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LOCAL MULTIGRID IN H(curl) RALF HIPTMAIR∗ AND WEIYING ZHENG † Abstract.WeconsiderH(curl,Ω)-ellipticvariationalproblemsonboundedLipschitzpolyhedra andtheirfiniteelementGalerkindiscretizationbymeansof lowestorderedgeelements.Weassume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop 9 aconvergencetheoryforthetheso-calledlocalmultigridcorrectionschemewithhybridsmoothing. 0 Weestablishthatitsconvergencerateisuniformwithrespecttothenumberofrefinementsteps.The 0 proofreliesoncorrespondingresultsforlocalmultigridinaH1(Ω)-contextalongwithlocaldiscrete Helmholtz-typedecompositions oftheedgeelementspace. 2 n Key words. Edge elements, local multigrid, stable multilevel splittings, subspace correction a theory,regulardecompositionsofH(curl,Ω),Helmholtz-typedecompositions,localmeshrefinement J 7 AMS subject classifications. 65N30, 65N55,78A25 ] 1. Introduction. On a polyhedron Ω⊂R3, scaled such that diam(Ω) = 1, we A consider the variational problem: seek u∈H (curl,Ω) such that N ΓD . (1.1) (curlu,curlv) +(u,v) =(f,v) ∀v∈H (curl,Ω). h L2(Ω) L2(Ω) L2(Ω) ΓD t a =:a(u,v) m For the H|ilbert space of {szquare integrable}vector fields with square integrable curl [ andvanishingtangentialcomponentsonΓ weusethesymbolH (curl,Ω),see[24, D ΓD 1 Ch.1]fordetails.Thesourcetermf in(1.1)isavectorfieldin(L2(Ω))3.Thelefthand v side of (1.1) agrees with the inner product of H (curl,Ω) and will be abbreviated 4 ΓD by a(u,v) (“energy inner product”). 6 7 Further,ΓD denotesthepartoftheboundary∂ΩonwhichhomogeneousDirichlet 0 boundaryconditions in the formofvanishing tangentialtracesof uareimposed. The . geometryofthe DirichletboundarypartΓ issupposedto be simple inthe following 1 D 0 sense: for each connected component Γi of ΓD we can find an open Lipschitz domain 9 Ω ⊂R3 such that i 0 : (1.2) Ω ∩Ω=Γ , Ω ∩Ω=∅, v i i i i X and Ω and Ω have positive distance for i 6= j. Further, the interior of Ω ∪Ω ∪ i j 1 r Ω ... is expected to be a Lipschitz-domain, too (see Fig. 5.2). This is not a severe 2 a restriction, because variational problems related to (1.1) usually arise in quasi-static electromagnetic modelling, where simple geometries are common. Of course, Γ =∅ D is admitted. Lowest order H (curl,Ω)-conforming edge elements are widely used for the ΓD finite element Galerkin discretization of variational problems like (1.1). Then, for a solutionu∈(H1(Ω))3 withcurlu∈(H1(Ω))3 wecanexpecttheoptimalasymptotic convergence rate (1.3) ku−u k ≤CN−1/3 , h H(curl,Ω) h ∗SAM,ETHZu¨rich,CH-8092Zu¨rich,[email protected] †LSEC,InstituteofComputationalMathematics,AcademyofMathematicsandSystemSciences, ChineseAcademyofSciences,Beijing,100080,People’sRepublicofChina.Thisauthorwassupported inpartbyChinaNSFunderthegrant10401040 ([email protected]). 1 2 R.HiptmairandW.-Y.Zheng H(curl,Ω) : Sobolev space of square integrable vector fields on Ω⊂R3 with square integrable curl H (curl,Ω): vector fields in H(curl,Ω) with vanishing tangential compo- ΓD nents on Γ ⊂∂Ω D M, T : tetrahedral finite element meshes, may contain hanging nodes N(M) : set of vertices (nodes) of a mesh M E(M) : set of edges of a mesh M ρ ,ρ : shape regularity measures K M h : – local meshwidth function for a finite element mesh – (as subscript) tag for finite element functions U(M) : lowest order edge element space on M b : nodal basis function of U(M) associated with edge E E V(M) : space of continuous piecewise linear functions on M V (M) : quadratic Lagrangianfinite element space on M 2 V (M) : quadratic surplus space, see (2.19) 2 b : nodal basis function of V(M) (“tent function”) associatedwith p e vertex p B (M) : set of nodal basis functions for finite element space X on mesh X M Π : nodal edge interpolation operator onto U(M), see (2.7) h I : vertex based piecewise linar interpolation onto V(M) h P : space of 3-variate polynomials of total degree ≤p p U(M), V(M): finite element spaces oblivious of zero boundary conditions ≺ : nesting of finite element meshes ℓ(K) : level of element K in hierarchy of refined meshes ω : refinement zone, see (4.1) l Σ : refinement strip, see (5.57) l Bl , Bl : sets of basis functions supported inside refinement zones, see V U (4.9) Q : quasi-interpolationoperator, Def. 5.1 h Table0.1 Important notation used inthis paper on families of finite element meshes arising from global refinement. Here, u is the h finite element solution, N the dimension of the finite element space, and C > 0 h does not depend on N . However, often u will fail to possess the required regularity h due to singularities arising at edges/corners of ∂Ω and material interfaces [22,23]. Fortunately, it seems to be possible to retain (1.3) by the use of adaptive local mesh refinement based on a posteriori errorestimates, see [10,55]for theory in H1-setting, [7,17]for numericalevidence in the case of edge element discretization,and [8,34,52] for related theoretical investigations. We alsoneed waysto compute the asymptotically optimalfinite element solution with optimal computational effort, that is, with a number of operations proportional to N . This can only be achieved by means of iterative solvers, whose convergence h remains fast regardless of the depth of refinement. Multigrid methods are the most prominentclassofiterativesolversthatachievethisgoal.Bynow,geometricmultigrid methods for discrete H(curl,Ω)-elliptic variational problems like (1.1) have become well established [20,29,54,58]. Their asymptotic theory on sequencies of regularly LocalMultigridinH(curl) 3 refined meshes has also matured [2,25,29,31,51].It confirms asymptotic optimality: the speed of convergence is uniformly fast regardless of the number of refinement levels involved. In addition, the costs of one step of the iteration scale linearly with the number of unknowns. Yet, the latter property is lost when the standard multigrid correction scheme is applied to meshes generatedby pronouncedlocalrefinement. Optimal computational costs can only be maintained, if one adopts the local multigrid policy, which was pioneered by A. Brandt et al. in [5], see also [41]. Crudely speaking, its gist is to confine relaxations to “new” degrees of freedom located in zones where refinement has changed the mesh. Thus an exponential increase of computational costs with the number of refinement levels can be avoided: the total costs of a V-cycle remain proportional to the number of unknowns. An algorithm blending the local multigrid idea with the geometric multigrid correction scheme of [29] is described in [54]. On the other hand, a proof of uniform asymptotic convergence has remained elusive so far. It is the objective of this paper to provide it, see Theorem 4.2. We recall the key insight that (1.1) is one member of a family of variational problems. Its kin is obtained by replacing curl with grad or div, respectively. All these differential operators turn out to be incarnations of the fundamental exterior derivative of differential geometry, cf. [29, Sect. 2]. They are closely connected in the deRham complex [3] and, thus, it is hardly surprising that results about the related H1 (Ω)-elliptic variational problem, which seeks u∈H1 (Ω) such that ΓD ΓD (1.4) (gradu,gradv) +(u,v) =(f,v) ∀v ∈H1 (Ω), L2(Ω) L2(Ω) L2(Ω) ΓD prove instrumental in the multigrid analysis for discretized versions of (1.1). Here H1 (Ω) is the subspace of H1(Ω) whose functions have vanishing traces on Γ . ΓD D Thus, when tackling (1.1), we take the cue from the local multigrid theory for (1.4) discretized by means of linear continuous finite elements. This theory has been developedinvarioussettings,cf. [5,11,14,15,62].In[1]localrefinementwithhanging nodesistreated.Recently,H.WuandZ.Chen[60]provedtheuniformconvergenceof local multigrid V-cycles on adaptively refined meshes in two dimensions. Their mesh refinements are controlled by a posteriori error estimators and carried out according to the “newest vertex bisection” strategy introduced, independently, in [6,40]. As in the case of global multigrid, the essential new aspect of local multigrid theory for (1.1) compared to (1.4) is the need to deal with the kernel of the curl- operator, cf. [29, Sect. 3]. In this context, the availability of discrete scalar potential representations for irrotational edge element vector fields is pivotal. Therefore, we devote the entire Sect. 2 to the discussion of edge elements and their relationship withconventionalLagrangianfinite elements.Mesheswithhangingnodeswillreceive particular attention. Next, in Sect. 3 we present details about local mesh refinement, because some parts of the proofs rest onthe subtleties of how elements are split. The followingSect. 4 introduces the localmultigrid method fromthe abstractperspective of successive subspace correction. Theproofofuniformconvergence(Theorem4.2)istackledinSects.5and6,which formthecoreofthearticle.Inparticular,theinvestigationofthestabilityofthelocal multilevelsplittingrequiresseveralsteps,thefirstofwhichaddressestheissueforthe bilinearformfrom(1.4)andlinearfinite elements.These resultsarealreadyavailable in the literature, but are re-derived to make the presentation self-contained. This alsoappliesto the continuousanddiscrete Helmholtz-typedecompositionscoveredin Sect. 5.3. Many developments are rather technical and to aid the reader important 4 R.HiptmairandW.-Y.Zheng notations are listed in Table 0.1. Eventually, in Sect. 7, we report two numerical experiments to show the competitive performance of the local multigrid method and the relevance of the convergence theory. Remark 1.1. In this article we forgo generality and do not discuss the more general bi-linear form (1.5) a(u,v):=(αcurlu,curlv) +(βu,v) , ∀u,v∈H (curl,Ω), L2(Ω) L2(Ω) ΓD with uniformly positive coefficient functions α,β ∈ L∞(Ω). We do this partly for the sake of lucidity and partly, because the current theory cannot provide estimates that are robust with respect to large variations of α and β, cf. [33]. We refer to [63] for further information and references. 2. Finite element spaces. Whenever we refer to a finite element mesh in this article, we have in mind a tetrahedral triangulation of Ω, see [19, Ch. 3]. In certain settings,itmayfeaturehangingnodes,thatis,thefaceofonetetrahedroncancoincide with the union of faces of other tetrahedra. Further, the mesh is supposed to resolve the Dirichlet boundary in the sense that Γ is the union of faces of tetrahedra. The D symbol M with optional subscripts is reservedfor finite element meshes and the sets of their elements alike. We write h ∈ L∞(Ω) for the piecewise constant function, which assumes value h := diam(K) in each element K ∈ M. The ratio of diam(K) to the radius of the K largestballcontainedinK iscalledtheshaperegularitymeasureρ [19,Ch.3,§3.1]. K The shape regularity measure ρ of M is the maximum of all ρ , K ∈M. M K 2.1. Conforming meshes. Provisionally, we consider only finite element meshes M that are conforming, that is, each face of a tetrahedron is either contained in ∂Ω or a face of another tetrahedron, see [19, Ch. 2, § 2.2]. In particular, thisrulesouthangingnodes.Following[12,43],weintroducethespaceoflowestorder H (curl,Ω)-conforming edge finite elements, also known as Whitney-1-forms [59], ΓD (2.1) U(M):={v ∈H (curl,Ω): ∀K ∈M:∃a,b∈R3 : h ΓD (2.2) v (x)=a+b×x, x∈K}. h Foradetailedderivationanddescriptionpleaseconsult[30,Sect.3]orthemonographs [13,42].Notice that curlU(M) is a spaceofpiecewise constantvectorfields. We also remark that appropriate global degrees of freedom (d.o.f.) for U(M) are given by U(M) 7→ R (2.3) , E ∈E(M), v 7→ v ·d~s (cid:26) h E h where E(M) is the set of edges of MRnot contained in Γ . We write B (M) for the D U nodal basis of U(M) dual to the global d.o.f. (2.3). Basis functions are associated with active edges. Hence, we can write B (M) = {b } . The support of the U E E∈E(M) basis function b is the union of tetrahedrasharingthe edgeE. We recallthe simple E formula for local shape functions (2.4) b =λ gradλ −λ gradλ E =[a ,a ]⊂K E|K i j j i i j for any tetrahedronK ∈M with vertices a , i=1,2,3,4,and associatedbarycentric i coordinate functions λ . i LocalMultigridinH(curl) 5 TheedgeelementspaceU(M)withbasisB (M)isperfectlysuitedforthefinite U elementGalerkindiscretizationof (1.1).The discreteproblembasedonU(M) reads: seek u ∈U(M) such that h (2.5) (curlu ,curlv ) +(u ,v ) =(f,v ) ∀v ∈U(M). h h L2(Ω) h h L2(Ω) h L2(Ω) h ThepropertiesofU(M)willbekeytoconstructingandanalyzingthelocalmultigrid method for the large sparse linear system of equations resulting from (2.5). Next, we collect important facts. The basis B (M) enjoys uniform L2-stability, meaning the existence of a con- U stant1 C =C(ρ )>0 such that for all v = α b ∈U(M), α ∈R, M h E E E E∈E(M) P (2.6) C−1kv k2 ≤ α2 kb k2 ≤Ckv k2 . h L2(Ω) E E L2(Ω) h L2(Ω) E∈XE(M) The global d.o.f. induce a nodal edge interpolation operator dom(Π )⊂H (curl,Ω) 7→ U(M) h ΓD (2.7) Πh : v 7→ Ev·d~s ·bE .  E∈E(M) P (cid:16)R (cid:17) Obviously, Π provides a local projection, but it turns out to be unbounded even h on (H1(Ω))3. Only for vector fields with discrete rotation the following interpolation error estimate is available, see [30, Lemma 4.6]: Lemma 2.1. The interpolation operator Π is bounded on {Ψ ∈ h (H1(Ω))3, curlΨ ∈ curlU(M)}⊂(H1(Ω))3, and for any conforming mesh there is C =C(ρ )>0 such that M h−1(Id−Π )Ψ ≤C|Ψ| ∀Ψ∈(H1(Ω))3, curlΨ∈curlU(M). h L2(Ω) H1(Ω) (cid:13) (cid:13) (cid:13) (cid:13) If Ω is homeomorphic to a ball, then gradH1(Ω) = H(curl0,Ω) := {v ∈ H(curl,Ω), curlv = 0}, that is, H1(Ω) provides scalar potentials for H(curl,Ω). To state a discrete analogue of this relationship we need the Lagrangian finite element space of piecewise linear continuous functions on M (2.8) V(M):={u ∈H1 (Ω): u ∈P (K)∀K ∈M}, h ΓD h|K 1 where P (K) is the space of 3-variate polynomials of degree ≤ p on K. The global p degrees of freedom for V(M) boil down to point evaluations at the vertices of M away from Γ (set N(M)). The dual basis of “tent functions” will be denoted by D B (M)={b } .ItsunconditionalL2-stabilityiswellknown:withauniversal V p p∈N(M) constant C >0 we have for all u = α b ∈V(M), α ∈R, h p p p p∈N(M) P (2.9) C−1ku k2 ≤ α2kb k2 ≤Cku k2 . h L2(Ω) p p L2(Ω) h L2(Ω) p∈XN(M) 1The symbol C will stand for generic positive constants throughout this article. Its value may varybetween differentoccurrences. Wewillalwaysspecifyonwhichquantities these constants may depend. 6 R.HiptmairandW.-Y.Zheng For the nodal interpolation operator related to B we write I : dom(I ) ⊂ V h h H1 (Ω)7→V(M).Recallthestandardestimateforlinearinterpolationonconforming ΓD meshes (i.e., no hanging nodes allowed), [19, Thm. 3.2.1], that asserts the existence of C =C(k,ρ )>0 such that M (2.10) hk−2(Id−I )u ≤C|u| ∀u∈H2(Ω)∩H1 (Ω), k ∈{0,1}. h Hk(Ω) H2(Ω) ΓD (cid:13) (cid:13) Obviousl(cid:13)y, gradV(M) ⊂(cid:13)U(M), and immediate from Stokes theorem is the crucial commuting diagram property (2.11) Π ◦grad=grad◦I on dom(I ). h h h This enables us to give an elementary proof of Lemma 2.1. Proof. [of Lemma 2.1] Pick one K ∈ M and, without loss of generality, assume 0∈K. Then define the lifting operator, cf. the “Koszul lifting” [3, Sect. 3.2], (2.12) w7→Lw, Lw(x):= 1w(x)×x, x∈K . 3 Elementary calculations reveal that for any constant vectorfield w∈(P (K))3 0 (2.13) curlLw=w, (2.14) kLwk ≤h kwk , L2(K) K L2(K) (2.15) Lw∈U(K). The continuity (2.14) permits us to extend L to (L2(K))3. Given Ψ ∈ (H1(K))3 with curlΨ ≡ const3, by (2.15) we know LcurlΨ ∈ (P (K))3. Thus, an inverse inequality leads to 1 (2.14) (2.16) |LcurlΨ| ≤Ch−1kLcurlΨk ≤ CkcurlΨk , H1(K) K L2(K) L2(K) with C =C(ρ )>0. Next, (2.13) implies K (2.17) curl(Ψ−LcurlΨ)=0 ⇒ ∃p∈H1(K): Ψ−LcurlΨ=gradp. From (2.16) we conclude that p ∈ H2(K) and |p| ≤ C|Ψ| . Moreover, H2(K) H1(K) thanks to the commuting diagram property we have (2.18) Ψ−Π Ψ=LcurlΨ−Π LcurlΨ+grad(p−I p), h h h =0by(2.15) which means, by the standa|rd estimate{z(2.10) for li}near interpolation on K, kΨ−Π Ψk =|p−I p| ≤Ch |p| ≤Ch |Ψ| . h L2(K) h H1(K) K H2(K) K H1(K) Summation over all elements finishes the proof. As theoretical tools we need “higher order” counterparts of the above finite element spaces. We recall the quadratic Lagrangianfinite element space (2.19) V (M):={u ∈H1 (Ω): u ∈P (K)∀K ∈M}, 2 h ΓD h|K 2 and its subspace of quadratic surpluses (2.20) V (M):={u ∈V (M): I u =0}. 2 h 2 h h e LocalMultigridinH(curl) 7 This implies a direct splitting (2.21) V (M)=V(M)⊕V (M), 2 2 which is unconditionally H1-stable: there is a Ce=C(ρM)>0 such that (2.22) C−1|u |2 ≤|(Id−I )u |2 +|I u |2 ≤C|u |2 , h H1(Ω) h h H1(Ω) h h H1(Ω) h H1(Ω) for all u ∈V (M). h 2 Next, we examine the space (V(M))3 of continuous piecewise linear vector fields that vanish on Γ . Standard affine equivalence techniques for edge elements, see [30, D Sect. 3.6], confirm (2.23) ∃C =C(ρ )>0: kΠ Ψ k ≤CkΨ k ∀Ψ ∈(V(M))3 . M h h L2(Ω) h L2(Ω) h Lemma 2.2. For all Ψ ∈(V(M))3 we can find v ∈V (M) such that h h 2 Ψh =ΠhΨh+gradvhe, e and, with C =C(ρ )>0, M e C−1kΨ k2 ≤kΠ Ψ k2 +kgradv k2 ≤CkΨ k2 . h L2(Ω) h h L2(Ω) h L2(Ω) h L2(Ω) e For the proof we rely on a very useful insight, which relieves us from all worries concerning the topology of Ω: Lemma 2.3. If v∈H (curl0,Ω) and Π v=0, then v∈gradH1 (Ω). ΓD h ΓD Proof. Since the mesh covers Ω, the relative homology group H (Ω;Γ ) is gen- 1 D erated by a set of edge paths. By definition (2.3) of the d.o.f. of U(M), the path integrals of v along all these paths vanish. As an irrotational vector field with vanishing circulation along a complete set of Γ -relative fundamental cycles, v must be D a gradient. Proof. [of Lemma 2.2] Given Ψ ∈(V(M))3, we decompose it according to h (2.24) Ψ =Π Ψ +(Id−Π )Ψ . h h h h h =:gradvh e | {z } Note that curl(Id− Π )Ψ is piecewise constant with vanishing flux through all h h triangular faces of M. Then Stokes’ theorem teaches that curl(Id−Π )Ψ =0. h h By the projector property of Π , (Id − Π )Ψ satisfies the assumptions of h h h Lemma 2.3. Taking into account that, moreover, the field is piecewise linear, it is clear that (Id−Π )Ψ = gradψ with ψ ∈ V (M). Along an arbitrary edge path γ h h 2 in M we have (Id−Π )Ψ ·d~s=0 so that ψ attains the same value (w.l.o.g.=0) γ h h on all vertices of M. The stability of the splitting is a consequence of (2.23). R By definition, the spaces U(M) and V(M) accommodate the homogeneous boundary conditions on Γ . Later, we will also need finite element spaces oblivi- D ous of boundary conditions, that is, for the case Γ = ∅. These will be tagged by a D barontop,e.g.,U(M),V(M),etc.Thesameconventionwillbeemployedfornotions andoperatorsassociatedwithfiniteelementspaces:iftheyrefertotheparticularcase Γ =∅, they will be endowed with an overbar,e.g. Π , I , B (M), N(M), etc. D h h U 8 R.HiptmairandW.-Y.Zheng 2.2. Meshes with hanging nodes. Now, general tetrahedral meshes with hanging nodes are admitted. We simply retain the definitions (2.8) and (2.19) of the spaces V(M) and V (M) of continuous finite element functions. Degrees of freedom 2 for V(M) are point evaluations at active vertices of M. A vertex is called active, if it is not located in the interior of an edge/face of M or on Γ . A 2D2 illustration is D given in Fig. 2.1. M M M M 0 1 2 3 Fig.2.1.Activevertices(•)of2Dtriangularmesheswithhangingnodes,Ω=]0,1[2,ΓD =∂Ω. In M1,M2,M3 active edges are marked withgreen arrows. The values of a finite element function at the remaining (“slave”) vertices are determined by recursive affine interpolation. A dual nodal basis B (M) and corre- V sponding interpolation operator I can be defined as above. h In principle, the definition (2.1) of the edge element space could be retained on non-conforming meshes, as well. Yet, for this choice an edge interpolation operator Π that satisfies the commuting diagram property (2.11) is not available. Thus, we h construct basis functions directly and rely on the notion of active edges, see Fig. 2.1. Definition 2.4. An edge of M is active, if it is an edge of some K ∈ M, not contained in Γ , and connects two vertices that are either active or located on Γ . D D We keep the symbol E(M) to designate the set of active edges of M. To each E ∈E(M)weassociateabasisfunctionb ,which,locallyonthetetrahedraofM,is E apolynomialoftheform(2.2).Inordertofixthisbasisfunctioncompletely,itsuffices to speficify its path integrals (2.3) along all edges of M. In the spirit of duality, we demand 1 , if F =E , (2.25) b ·d~s= E ZF (0 , if F ∈E(M)\{E}. Forthe non-active(“slave”)edgesofMthepathintegralsofb (subsequentlycalled E “weights”) are chosen to fit (2.11), keeping in mind that B (M) := {b } , U E E∈E(M) and that the d.o.f. and Π are still defined according to (2.3) and (2.7), respectively. h Ultimately, we set U(M):=Span{B (M)}. U Let us explain the policy for setting the weights in the case of the subdivided tetrahedronofFig.2.2with hangingnodes atthe midpoints ofedges,whichwillturn outto be the only relevantsituation,cf. Sect.5.3.Weights haveto be assignedto the “small edges” of the refined tetrahedron, some of which will be active, and some of which will have “slave” status, see the caption of Fig. 2.2. 2Foreaseofvisualization,wewilloftenelucidategeometricconceptsintwo-dimensionalsettings. Theirunderlyingideasarethesamein2Dand3D. LocalMultigridinH(curl) 9 We write the direction vectors of slave edges as linear combinations of active edges, for instance, q −p = 1(p −p ), 1 3 2 4 3 q −q = 1(p +p )− 1(p +p )= 1(p −p ), 1 2 2 4 3 2 2 3 2 4 2 p −q =p − 1(p +p )=p −p + 1(p −p ), 5 4 5 2 1 3 5 1 2 1 3 q −q = 1(p +p )− 1(p +p )= 1(p −p )+ 1(p −p ). 4 3 2 1 3 2 4 2 2 1 2 2 3 4 In a sence, we express slave edges as “linear combinations” of active edges. In a different context, this policy is explained in more detail in [26]. p 4 E q 1 p 5 p 3 q 3 q 4 q 2 p 1 p 6 p 2 Fig. 2.2. Subdivided tetrahedron, active vertices (•) p ,...,p , slave vertices (◦) q ,...,q , 1 6 1 4 active edges [p ,p ], [p ,p ], [p ,p ], [p ,p ], [p ,p ], [p ,p ], [p ,p ], [p ,p ], [p ,p ], slave 1 5 1 6 4 5 2 6 2 3 2 4 3 4 5 6 1 3 edges[p ,q ],[q ,p ],[p ,q ],[q ,p ],[p ,q ],[q ,p ],[p ,q ],[q ,q ],[p ,q ],[q ,q ],[q ,q ], 1 4 4 3 2 2 2 3 3 1 1 4 6 4 2 4 6 2 1 2 2 3 [q ,q ], [q ,p ], [q ,p ], [q ,p ] 1 3 1 5 3 5 4 5 Thecoefficientsinthecombinationstellustheweights.Forexample,fortheactive edge E =[p ,p ] in Fig. 2.2 they are givenin Table 2.1. Using these weights and the 3 4 formula(2.4),b canbe assembledonthe tetrahedronbyimposing(seeTable2.1for E notations) w for any contributing slave edge S , b ·d~s= S , S ∈{“small edges”}. E ZS (0 for all other (slave) edges. Firstly, the procedure for the selection of weight guarantees that gradV(M) ⊂ U(M). For illustration, we single out the gradient w of the nodal basis func- h tion belonging to vertex p in Fig. 2.2. Its path integral equals 1 along the (ori- 5 ented) edges [p ,p ],[p ,p ],[p ,p ],[q ,p ],[q ,p ],[q ,p ], and vanishes on all 1 5 3 5 6 5 4 5 3 5 1 5 10 R.HiptmairandW.-Y.Zheng Slave edge S [q ,p ] [p ,q ] [q ,q ] [q ,p ] [q ,q ] 1 4 3 1 2 3 4 5 3 4 weight w 1 1 1 1 −1 S 2 2 2 2 2 Table2.1 Weights for slave edges in Fig. 2.2 relative to active edge E =[p ,p ]. Only slave edges with 3 4 non-zero weights are listed. other edges. Hence we expect (2.26) w =b +b +b . h [p ,p ] [p ,p ] [p ,p ] 1 5 3 5 6 5 This can be verified through showing equality of path integrals along slave edges. We take a close look at the slave edge [q ,p ]. By construction the basis functions 4 5 belonging to active edges satisfy b ·d~s=1, b ·d~s=0, b ·d~s=0, [p ,p ] [p ,p ] [p ,p ] 1 5 5 4 3 4 Z Z Z [q ,p ] [q ,p ] [q ,p ] 4 5 4 5 4 5 b ·d~s=−1 , b ·d~s=0 b ·d~s=0, [p1,p3] 2 [p1,p6] [p2,p6] Z Z Z [q ,p ] [q ,p ] [q ,p ] 4 5 4 5 4 5 b ·d~s=0. [p ,p ] 2 3 Z [q ,p ] 4 5 Then, evidently, 1= w ·d~s h Z [q ,p ] 4 5 = b ·d~s+ b ·d~s+ b ·d~s=1+0+0. [p ,p ] [p ,p ] [p ,p ] 1 5 3 5 6 5 Z Z Z [q ,p ] [q ,p ] [q ,p ] 4 5 4 5 4 5 The same considerations apply to all other slave edges and (2.26) is established. Secondly, the construction ensures the commuting diagram property (2.11): again appealing to Fig. 2.2 we find, for example, gradI u·d~s=I u(q )−I u(q )= h h 4 h 3 Z [q ,q ] 3 4 1(u(p )+u(p ))− 1(u(p )+u(p ))= 1 gradu·d~s+ 1 gradu·d~s= 2 4 2 2 1 3 2 2 Z Z [p ,p ] [p ,p ] 3 4 1 2 1 gradu·d~s+ 1 gradu·d~s+ 1 gradu·d~s. 2 4 4 Z Z Z [p ,p ] [p ,p ] [p ,p ] 3 4 1 6 6 2 In words,combining the path integrals of gradu along active edges with the relative weightsoftheslaveedge[q ,q ]yieldsthesameresultasevaluatingthepathintegral 3 4 of the gradient of the interpolant I u along [q ,q ]. h 3 4 The definitions (2.19) and (2.20) also carry over to meshes with hanging nodes. This remains true for the splitting asserted in Lemma 2.2. However, though the al- gebraic relationships like (2.11) remainvalid, the estimates and normequivalences of