q − r Λ ( T h ) grad ✲ Q − r Λ ( T h ) curl ✲ Q − r Λ ( T h ) div ✲ Q − r Λ ( T h ) ✲ , ✲ R ✲ S r +3 Λ ( T h ) grad ✲ S r +2 Λ ( T h ) curl ✲ S r +1 Λ ( T h ) div ✲ S r Λ ( T h ) ✲ , ✲ R ✲ S − r Λ ( T h ) grad ✲ S − r Λ ( T h ) curl ✲ S − r Λ ( T h ) div ✲ S − r Λ ( T h ) ✲ . ( . )Some problems involve de Rham complexes with enhanced smoothness. For example, for Stokesflows, the velocity belongs to (cid:2) H (cid:3) n while the pressure is in L with vanishing integration, and div : (cid:2) H (cid:3) n L (Ω) is onto. In this case, we consider the 2D complex0 −−−−→ R −−−−→ H (Ω) curl −−−−→ (cid:2) H (Ω) (cid:3) −−−−→ L (Ω) −−−−→ , ( . )and the 3D version0 −−−−→ R −−−−→ H (Ω) grad −−−−→ H (curl , Ω) curl −−−−→ (cid:2) H (Ω) (cid:3) −−−−→ L (Ω) −−−−→ , ( . )where H (curl , Ω) := n u ∈ (cid:2) H (Ω) (cid:3) : curl u ∈ (cid:2) H (Ω) (cid:3) o . The subsequent discussions involve another 3D smooth de Rham complex0 −−−−→ R −−−−→ H (Ω) grad −−−−→ (cid:2) H (Ω) (cid:3) −−−−→ H (div , Ω) div −−−−→ L (Ω) −−−−→ . ( . )The sequence ( . ) is exact on any contractible domain Ω. In this section, we propose two operations for reducing piecewise-defined approximation spaces on cu-bical elements: a “serendipity” operation, which reduces local bubble functions, and a “DoF-transfer”operation, which enhances inter-element continuity by moving degrees of freedom. All the abovemen-tioned properties are achieved, including preservation of the exactness.The serendipity operation, denoted by Q , eliminates some interior DoFs and corresponding shapefunctions on individual elements. Figure 1 shows an example of the serendipity operation applied tothe tensor product element Q − Λ ( (cid:3) ), resulting in the serendipity element S Λ ( (cid:3) ). On 0-forms,6 rr rrr rrr rr rr rr r ✲ Q r rr rrr rrr rr r Figure 1: An instance of the serendipity operation, taking Q − Λ ( (cid:3) ) to S Λ ( (cid:3) ). r rr rrr rrr rr rr rr r ✲ T r❞ r❞r❞ r❞r rr r ✻ ✻✻❄ ❄❄ ✲✲✲✛✛✛ +12 ✲ T rr rrrr rr ❄✻ ✲✛ +12Figure 2: An instance of the DoF-transfer operation, taking Q − Λ ( (cid:3) ) to G − Λ ( (cid:3) ) and taking Q − Λ ( (cid:3) ) to G − Λ ( (cid:3) ).there is only one kind of serendipity operation, but on k -forms with k ≥
1, there are serendipityoperations to both the regular and trimmed serendipity sequences; we denote both by Q as it will beclear from context which one is being used.The DoF-transfer operation, denoted by Q , “moves” two DoFs per edge from edge-association tovertex-association, interpolating either a directional derivative at the vertex (in the case of 0-forms)or a coordinate value of the vector field (in the case of 1-forms). Figure 2 shows an example of eachtype. The serendipity operation reduces local DoFs while the DoF-transfer operation reduces globalDoFs and, as we will show, T and Q both preserve unisolvence and exactness.Our subsequent constructions of finite element sequences are summarized in diagrams ( . ) and ( . ).They employ the finite element sequences for Q − Λ • , S Λ • , and S − Λ • given in ( . ). Q − r +2 Λ Q − r +2 Λ Q − r +2 Λ G − r +2 Λ G − r +2 Λ G − r +2 Λ S r +2 Λ S r +1 Λ S r Λ A r +2 Λ A r +1 Λ A r Λ T curl Q T div
Q T Q curl Q div T T T curl Q div Q ( . )In diagram ( . ), we start from the standard tensor product complex Q − Λ • (top, back), with thedifferential operators as shown. By a serendipity operation Q , we obtain the BDM-type complex S Λ • T , we get the Hermite complex G − Λ • (top, front).Applying the serendipity operation to G − Λ • or DoF-transferring S Λ • , we obtain the Adini complex A Λ • (bottom, front). We will show that the diagram ( . ) commutes, i.e. that the operations Q , T and the differential operators commute with each other. The extension of this diagram to complexesin 3D is straightforward.A variant of this construction is described in diagram ( . ). Here, we start again with Q − Λ • butuse a serendipity operation that results in the trimmed serendipity sequence S − Λ • . Applying both Q and T gives a “trimmed-Adini” complex, denoted A − Λ • . As indicated in the diagram, there are someequalities between trimmed and non-trimmed-Adini elements when k = 0 or n , based on identities forserendipity-type elements, but for 0 < k < n , the Adini and trimmed-Adini elements are truly distinct. Q − r +2 Λ Q − r +2 Λ Q − r +2 Λ G − r +2 Λ G − r +2 Λ G − r +2 Λ S − r +2 Λ S − r +2 Λ S − r +2 Λ A − r +2 Λ A − r +2 Λ A − r +2 Λ = = A r +2 Λ A r +1 Λ T curl Q T div
Q T Q curl Q div T T T curl Q div Q ( . )We now formalize the operations T and Q . Degrees of freedom on an n -cube (cid:3) n when P is aspace of polynomial differential k -forms are typically described by u Z f (tr f u ) ∧ q, q ∈ I f , ( . )where f ≺ (cid:3) n is a sub-face of the cube and I f is a space of polynomial differential ( n − k )-forms. Wecall I f the index space associated to f and there is a canonical association D ←→ n M d = k M f ≺ (cid:3) n dim f = d I f ( . )For instance, DoFs associated to Q − r Λ k ( (cid:3) n ) have I f := Q − r − Λ d − k ( f ) for each f ≺ (cid:3) n [4, equation(13)]. For finite elements conforming with respect to the standard deRham sequence, i.e. ( . ) and( . ), DoFs are associated to faces of dimension 0 only in the case of 0-forms ( k = 0). In those cases,dim I f = 1 and ( . ) is interpreted as evaluation of the 0-form at the vertex.The non-standard elements considered here have two additional kinds of DoFs. For a 0-form u andvertex x , we allow partial derivative evaluation at vertices, i.e. u ∂ i u ( x ) ( . )8here i indicates the direction of an edge e i incident to x . For a 1-form v and vertex x , we allowevaluation of the 1-form at the vertex, i.e. v v ( x ) , ( . )where v ( x ) is a vector in R or R . While these DoFs are associated to a vertex of the geometry inthe global setting, we observe that they require both a vertex and an incident edge direction to becharacterized locally, e.g. the component of the vector v ( x ) in the 1-form case. Therefore, we continueto treat these as edge DoFs to simplify the upcoming formalism.Let n = 2 or 3. The serendipity operation Q : ( P , (cid:3) n , D ) → ( P , (cid:3) n , D ) can be thought of asacting component-wise on P and D and should satisfy the following conditions: • ( P , (cid:3) n , D ) and ( P , (cid:3) n , D ) are finite elements, and Q is onto. • P ⊂ P and Q : P → P is a projection. • Q is the identity on DoFs associated to vertices or edges of (cid:3) n . For max( k, ≤ d ≤ n and each f ≺ (cid:3) n with dim f = d , let I f, and I f, denote the index spaces from ( . ) from D and D ,respectively. Then I f, ⊆ I f, and Q : I f, → I f, is a projection. Q : D → D is the linearmorphism that is determined by these conditions via ( . ).The DoF-transferring operation T : ( P , (cid:3) n , D ) → ( P , (cid:3) n , D ) can be thought of as actingcomponent-wise on P and D , and should satisfy the following conditions: • ( P , (cid:3) n , D ) and ( P , (cid:3) n , D ) are finite elements, and Q is onto. • P = P and T : P → P is the identity map. • D only has DoFs of type ( . ). • T : D → D only affects DoFs associated to edges e ≺ (cid:3) n with dim I e ≥
2, and leaves all otherDoFs fixed. On each such edge e , T changes two DoFs from type ( . ) to type ( . ) or ( . ),according to whether P is a space of 0-forms or 1-forms, respectively. Lemma 1. If Q leads to the same local shape functions for both the DoF-transferred and the originalelements, then diagrams ( . ) and ( . ) are commutative.Proof. The result holds because T ◦ Q and Q ◦ T yield elements with same local spaces and sameinter-element continuity.The condition for Q in Lemma 1 is fulfilled in all the examples below for the Hermite, Adini andtrimmed-Adini families. We now verify some properties of the two operations. Lemma 2.
The DoF-transfer operation preserves unisolvence, i.e. if ( P, (cid:3) n , D ) defines a unisolventfinite element, then T ( P, (cid:3) n , D ) is also unisolvent.Proof. The operation T only changes the type of some edge DoFs, from type ( . ) to type ( . ) or( . ), but preserves association of DoFs to edges, as discussed after ( . ). The resulting set of DoFs isstill linearly independent on each edge, so unisolvence of the element is preserved by the operation.9 emma 3. The DoF-transfer operation preserves exactness, i.e. if ( P • , (cid:3) n , D • ) defines an exact se-quence of finite elements, then T ( P • , (cid:3) n , D • ) is also exact.Proof. Since the DoF-transfer only changes some 0- and 1-forms, the exactness at T ( P k , (cid:3) n , D k ), k ≥
3, holds due to the exactness of the original sequence ( P • , (cid:3) n , D • ). For the exactness at k = 1,we observe that if u ∈ T ( P , (cid:3) n , D ) and du = 0, then there exists φ ∈ ( P , (cid:3) n , D ) such thatgrad φ = u , by the exactness of ( P • , (cid:3) n , D • ) at index 1 and the fact that T ( P • , (cid:3) n , D • ) are subspacesof ( P • , (cid:3) n , D • ). Since u is C at vertices, we conclude that φ is correspondingly C at vertices. Thisimplies that φ ∈ T ( P , (cid:3) n , D ) and shows the exactness of T ( P • , (cid:3) n , D • ) at index 1. It only remainsto check the exactness at index 2. This follows by a dimension count. In n dimensions, on eachelement, n DoFs are added per vertex and n DoFs are removed per edge for 0-forms and likewise for 1-forms. Thus, the alternating sum of dimension counts of spaces in the sequence is unaffected, meaningexactness is preserved. Therefore the exactness of the whole sequence T ( P • , (cid:3) n , D • ) holds. Remark 1.
Starting with 2-forms, the DoF-transferred sequence branches into the standard finiteelement sequence. A direct check of the exactness at the space of two forms is not trivial. For example,the 2D 1-forms ( H (div) element) with vertex continuity and the 2-forms of piecewise polynomials havebeen analyzed in Stenberg [38] by a macroelement technique. From the point of view presented in thispaper and [15], the exactness seems more clear and natural. In subsequent discussions, we use the term “bubble function” to mean a shape function on (cid:3) n forwhich all DoFs vanish on all ℓ -dimensional facets of (cid:3) n for 0 ≤ ℓ ≤ n − Lemma 4.
The DoF-transfer operation does not change the local space of bubble functions on facetsof dimension ℓ ≥ .Proof. Since the DoF-transferred sequence T ( P • , (cid:3) n , D • ) has stronger inter-element continuity than( P • , (cid:3) n , D • ), we conclude that the bubble spaces of T ( P • , (cid:3) n , D • ) are contained in those of ( P • , (cid:3) n , D • ).On the other hand, the DoF-transfer operation does not change the interior DoFs associated to facetswith dimension ℓ ≥
2. So, the spaces of bubble functions of T ( P • , (cid:3) n , D • ) and ( P • , (cid:3) n , D • ) have thesame dimension. Therefore, the spaces of bubble functions for these two families are the same.We now examine properties of the serendipity operation. First, we address the issue of unisolvence.Degrees of freedom of type ( . ) are unisolvent if and only if on each k dimensional face f ∈ (cid:3) m , k ≤ m ≤ n , the trace space with vanishing boundary conditions on f (the space of bubble functionson f ) coincides with the local space I f used in the definition of degrees of freedom in ( . ), i.e., ∗ tr f ( P k ∩ H Λ k ) = I f . Here, the Hodge star is used since in the language of differential forms, theinner product is defined by ( u, v ) f := Z f ∗ u ∧ v dx . In terms of vector proxies, the test space I f is justthe same as the space of bubbles on f . We assume that the serendipity operation only affects degreesof freedom of type ( . ). If the image of a the serendipity operation Q : ( P , (cid:3) n , D ) → ( P , (cid:3) n , D )satisfies the corresponding relation ∗ tr f ( P • ∩ H Λ • ) = I f, , then unisolvence will be preserved.10or exactness, consider the diagram ( . ):0 0 0 00 ✲ ker( Q ) ❄ d ✲ ker( Q ) ❄ d ✲ · · · ❄ d ✲ ker( Q n ) ❄ ✲ ✲ R ✲ V ι ❄ d ✲ V ι ❄ d ✲ · · · ι ❄ d ✲ V nι ❄ ✲ ✲ R ✲ ˜ V Q ❄ d ✲ ˜ V Q ❄ d ✲ · · · Q ❄ d ✲ ˜ V n Q ❄ ✲ ❄ ❄ ❄ ❄ ( . )The first row of ( . ) consists of the kernel spaces of the serendipity operation, indicating thesubspaces that have been eliminated. The second row is the finite element sequence that we start withand the last row is the reduced sequence. By definition, each column0 ✲ ker( Q k ) ι ✲ V k Q k ✲ ˜ V k ✲ , ( . )is a short exact sequence, where ι is the inclusion map. Assume that the row in the middle, i.e.0 ✲ R ✲ V d ✲ V d ✲ · · · d ✲ V n ✲ , ( . )is exact. From a general algebraic result, the exactness of either the first or last row of ( . ) impliesexactness of the other (c.f. [20] for the general result and [17, Proposition 5.16, Proposition 5.17] foran application in the context of finite element systems). In particular, the exactness of the kernelsequence 0 ✲ ker( Q ) d ✲ ker( Q ) d ✲ · · · d ✲ ker( Q n ) ✲ , ( . )implies that of the reduced sequence0 ✲ R ✲ ˜ V d ✲ ˜ V d ✲ · · · d ✲ ˜ V n ✲ , ( . )and vice versa. The following lemma summarizes this result. Lemma 5.
In any setting of the form ( . ) , if any two rows of the diagram are exact then the third isexact as well. In particular, exactness of the first and second rows implies that the serendipity operationpreserves exactness. ❢ r❢r❢ r❢ +4 ✲ curl rr rrrr rr ✻❄✛ ✲ +12 ✲ div +9Figure 3: The lowest order Hermite complex, with regularity H (curl) → H (div) → L . The local shapefunction spaces are Q → P , × P , → Q , or, equivalently, Q Λ ( (cid:3) ) → Q Λ ( (cid:3) ) → Q Λ ( (cid:3) ). Remark 2.
From the perspective of finite element systems, the serendipity operation reduces local(bubble) spaces on various dimensions and the DoF-transfer only changes the spaces carried by zerodimensional cells (vertex spaces) and the trace/restriction from one dimensional cells to their bound-aries. Although a rigorous theory has not been studied in the literature, we observe that the structuresof the FES are not changed for two and higher dimensions. The DoF-transfer defined in this sectionchanges the system associated with one dimensional cells from Lagrange to Hermite type. This partlyexplains the fact that face and interior bubble functions are not affected by the DoF-transfer.
In this section we introduce the Hermite, Adini and trimmed-Adini complexes obtained from standardcomplexes by the serendipity and DoF-transfer operations. We describe the function spaces and degreesof freedom definitions in two and three space dimensions.
The 2D Hermite sequence is denoted by0 ✲ R ✲ G − r Λ ( T h ) curl ✲ G − r Λ ( T h ) div ✲ G − r Λ ( T h ) ✲ . ( . )The shape function spaces coincide with those of the tensor product family on squares, i.e. Q − Λ k ,or Raviart-Thomas elements. However, the Hermite family is only defined when r ≥
3, as smaller r values do not provide enough degrees of freedom to ensure the requisite continuity at vertices for k = 0and k = 1. The lowest order case is shown in Figure 3. Space G − r Λ ( (cid:3) ) . The shape function space of G − r Λ ( (cid:3) ) is Q r . The DoFs can be given by • function evaluation and first order derivatives at each vertex x ∈ V : u ( x ) , ∂ i u ( x ) , i = 1 , , • moments on each edge e ∈ E : Z e uq dx, q ∈ P r − ( e ) , interior DoFs in each K ∈ F : Z K up dx, p ∈ Q r − ( K ) . Space G − r Λ ( (cid:3) ) . The shape function space of G − r Λ ( (cid:3) ) is P r,r − × P r − ,r , the same as Q − r Λ ( (cid:3) ).The DoFs can be given by • function evaluation at each vertex x ∈ V : v ( x ) , • moments of the normal components on each edge e ∈ E : Z e v · ν e q dx, q ∈ P r − ( e ) , • interior degrees of freedom in each K ∈ F : Z K v · ψ dx, ψ ∈ P r − ,r − ( K ) × P r − ,r − ( K ) . Space G − r Λ ( (cid:3) ) . The space G − r Λ ( (cid:3) ) consists of piecewise Q r − polynomials on each element. The 3D Hermite sequence is denoted by0 ✲ R ✲ G − r Λ ( T h ) grad ✲ G − r Λ ( T h ) curl ✲ G − r Λ ( T h ) div ✲ G − r Λ ( T h ) ✲ . ( . ) Space G − r Λ ( (cid:3) ) . The shape function space of G − r Λ ( (cid:3) ) is Q r . The DoFs can be given by • function evaluation and first order derivatives at each vertex x ∈ V : u ( x ) , ∂ i u ( x ) , i = 1 , , • moments on each edge ∀ e ∈ E : Z e uq dx, q ∈ P r − ( e ) , • moments on each face f ∈ F : Z f up dx, p ∈ Q r − ( f ) , • interior DoFs in each K ∈ K : Z K uw dx, w ∈ Q r − ( K ) . pace G − r Λ ( (cid:3) ) . The discrete H (curl) space of the 3D Hermite complex has the shape functionspace P r,r +1 ,r +1 × P r +1 ,r,r +1 × P r +1 ,r +1 ,r . DoFs for a function v ∈ G − r Λ ( (cid:3) ) can be given by • function evaluation at each vertex x ∈ V : v ( x ) , • moments of the tangential components on each edge e ∈ E : Z e v · τ e q dx, q ∈ P r − ( e ) , • moments of the tangential components on each face f ∈ F : Z f ( v × ν f ) · p dx, p ∈ P r − ,r − ( f ) × P r − ,r − ( f ) , where v × ν f is treated as a 2-vector in the plane of the face, • interior DoFs in each K ∈ K : Z K v · s dx, s ∈ P r − ,r − ,r − ( K ) × P r − ,r − ,r − ( K ) × P r − ,r − ,r − ( K ) . Spaces G − r Λ ( (cid:3) ) and G − r Λ ( (cid:3) ) . The last two spaces in the sequence coincide with the standardRaviart-Thomas element of degree r and the piecewise polynomial tensor product element Q r − , i.e. Q − r Λ ( (cid:3) ) and Q − r Λ ( (cid:3) ). The DoFs of G − r Λ ( (cid:3) ), can be given by (c.f. [9]) • moments on each face f ∈ F : Z f u · ν f q dx, q ∈ Q r − ( f ) , • interior DoFs for each K ∈ K : Z K u · s dx, s ∈ P r − ,r − ,r − ( K ) × P r − ,r − ,r − ( K ) × P r − ,r − ,r − ( K ) . The Adini element for the plate problem [1] can be regarded as a serendipity version of the cubicalHermite element. The shape function space of the 2D Adini element in the lowest order reads Q + span (cid:8) x i q : 1 ≤ i ≤ , q ∈ Q (cid:9) , which coincides with the shape function space of the serendipity element S Λ ( (cid:3) ). The degrees offreedom in this case can be given as the function evaluation and the first order derivatives at eachvertex x ∈ V . We now explain how this definition can be expanded to an entire exact seqeuence offinite elements. 14 ❢ r❢r❢ r❢ ✲ curl rr rrrr rr ✻❄✛ ✲ +2 ✲ div +3Figure 4: The lowest order Adini complex, with regularity H (curl) → H (div) → L . The local shapefunction spaces are S Λ ( (cid:3) ) → S Λ ( (cid:3) ) → S Λ ( (cid:3) ). The 2D Adini sequence is denoted by0 ✲ R ✲ A r +2 Λ ( T h ) curl ✲ A r +1 Λ ( T h ) div ✲ A r Λ ( T h ) ✲ , ( . )The space A r +2 Λ ( (cid:3) ) represents the Adini element of degree r + 2 and A r Λ ( (cid:3) ) = S r Λ ( (cid:3) ) = P r , the space of piecewise polynomials of total degree at most r . The lowest order case of the sequenceis shown in Figure 4. Space A r Λ ( (cid:3) ) . The shape function space is S r Λ ( (cid:3) ) = P r + span { x r y, xy r } . The DoFs aregiven by • function evaluation and first order derivatives at each vertex x ∈ V : u ( x ) , ∂ i u ( x ) , i = 1 , , • moments on each edge e ∈ E : Z e uq dx, q ∈ P r − ( e ) , • interior DoFs in each K ∈ F : Z K up dx, p ∈ P r − ( K ) . Space A r Λ ( (cid:3) ) . The shape function space is S r Λ ( (cid:3) ) = P r + span (cid:8) curl (cid:0) x r +1 y (cid:1) , curl (cid:0) xy r +1 (cid:1)(cid:9) , ( . )which coincides with the BDM element of degree r . The degrees of freedom can be given by • function evaluation at each vertex x ∈ V : u i ( x ) , i = 1 , , • moments on each edge e ∈ E : Z e u · ν e q dx, q ∈ P r − ( e ) , • interior DoFs in each K ∈ K : Z K u · p dx, p ∈ [ P r − ( K )] . .2.2 Three dimensions The 3D Adini sequence is denoted by0 ✲ R ✲ A r +3 Λ ( T h ) grad ✲ A r +2 Λ ( T h ) curl ✲ A r +1 Λ ( T h ) div ✲ A r Λ ( T h ) ✲ . ( . ) Space A r Λ ( (cid:3) ) . The shape function space is the same as that of the serendipity element S r Λ , i.e.polynomials with superlinear degree at most r . For r = 3, the shape function space can be equivalentlyrepresented as Q + span (cid:8) x i q : 1 ≤ i ≤ , q ∈ Q (cid:9) , and A r Λ ( T h ) coincides with the 3D Adini element. The degrees of freedom can be given by • function evaluation and first order derivatives at each vertex x ∈ V : u ( x ) , ∂ i u ( x ) , i = 1 , , , • moments on each edge e ∈ E : Z e uq dx, q ∈ P r − ( e ) , • moments on each face f ∈ F : Z f uv dx, v ∈ P r − ( f ) , • interior DoFs in each K ∈ K : Z K uw dx, w ∈ P r − ( K ) . Space A r Λ ( (cid:3) ) . The shape function space coincides with that of the H (curl) finite element givenin [2], i.e. S r Λ ( K ) := [ P r ( K )] + span yz ( w ( x, z ) − w ( x, y )) zx ( w ( x, y ) − w ( y, z )) xy ( w ( y, z ) − w ( x, z )) + grad q ( x, y, z ) , where w i ∈ P r ( K ) and q is a polynomial on K with superlinear degree at most k + 1. The DoFs for avector function u can be given by • function evaluation at each vertex x ∈ V : u ( x ) , • moments on each edge e ∈ E : Z e u · τ e q dx, q ∈ P r − ( e ) , • moments on each face f ∈ F : Z f ( u × ν f ) · p dx, p ∈ [ P r − ( f )] , where u × ν f is considered as a 2D vector, 16 ❢ r❢r❢ r❢ ✲ curl rr rrrr rr ✻❄✛ ✲ +5 ✲ div +6Figure 5: The lowest order trimmed-Adini complex, with regularity H (curl) → H (div) → L . Thelocal shape function spaces are S − Λ → S − Λ → S − Λ . • interior DoFs in K ∈ K : Z K u · s dx, s ∈ [ P r − ( K )] . The last two spaces in the sequence are the same as the BDM type H (div) element and the piecewisepolynomial space in [2]. For completeness, we include the definitions of these spaces. Space A r Λ ( (cid:3) ) . The shape function space of the cubical BDM space is given byBDM r := [ P r ( K )] + span yz ( w ( x, z ) − w ( x, y )) zx ( w ( x, y ) − w ( y, z )) yz ( w ( y, z ) − w ( x, z )) . The DoFs for a vector function v can be taken as • moments on each face f ∈ F : Z f v · ν f q dx, q ∈ P r ( f ) , • interior DoFs in K ∈ K : Z K v · p dx, p ∈ [ P r − ( K )] . Space A r Λ ( (cid:3) ) . Piecewise polynomials of degree r in 3 variables. The Adini element can also be treated as the first element in a different finite element sequence, whichwe call the trimmed-Adini complex. The shape function spaces are those of the trimmed serendipityfamily, defined in [23] as S − r Λ k := S r − Λ k + κ S r − Λ k +1 . The operator κ is the Koszul operator, which is discussed in detail in a finite element context in [5].As shown in [23], the trimmed serendipity spaces “nest” in between the regular serendipity spacesvia the inclusions S r Λ k ⊂ S − r +1 Λ k ⊂ S r +1 Λ k and satisfy identities at form order 0 and n , given by S − r Λ = S r Λ and S − r Λ n = S r − Λ n . The associated complexes in 2D and 3D are stated in ( . ) and( . ). Figure 5 shows the lowest order trimmed-Adini complex in two dimensions.17 .3.1 Two dimensions The 2D trimmed Adini sequence is denoted by0 ✲ R ✲ A − r Λ ( T h ) curl ✲ A − r Λ ( T h ) div ✲ A − r Λ ( T h ) ✲ . ( . )Given the identities on the serendipity and trimmed serendipity spaces for 0-forms and n -forms justmentioned, the sequence could also be written as A r Λ → A − r Λ → A r − Λ . Thus, only the space A − r Λ ( (cid:3) ) is distinct from a 2D Adini-type element already defined. Space A − r Λ ( (cid:3) ) . The degrees of freedom can be given by • function evaluation at each vertex x ∈ V : u i ( x ) , i = 1 , , • moments on each edge e ∈ E : Z e u · n e q dx, q ∈ P r − ( e ) , • interior DoFs in K ∈ K : Z K u · p dx, p ∈ [ P r − ( K )] ⊕ curl H r − Λ ( K ) , where H r − Λ ( K ) denotes the space of (scalar) homogeneous polynomials of degree r − K . The 3D trimmed Adini complex is denoted by0 ✲ R ✲ A − r Λ ( T h ) grad ✲ A − r Λ ( T h ) curl ✲ A − r Λ ( T h ) div ✲ A − r Λ ( T h ) ✲ . ( . )Using the identities for 0-forms and n -forms, we can re-write this sequence as0 ✲ R ✲ A r Λ ( T h ) grad ✲ A − r Λ ( T h ) curl ✲ A − r Λ ( T h ) div ✲ A r − Λ ( T h ) ✲ . ( . )Thus, we will only describe the 1-form and 2-form spaces in 3D. Space A − r Λ ( (cid:3) ) . The DoFs for a vector valued function u can be given by • value at each vertex x ∈ V : u ( x ) , • moments on each edge e ∈ E : Z e u · τ e q dx, q ∈ P r − ( e ) , • moments on each face f ∈ F : Z f ( u × ν f ) · p dx, p ∈ [ P r − ( f )] ⊕ grad H r − Λ ( f ) , where u × ν f is considered as a 2D vector, • interior DoFs in K ∈ K : Z K u · s dx, s ∈ [ P r − ( K )] ⊕ curl H r − Λ ( K ) . pace A − r Λ ( (cid:3) ) . The DoFs for a vector function v can be taken as • moments on each face f ∈ F : Z f v · ν f q dx, q ∈ P r − ( f ) , • interior DoFs in K ∈ K : Z K v · p dx, p ∈ [ P r − ( K )] ⊕ grad H r − Λ ( K ) . The unisolvence of the elements and the exactness of the various complexes introduced in this sectionfollow from the properties of the serendipity and the DoF-transfer operations shown in Lemma 2 –Lemma 5. For instance, since the Hermite and Adini complexes are the images of the T applied tothe tensor product and serendipity complexes, respectively (recall the diagram in ( . )), and since T preserves unisolvence by Lemma 2 and exactness by Lemma 3, it follows that the Hermite and Adinicomplexes are are unisolvent and exact.It is possible to prove exacness without applying the DoF-transfer to the S • Λ • sequence. Considerthe diagram shown in ( . ), which does not assume any relation between S • Λ • and A • Λ • .ker( D , Q ) ker( D , Q ) ker( D , Q )ker( D , G ) ker( D , G ) ker( D , G ) Q − r +2 Λ Q − r +2 Λ Q − r +1 Λ G − r +2 Λ G − r +2 Λ G − r +1 Λ S r +2 Λ S r +1 Λ S r Λ A r +2 Λ A r +1 Λ A r Λ
2= curl ι = div ι = ι curl ι div ι ι T curl Q T div
Q T Q curl Q divcurl Q div Q ( . )By Lemma 2 and Lemma 3, the Hermite family G − • Λ • is unisolvent and exact. Since the serendipityfamily S • Λ • is known to be exact on contractible domains [2], we know that the kernel sequence0 ✲ ker( D , Q ) curl ✲ ker( D , Q ) div ✲ ker( D , Q ) ✲ , is exact by Lemma 5. Since the DoF-transfer does not change bubble spaces (by Lemma 4) the kernelsequences are in fact equal, i.e. ker( D • , Q ) = ker( D • , G ). Using Lemma 5 again, we conclucde theAdini sequence A • Λ • is also exact. 19 Approximation and convergence properties
The DoF-transfer and serendipity operations aid in establishing approximation and convergence prop-erties. The DoF-transfer operation ensures a discrete Korn inequality and non-conforming convergence.The serendipity operation preserves these properties since it is a local space reduction that does notchange inter-element continuity.
The Korn inequality is an indispensable tool in linear elasticity, asserting that if a vector field isorthogonal to the rigid body motion, then its H norm can be controlled by its symmetric gradient.Specifically, the Korn inequality reads: if u and v are vector valued functions in a domain Ω, thenthere exists a positive constant C > k u k ≤ C (cid:16) k ǫ ( u ) k + k u k (cid:17) , ∀ u ∈ (cid:2) H (Ω) (cid:3) n , k v k ≤ C k ǫ ( v ) k + n X i =1 Z ∂ Ω | v i | ds ! , ∀ v ∈ (cid:2) H (Ω) (cid:3) n , and k v k ≤ C k ǫ ( v ) k , ∀ v ∈ (cid:2) H (Ω) (cid:3) n . Here ǫ is the symmetric gradient mapping a vector to a symmetric matrix, defined by [ ǫ ( u )] ij :=1 / ∂ i u j + ∂ j u i ).For non-conforming finite element spaces, a discrete version of Korn’s inequalities is desirable. Inthis case, the derivatives appearing in the original Korn’s inequality are replaced by the piecewisederivatives and the broken Sobolev norms k u k m,h := X T ∈T h k u k m,T ! / . We first recall the results in [40]. Following the general results, we will see that the DoF-transferoperation leads to finite elements satisfying the discrete Korn inequality. The following two conditionsfor a finite dimensional space V h are crucial for the discrete Korn inequality in [40].(H1) There exists an integer r ≥ v ∈ V h , v | T ∈ P r ( T ) , ∀ T ∈ T h .(H2) For any v ∈ V h , T ∈ T h , let F be an arbitrary ( n −
1) dimensional face of T . Then v is continuouson a set consisting of at least n points that are not on a common n − Theorem 1 ([40]) . Assume that the assumptions (H1) and (H2) are true. Then there exists a constant C independent of the mesh size h , such that the following discrete Korn inequalities hold: k v k ,h ≤ C X T ∈T h k ǫ ( v ) k ,T + k v k , Ω ! , ∀ v ∈ V h , k v k ,h ≤ C X T ∈T h k ǫ ( v ) k ,T + n X i =1 Z ∂ Ω | v i | ds ! , ∀ v ∈ V h , nd k v k ,h ≤ C X T ∈T h k ǫ ( v ) k ,T , ∀ v ∈ V h , where V h is the subspace of V h with vanishing degrees of freedom on the boundary. In some cases, the condition (H2) is a strong assumption. For example, generally the shape functionsof the Crouzeix-Raviart non-conforming element [11] are only continuous at n − n − n − n points. Thereforethe assumption (H2) is fulfilled and we conclude the following result. Theorem 2.
The 0- and 1-forms in the Hermite, Adini and trimmed-Adini complexes satisfy thediscrete Korn inequalities in Theorem 1.
The DoF-transfer operation ensures extra continuity at vertices compared to standard elements,namely, C continuity for 0-forms and C continuity for 1-forms. This order of continuity allowsprovable convergence results for problems requiring more regularity in the solution. Specifically, theresulting 0-forms can be regarded as convergent non-conforming elements for the biharmonic equations[1, 27] while the 1-forms are convergent non-conforming elements for Poisson type equations involvingthe scalar Laplacian operator.First, recall the Strang lemma (c.f. [12]). Lemma 6 (Strang Lemma) . Let V and V h be Hilbert spaces and dim V h < ∞ . Let a h ( · , · ) be asymmetric positive-definite bilinear form on V + V h that reduces to a ( · , · ) on V . Let u ∈ V solve a ( u, v ) = F ( v ) , ∀ v ∈ V, where F ∈ V ∗ ∩ V ∗ h . Let u h ∈ V h solve a h ( u h , v ) = F ( v ) , ∀ v ∈ V h . Then k u − u h k h ≤ inf v h ∈ V h k u − v h k h + sup w h ∈ V h \{ } | a h ( u − u h , w h ) |k w h k h . ( . )The first term on the right hand side of ( . ) is the local approximation error, which is determinedby the local shape function spaces. The second term, which does not appear in conforming methods,is the consistency error. The consistency error is generally determined by the inter-element continuity,although not strong enough to guarantee conformity. As we shall see, for the new elements proposedin this paper, the consistency error is controlled also because of the proper local shape function spacesand the geometric symmetry. In particular, we note that if the consistency error of a space V h tendsto zero asymptotically as h →
0, then any of its subspaces inherit this property. This means thatthe space reductions, either by the serendipity or by the DoF-transfer operations, do not break theconvergence of non-conforming methods. 21n this section, we prove the convergence of the DoF-transferred 1-forms for Poisson equations.Since one often seeks simple nonconforming elements and one cannot improve the consistency errorby increasing local polynomial degrees, we focus on the lowest order Adini 1-form A Λ . The space A Λ is not a tensor product of copies of scalar elements, meaning that one cannot collect the shapefunctions and degrees of freedom of one component to get a unisolvent scalar element. However, onecan obtain a scalar element with the same inter-element continuity by adding some interior degrees offreedom. We now show that each component of A Λ (in the above sense) yields a convergent scalarelement for the Poisson problem. Then the convergence of each component implies that of the element A Λ for vector Poisson problems. We start the analysis with the following explicit form of the bases. Lemma 7.
The shape function space of the Adini 1-form A Λ has a direct sum decomposition A Λ = P Λ ⊕ J Λ ⊕ d J Λ , where in three space dimensions J Λ = span − xyzx z , − y zxyz , − xyzxy , − xz xyz , − xyz x y , − yz xyz , and d J Λ = span x yx , x z x , xyzx zx y , x yzx zx y , y zy , y xy , y z xyzxy , y z xy zxy , z yz , z xz , yz xz xyz , yz xz xyz . Without loss of generality, we focus on the details of the x -component of the Adini 1-forms in 3Don a reference cubic K = ( − , . We use E xK to denote the edges of K parallel to the x -axis, and F xK for the faces of K parallel to the x -axis.Collecting the shape functions of the x-component, we define P xK := P ( K ) + span { xyz, x y, x z, yz , y z, y , z , zy , z y, x yz } . Equivalently, P xK = span { , x, x }{ , y }{ , z } + span { y , z , y z, yz , y , z , y z, yz } , where the three sets of curly braces indicate one choice is to be made from each set of braces beforemultiplying together. The following degrees of freedom define a unisolvent finite element with localshape function space P xK : • function evaluation at each vertex x ∈ V : v ( x ) , • moments on edges parallel to the x -axis: Z e v dx, ∀ e ∈ E xK , moments on faces parallel to the x -axis: Z f v dx, ∀ f ∈ F xK , • interior degrees of freedom on K : Z K vq dx, q ∈ P xK ∩ H ( K ) . We note that the above degrees of freedom on faces (including vertices and edges) are those for the x -component of A Λ , which are linearly independent for P xK and have dimension 16. On the otherhand, the space P xK has dimension 20. This difference is due to the non-tensor-product nature of A Λ as explained above. Therefore we also include the interior degrees of freedom with dimension20 −
16 = 4.
Lemma 8.
The above degrees of freedom are unisolvent for A Λ . The local shape function space P xK and the above degrees of freedom give a finite element spaceon a mesh T h on Ω, which we denote as V h . Define V h as the corresponding space with homogeneousboundary conditions: V h := (cid:26) v ∈ V h : v ( a ) = 0 , a ∈ V b ; Z e v = 0 , e ∈ E xb , Z f v = 0 , f ∈ F xb (cid:27) , where V b is the set of boundary vertices, E xb the boundary edges parallel to the x -axis and F xb theboundary faces parallel to the x -axis.Consider the variational form of the Poisson equation: given f ∈ L (Ω), find u ∈ H (Ω), such that( ∇ u, ∇ v ) = ( f, v ) , ∀ v ∈ H (Ω); ( . )and its non-conforming element discretization: find u h ∈ V h , such that( ∇ h u h , ∇ h v h ) = ( f, v h ) , ∀ v h ∈ V h . ( . )Here ∇ h is the piecewise gradient defined on T h .To show the convergence of ( . ), we introduce some auxiliary spaces, again using the conventionthat a list inside curly braces indicates one of the options should be chosen: B xK = span (cid:8) ( x − (cid:2) ( y − { , x } + { , y } ( z − (cid:3)(cid:9) ,Q xK = span { , x, x }{ , y }{ , z } ;and A xK = Q xK + B xK . The auxiliary space A xK is defined so that it is unisolvent with the vertex, edge and faces DoFs of V h ,and also forms a C -conforming global space. Denote by Π Q Π B and Π A the interpolations such thatΠ Q v ∈ Q xK ; (Π Q v )( a ) = v ( a ) , ∀ a ∈ V ( K ); Z e (Π Q v ) = Z e v, ∀ e ∈ E xK ;23 B v ∈ B xK ; Z f (Π B v ) = Z f v, ∀ f ∈ F xK , and Π A v ∈ A xK ; (Π A v )( a ) = v ( a ); Z e (Π A v ) = Z e v, ∀ e ∈ E xK ; Z F (Π A v ) = Z f v, ∀ f ∈ F xK . It is straightforward to check that the interpolations are all well-defined.
Lemma 9.
For any u ∈ C ( K ) , it holds that (Π A − Π B )( I − Π Q ) u = 0 . Proof.
For any vertex a of K , [( I − Π Q ) u ]( a ) = 0, and therefore Π A ( I − Π Q ) u ∈ B xK ; actually, { v ∈ A K : v ( a ) = 0 } = B xK . The assertion follows. Lemma 10.
For any v ∈ C ( K ) , it holds that Z ∂K ( v − Π A v ) dx = 0 , v ∈ P xK .Proof. Note that Π A Π Q = Π Q , and we have by Lemma 9 that u − Π A u = ( u − Π Q u ) − Π A ( u − Π Q u ) = ( u − Π Q u ) − Π B ( u − Π Q u ) = ( I − Π B )( I − Π Q ) u. By direct calculations, we get Table 1. u y z y y z yz z y z yz Π Q u y z y z yz yzu − Π Q u y − z − y − y y z − z yz − y z − z y z − yz yz − yz Table 1: Interpolations of shape functions.Denote faces by (Left, Right, Out, Interior, Bottom, Top) F L ∼ ( − , y, z ); F R ∼ (1 , y, z ); ( y, z ) ∈ ( − , F O ∼ ( x, − , z ); F I ∼ ( x, , z ); ( x, z ) ∈ ( − , F B ∼ ( x, y, − F T ∼ ( x, y, x, y ) ∈ ( − , . Here, for instance, F L ∼ ( − , y, z ) means the face { x = 1 , y, z ∈ ( − , } .For any v ∈ P xK , it can be verified that Z F L u − Π A u = Z F R u − Π A u, Z F I u − Π A u = Z F O u − Π A u, Z F T u − Π A u = Z F B u − Π A u. ( . )Actually,1. if v ∈ Q xK , then ( I − Π A ) v = ( I − Π B )( I − Π Q ) v = 0;2. if v ∈ span { y , z , y , y z, yz , z , y z, yz } (1) on { F I , F O , F T , F B } , by definition of Π B , Z F I ,F O ,F T ,F B ( I − Π B )(( I − Π Q ) v ) = 0;(2) on { F L , F R } , Z F L ( I − Π Q ) v = Z F R ( I − Π Q ) v , for v ∈ span { y , z , y , y z, yz , z , y z, yz } and Z F L ,F R w = 0 for w ∈ B K , thus Z F L ( I − Π B )(( I − Π Q ) v ) = Z F R ( I − Π B )(( I − Π Q ) v ).24he assertion of the lemma then follows.Define an auxiliary space A h with the local shape function space A xK and the same vertex, edgeand face degrees of freedom as V h . Define the nodal interpolation Π Ah with respect to A h . We use J u K f to denote the jump of u on a face, i.e. if f = T ∩ T , then J u K f := u T − u T . When the context isclear, we omit the subscript f and just write J u K instead. Theorem 3.
Let u ∈ H (Ω) ∩ H (Ω) and u h be the solutions of ( . ) and ( . ) , respectively. Then k u − u h k ,h Ch k u k , Ω . ( . ) Proof.
By the Strang lemma, we can estimate the consistency error. We have( ∇ u, ∇ h v h ) − ( f, v h ) = X f ∈F Z f ∂ n u J v h K = X f ∈F Z f ∂ n u J v h − Π Ah v h K = X K ∈T h Z ∂K ∂ n u ( v h − Π Ah v h )= X K ∈T h X i =1 Z ∂K ∂ x i u ( v h − Π Ah v h ) n i = X K ∈T h X i =1 Z ∂K ( ∂ x i u − C iK )( v h − Π Ah v h ) n i , where the second equality is due to the C conformity of A h and the last is by Lemma 10. Here C iK is an arbitrary constant.This way, | ( ∇ u, ∇ h v h ) − ( f, v h ) | X K ∈T h X i =1 inf C iK ∈ R T h ) × ( k ∂ x i u − C iK k ∂K k ( v h − Π Ah v h ) k ∂K ) Ch k u k , Ω k v h k ,h , where T h ) is the number of elements in T h . The consistency error is controlled, and the assertionfollows.Now we have proved that the space V h , which consists of the local space P xK and the same inter-element continuity as the x -component of A Λ , converges as a non-conforming element for the Poissonequation. This means that each component (in a proper sense) of A Λ converges as a nonconformingelement. However, since we have added interior degrees of freedom to V h , there is one more step toconclude with the convergence of A Λ as a non-conforming element for the vector Poisson equation.Consider the vector Poisson equation: given f ∈ [ L (Ω)] , find u ∈ (cid:2) H (Ω) (cid:3) , such that( ∇ u , ∇ v ) = ( f , v ) , ∀ v ∈ (cid:2) H (Ω) (cid:3) ; ( . )and its non-conforming element discretization: find u h ∈ ˚ A Λ , such that( ∇ h u h , ∇ h v h ) = ( f , v h ) , ∀ v h ∈ ˚ A Λ . ( . )Here ˚ A Λ is the subspace A Λ with vanishing boundary degrees of freedom. Theorem 4.
Let u ∈ [ H (Ω) ∩ H (Ω)] and u h be the solutions of ( . ) and ( . ) , respectively. Then k u − u h k ,h Ch k u k , Ω . ( . ) Proof.
The theorem follows from the Strang lemma. In fact, the local approximation error satisfiesthe above estimate since piecewise constants are contained in A Λ . Moreover, the consistency erroralso satisfies the estimate as we have shown for each component of A Λ with the same local spacesand inter-element continuity. Remark 3.
The two dimensional results follow the same way. ❢ r❢r❢ r❢ ✲ curl rr rrrr rr ✻❄✛ ✲ ✲ div +1Figure 6: The reduced Adini complex. By the Strang lemma, the errors of solving PDEs using non-conforming elements come from twosources: one is the local approximation error and another is the consistency error due to the noncon-formity. In most cases, the consistency error renders only first order convergence. In this respect,higher local polynomial degree does not increase the convergence order of non-conforming elements;and serendipity operations for non-conforming elements should be defined not to preserve the full poly-nomial degree, but to preserve the constants . From this point of view, we can further simplify the2D Adini complex by a new serendipity operation. The construction yields a new complex which isconforming for H - H (div)- L , convergent as non-conforming elements for H -[ H ] - L , and minimalin the sense that no interior bubbles are involved.We define the reduced space ˜ A Λ by˜ A Λ ( K ) : = [ P ( K )] + span (cid:8) curl( x ) , curl( y ) curl( x y ) , curl( xy ) , curl( x y ) , curl( xy ) (cid:9) = curl A Λ ( K ) + span { ( x, T , (0 , y ) T } , and define ˜ A Λ by ˜ A Λ ( K ) := P ( K ) . The degrees of freedom for u ∈ ˜ A Λ , as shown in Figure 6, are given by • function evaluation at each vertex x ∈ V , • moments on each edge e ∈ E : Z e u · ν e dx. The degrees of freedom for v ∈ ˜ A Λ are defined by the moments on each element K ∈ F : Z K v dx. Lemma 11.
The space ˜ A Λ with the degrees of freedom defined above is unisolvent.Proof. The local shape function space and the set of degrees of freedom have the same dimension.Therefore we only show that if all the degrees of freedom vanish on a function u ∈ ˜ A Λ , then u = 0.Actually, if all the degrees of freedom vanish, then we have Z ∂K u · ν ∂K dx = 0 , ∀ K ∈ F . By Stokes’26heorem, this implies that Z K div u dx = 0 and further div u = 0 since the image of div on ˜ A Λ is thespace of constants. Then u can be expressed as u = curl φ by some potential φ ∈ A Λ ( K ). Since u vanishes at vertices, φ have vanishing first order derivatives at vertices. Moreover, the function valuesof φ at the four vertices have to be equal, by the edge degrees of freedom and Stokes’ theorem on edges.The potential φ is determined up to a constant, so we are allowed to choose this common vertex valueto be zero. So φ = 0 and therefore u = 0.We verify the following properties. Lemma 12.
The reduced spaces satisfy • curl A Λ ( T h ) ⊂ ˜ A Λ ( T h ) , • div ˜ A Λ ( T h ) ⊂ ˜ A Λ ( T h ) , • the complex ✲ R ✲ A Λ ( T h ) curl ✲ ˜ A Λ ( T h ) div ✲ ˜ A Λ ( T h ) ✲ . ) is exact.Proof. The inclusions are trivial by definition. To show the exactness, we observe that if u ∈ ˜ A Λ ( T h )satisfies div u = 0, then u has to be in curl A Λ ( T h ) by the exactness of the Adini complex ( . ) andthe fact ˜ A Λ ⊂ A Λ .The local spaces of the reduced complex ( . ) can also be constructed by the Poincar´e/Koszuloperators [24, 5, 16]: ˜ A Λ ( K ) = curl ˜ A Λ ( K ) + κ ˜ A Λ ( K ) , ∀ K ∈ F , where κ : v v x maps a scalar function to a vector valued function by multiplying x with a chosenorigin. This also implies the local exactness and the approximation property that piecewise linearfunctions and piecewise constants are contained in the last two spaces respectively.Since the reduced space ˜ A Λ ( T h ) is a subspace of A Λ ( T h ) with vertex continuities, we have thediscrete Korn inequality and the convergence as a non-conforming element. Theorem 5.
The reduced space ˜ A Λ ( T h ) is H (div) -conforming, satisfies the discrete Korn inequalitystated in Theorem 1 and leads to convergent schemes as a non-conforming element for Poisson typeequations (as Theorem 3). A similar diagram as ( . ) also exists for this reduced and DoF-transferred complex. The DoF-transfer operation proposed in this paper increases the regularity of shape functions atvertices, which reduces global DoFs and leads to convergent non-conforming elements. On the otherhand, the serendipity operation reduces local DoFs. Both of these operations preserve the unisolvenceand exactness of the discrete complex on contractible domains. Based on the two operations and the27erendipity finite element families on cubes [2, 23], we constructed several complexes with vertex DoFsand fewer local and global DoFs than standard tensor product elements.There is always a trade-off between nodal (nonstandard) and non-nodal (standard) elements. For H (div), hybridization is possible for the standard face elements, and these standard elements are betterstudied from the perspective of solvers and software. On the other hand, nodal elements have fewerglobal DoFs and sometimes nodal-type interpolations admit a canonical construction of bases withmoderate condition numbers, sparsity and rotational symmetry (c.f. [15]). In summary, the seeminglysimple DoF-transfer operation changes the topology of the element connections, which in turn givenew opportunities for the study of bases, solvers and applications.We have seen that the DoF-transfer operation on cubical meshes often renders non-conformingconvergence. Therefore, in terms of non-conforming plate or Stokes elements, the nodal continuityseems natural and necessary when we require a discrete sequence that is both conforming as a deRham complex and non-conforming (convergent) as a Stokes complex. In contrast, another well knowncomplex of non-conforming elements, consisting of the Morley and the Crouzeix-Raviart elements, isnot conforming as a de Rham complex [11, 30].The DoF-transfer operation can also be applied to other element shapes. The discussions in [15] fornodal elements are only focused on finite elements with complete polynomial shape spaces. The generalapproach proposed in this paper yields some new elements and complexes on simplicial meshes, at leastfor the family with regularity 1 in [15]. For example, we can modify a 2D Lagrange → Raviart-Thomas → DG complex by the DoF-transfer operation to obtain a Stenberg type element with Raviart-Thomasshape functions, which is continuous at vertices. This complex is also exact on contractible domainsand from the viewpoint of differential complexes, we avoid the use of macroelement techniques.On simplicial meshes, complexes starting with the modified Zienkiewicz element [42] may also beexplored as an analogue of the discussions in this paper.Figure 7 shows a combination of cubical and simplicial elements. The edge modes of both familiesare the same. This renders a possibility for flexible meshes and varying polynomial degrees in highorder methods and adaptivity.As explained in [15], the 1-forms in the 2D Hermite complex can be represented by nodal basisfunctions. For the standard serendipity element (0-forms in the Adini complex), nodal basis functionshave been studied in [22]. Therefore the constructions in this paper could also shed some light oncomputational issues, especially for high order methods. We regard this respect as a further direction.The DoF-transfer operation is defined as changing vertex and edge degrees of freedom. In regardto the discussions in [15], this corresponds to the case ℓ = 1 where ℓ is a regularity index (denoted as r = 1 in [15]). More general DoF-transfer operations can be explored to cover the family ℓ = 2 andother global space reductions. Acknowledgements
AG was supported in part by National Science Foundation Award DMS-1522289. KH was supported inpart by the European Research Council under the European Union’s Seventh Framework Programme(FP7/2007-2013) / ERC grant agreement 339643. SZ was supported in part by the National NaturalScience Foundation of China with Grant No. 11471026.28 ❢ r❢r❢ r❢rr❢ ✁✁✁✁✁✁✁ ❆❆❆❆❆❆❆ ✲ curl rr rrrr rrrr ✻❄✛ ✲❅❅■ (cid:0)(cid:0)✒✁✁✁✁✁✁✁ ❆❆❆❆❆❆❆ +3+2 ✲ div ✁✁✁✁✁✁✁ ❆❆❆❆❆❆❆ +3+3Figure 7: Adini-Hermite-de Rham complex on flexible geometries References [1]
A. Adini and R. W. Clough , Analysis of plate bending by the finite element method . NSFReport G. 7337, University of California, Berkeley, CA, 1961.[2]
D. Arnold and G. Awanou , Finite element differential forms on cubical meshes , Mathematicsof Computation, 83 (2014), pp. 1551–1570.[3]
D. N. Arnold and G. Awanou , The serendipity family of finite elements , Foundations ofComputational Mathematics, 11 (2011), pp. 337–344.[4]
D. N. Arnold, D. Boffi, and F. Bonizzoni , Finite element differential forms on curvilinearcubic meshes and their approximation properties , Numerische Mathematik, 129 (2015), pp. 1–20.[5]
D. N. Arnold, R. S. Falk, and R. Winther , Finite element exterior calculus, homologicaltechniques, and applications , Acta numerica, 15 (2006), p. 1.[6] ,
Finite element exterior calculus: from Hodge theory to numerical stability , Bulletin of theAmerican Mathematical Society, 47 (2010), pp. 281–354.[7]
D. N. Arnold and A. Logg , Periodic table of the finite elements , SIAM News, 47 (2014).[8]
D. N. Arnold and J. Qin , Quadratic velocity/linear pressure Stokes elements , Advances incomputer methods for partial differential equations, 7 (1992), pp. 28–34.[9]
D. Boffi, F. Brezzi, and M. Fortin , Mixed Finite Element Methods and Applications ,Springer, 2013.[10]
R. Bott and L. W. Tu , Differential forms in algebraic topology , vol. 82, Springer Science &Business Media, 2013.[11]
S. C. Brenner , Forty Years of the Crouzeix-Raviart element , Numerical Methods for PartialDifferential Equations, 31 (2015), pp. 367–396.[12]
S. C. Brenner and R. Scott , The mathematical theory of finite element methods , vol. 15,Springer Science & Business Media, 2007. 2913]
F. Brezzi, J. Douglas Jr, and L. D. Marini , Two families of mixed finite elements for secondorder elliptic problems , Numerische Mathematik, 47 (1985), pp. 217–235.[14]
A. Buffa, J. Rivas, G. Sangalli, and R. V´azquez , Isogeometric discrete differential formsin three dimensions , SIAM Journal on Numerical Analysis, 49 (2011), pp. 818–844.[15]
S. H. Christiansen, J. Hu, and K. Hu , Nodal finite element de Rham complexes , NumerischeMathematik, (2016), pp. 1–36.[16]
S. H. Christiansen and K. Hu , Generalized Finite Element Systems for smooth differentialforms and Stokes problem , arXiv preprint arXiv:1605.08657, (2016).[17]
S. H. Christiansen, H. Z. Munthe-Kaas, and B. Owren , Topics in structure-preservingdiscretization , Acta Numerica, 20 (2011), pp. 1–119.[18]
L. B. Da Veiga, F. Brezzi, L. Marini, and A. Russo , Serendipity nodal VEM spaces ,Computers & Fluids, 141 (2016), pp. 2–12.[19]
L. B. Da Veiga, F. Brezzi, L. D. Marini, and A. Russo , Serendipity face and edge VEMspaces , arXiv preprint arXiv:1606.01048, (2016).[20]
D. S. Dummit and R. M. Foote , Abstract algebra , vol. 3, Wiley Hoboken, 2004.[21]
R. S. Falk and M. Neilan , Stokes complexes and the construction of stable finite elements withpointwise mass conservation , SIAM Journal on Numerical Analysis, 51 (2013), pp. 1308–1326.[22]
M. Floater and A. Gillette , Nodal bases for the serendipity family of finite elements , Foun-dations of Computational Mathematics, 17 (2017), pp. 879–893.[23]
A. Gillette and T. Kloefkorn , Trimmed serendipity finite element differential forms , Math-ematics of Computation, to appear (2016).[24]
R. Hiptmair , Canonical construction of finite elements , Mathematics of Computation of theAmerican Mathematical Society, 68 (1999), pp. 1325–1346.[25] ,
Finite elements in computational electromagnetism , Acta Numerica, 11 (2002), pp. 237–339.[26]
J. Hu , Finite element approximations of symmetric tensors on simplicial grids in R n : the higherorder case , Journal of Computational Mathematics, 33 (2015), pp. 283–296.[27] J. Hu, X. Yang, and S. Zhang , Capacity of the Adini element for biharmonic equations , Journalof Scientific Computing, 69 (2016), pp. 1366–1383.[28]
J. Hu and S. Zhang , A family of symmetric mixed finite elements for linear elasticity on tetra-hedral grids , Science China Mathematics, 58 (2015), pp. 297–307.[29]
V. John, A. Linke, C. Merdon, M. Neilan, and L. G. Rebholz , On the divergence con-straint in mixed finite element methods for incompressible flows , SIAM Review, 59 (2017), pp. 492–544. 3030]
A. Linke, C. Merdon, M. Neilan, and F. Neumann , Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes-Problem , Mathematics of Computation,(2018).[31]
K. A. Mardal, X.-C. Tai, and R. Winther , A robust finite element method for Darcy–Stokesflow , SIAM Journal on Numerical Analysis, 40 (2002), pp. 1605–1631.[32]
J.-C. N´ed´elec , Mixed finite elements in R , Numerische Mathematik, 35 (1980), pp. 315–341.[33] , A new family of mixed finite elements in R , Numerische Mathematik, 50 (1986), pp. 57–81.[34] M. Neilan , Discrete and conforming smooth de Rham complexes in three dimensions , Mathe-matics of Computation, (2015).[35]
M. Neilan and D. Sap , Stokes elements on cubic meshes yielding divergence-free approxima-tions , Calcolo, 53 (2016), pp. 263–283.[36]
T. Nilssen, X.-C. Tai, and R. Winther , A robust nonconforming H -element , Mathematicsof Computation, 70 (2001), pp. 489–505.[37] P.-A. Raviart and J.-M. Thomas , A mixed finite element method for 2-nd order elliptic prob-lems , in Mathematical aspects of finite element methods, Springer, 1977, pp. 292–315.[38]
R. Stenberg , A nonstandard mixed finite element family , Numerische Mathematik, 115 (2010),pp. 131–139.[39]
X.-C. Tai and R. Winther , A discrete de Rham complex with enhanced smoothness , Calcolo,43 (2006), pp. 287–306.[40]
M. Wang , The generalized Korn inequality on nonconforming finite element spaces , Chinese J.Numer. Math. Appl, 16 (1994), pp. 91–96.[41] ,
On the necessity and sufficiency of the patch test for convergence of nonconforming finiteelements , SIAM journal on numerical analysis, 39 (2001), pp. 363–384.[42]
M. Wang, Z.-C. Shi, and J. Xu , A new class of Zienkiewicz-type non-conforming element inany dimensions , Numerische Mathematik, 106 (2007), pp. 335–347.[43]
S. Zhang , A new family of stable mixed finite elements for the 3D Stokes equations , Mathematicsof computation, 74 (2005), pp. 543–554.[44]