Discrete Compactness for p-Version of Tetrahedral Edge Elements
aa r X i v : . [ m a t h . NA ] J a n DISCRETE COMPACTNESS FOR P -VERSION OF TETRAHEDRALEDGE ELEMENTS R. HIPTMAIR ∗ Report 2008-31, Seminar for Applied Mathematics, ETH Zurich
Abstract.
We consider the first family of H ( curl , Ω)-conforming Ned´el´ec finite elements ontetrahedral meshes. Spectral approximation ( p -version) is achieved by keeping the mesh fixed andraising the polynomial degree p uniformly in all mesh cells. We prove that the associated subspaces ofdiscretely weakly divergence free piecewise polynomial vector fields enjoy a long conjectured discretecompactness property as p → ∞ . This permits us to conclude asymptotic spectral correctness ofspectral Galerkin finite element approximations of Maxwell eigenvalue problems. Key words.
Edge elements, Maxwell eigenvalue problem, discrete compactness, Poincar´e lifting,projection based interpolation
AMS subject classifications.
1. Introduction.
Identifying spectrally correct conforming Galerkin approxi-mations of the Maxwell eigenvalue problem [15]: seek u ∈ H ( curl , Ω) and ω > (cid:0) µ − curl u , curl v (cid:1) L (Ω) = ω ( ǫ u , v ) L (Ω) ∀ v ∈ H ( curl , Ω) , (1.1)( ǫ , µ ∈ ( L ∞ (Ω)) , uniformly positive definite material tensors) has turned out tobe a highly inspiring challenge in numerical analysis. Obviously, eigenfunctions of(1.1) belong to H ( curl , Ω) ∩ H (div ǫ , Ω) and the compact embedding L (Ω) ֒ → H ( curl , Ω) ∩ H (div ǫ , Ω) [30] relates (1.1) to an eigenvalue problem for a com-pact selfadjoint operator. However, asymptotically dense families of finite elementsin H ( curl , Ω) ∩ H (div ǫ , Ω) are not known in general.Let us assume that a merely H ( curl , Ω)-conforming family (cid:0) W p ( M ) (cid:1) p ∈ N of finitedimensional trial and test spaces W p ( M ) ⊂ H ( curl , Ω) for (1.1) is employed forthe Galerkin discretization of (1.1). The corresponding discrete eigenfunctions u p ∈W p ( M ), if they exist, will satisfy u p ∈ X p ( M ) := { w p ∈ W p ( M ) : ( ǫ w p , v p ) L (Ω) = 0 ∀ v p ∈ Ker( curl ) ∩ W p ( M ) } . (1.2)We cannot expect X p ( M ) ⊂ H ( curl , Ω) ∩ H (div ǫ , Ω) and, thus, a standardGalerkin approximation of (1.1) boils down to an outer approximation of the eigen-value problem. Good approximation properties of the finite element space no longerautomatically translate into convergence of eigenvalues and eigenfunctions. As inves-tigated Caorsi, Fernandes and Rafetto in [12], an array of other requirements hasto be met by the finite element spaces, the most prominent of which is the discretecompactness property [3].
Definition 1.1.
The discrete compactness property holds for an asymptoti-cally dense family (cid:0) W p ( M ) (cid:1) p ∈ N of finite dimensional subspaces of H ( curl , Ω) , if any ∗ SAM, ETH Z¨urich, CH-8092 Z¨urich, [email protected] We use the customary notations for Sobolev spaces like H s (Ω), H ( curl , Ω), H (div , Ω), etc.,and write H ( curl , Ω), H ( curl , Ω), etc., for the kernels of differential operators. The reader isreferred to [24, § I.2] and [29, Sect. 2.4] for more information.1
R. Hiptmair bounded sequence in X p ( M ) ⊂ H ( curl , Ω) contains a subsequence that converges in L (Ω) . The same notion applies in the case of homogeneous Dirichlet boundary condi-tions, when (1.1) is considered in H ( curl , Ω). In this case the eigenfunctions willbelong to H ( curl , Ω) ∩ H (div ǫ , Ω) and zero tangential trace on ∂ Ω has to be im-posed on trial and test functions.The discrete compactness property of W p ( M ) is key to establishing spectral cor-rectness and asymptotic optimality of Galerkin approximations of (1.1), see [11,12,15]for details. Small wonder, that substantial effort has been spent on proving this prop-erty for various asymptotically dense families of H ( curl , Ω)/ H ( curl , Ω)-conformingfinite elements. For the h -version of Ned´el´ec’s edge elements Kikuchi [31–33] accom-plished the first proof, which was later generalized in [7,23,36], see [35, Sect. 7.3.2], [29,Sect. 4], and [15] for a survey. Conversely, spectral edge element schemes in 3D havelong defied all attempts to prove their discrete compactness property, though they per-form well for Maxwell eigenvalue problems [15, 17, 39]. Partial success was reportedfor edge elements in 2D: In [9] the analysis of the discrete compactness property fortriangular hp finite elements has been tackled, but the proof of the main result reliedon a conjectured L estimate, which had only been demonstrated numerically. Thefirst fully rigorous analysis of 2D hp edge elements on rectangles was devised in [8].In [29, Remark 15] an interpolation estimate was identified as crucial missing stepin the analysis. Since then, two major advances have paved the way for closing thegap: 1. In [16] M. Costabel and M. McIntosh discovered a construction of H (Ω)-stable vector potentials by means of a smoothed Poincar´e mapping. This willbe reviewed in Sect. 2 of the present paper.2. In the breakthrough paper [21] L. Demkowicz and A. Buffa achieved a com-prehensive analysis of commuting projection based interpolation operators.To maintain the article self-contained, their approach will be explained inSect. 4 and their interpolation error estimates will be presented in Sect. 5.In addition, we exploit the possibility to construct high order versions of Ned´el´ec’sfirst family of edge elements [38] by using Cartan’s Poincar´e map [4,27–29], see Sect. 3for details. Another important tool are stable polynomial preserving extension oper-ators developed, for example, in [1, 5, 22, 37]. In addition, we heavily rely on spectralpolynomial approximation estimates, see [6, 37, 40].Thus, standing on the shoulders of giants and combining all these profound the-ories of numerical analysis, this article manages to give the first proof for the discretecompactness property of the p -version for the first family of Ned´el´ec’s edge elementson tetrahedral meshes of Lipschitz polyhedra Ω, consult Sect. 6 for the proof. Theorem 1.2.
The sequence (cid:0) W p ( M ) (cid:1) p ∈ N of trial spaces generated by the p -version of the first family of Ned´el´ec’s H ( curl , Ω) - or H ( curl , Ω) -conforming finiteelements on a fixed tetrahedral mesh M of a bounded Lipschitz polyhedron Ω ⊂ R satisfies the discrete compactness property. The idea of the proof is to inspect the L (Ω)-orthogonal Helmholtz decomposition[24, § I.3] w p = e w p ⊕ L w p , w p ∈ Ker( curl ) , (1.3)whose so-called solenoidal components e w p belong to H ( curl , Ω) ∩ H (div 0 , Ω). Theabove mentioned compact embedding guarantees the existence of a subsequence of( e w p ) p ∈ N that converges in L (Ω). Hence, it “merely” takes to show k e w p − w p k L (Ω) → iscrete compactness
30 for p → ∞ in order to establish discrete compactness. Clever use of projectionoperators that enjoy a commuting diagram property, converts this task to a uniforminterpolation estimate. The core of this paper is devoted to this seemingly humbleprogram. Remark 1.1.
Generalizations of Thm. 1.2 to other families of tetrahedral edgeelements, and corresponding hp -finite element schemes are straightforward [8]. Forthe sake of readability, these extensions will not be pursued in the present paper.Since the Poncar´e map does not fit a tensor product structure, extending the resultsof this paper to 3D hexahedral edge elements will take some new ideas.
2. Poincar´e lifting.
Let D ⊂ R stand for a bounded domain that is star-shaped with respect to a subdomain B ⊂ D , that is, ∀ a ∈ B, x ∈ D : { t a + (1 − t ) x , < t < } ⊂ D . (2.1)
Definition 2.1.
The
Poincar´e lifting R a : C (Ω) C (Ω) , a ∈ B , is definedas R a ( u )( x ) := Z t u ( x + t ( x − a )) d t × ( x − a ) , x ∈ D , (2.2) where × designates the cross product of two vectors in R . This is a special case of the generalized path integral formula for differentialforms, which is instrumental in proving the exactness of closed forms on star-shapeddomains, the so-called “Poincar´e lemma”, see [13, Sect. 2.13].The linear mapping R a provides a right inverse of the curl -operator on divergence-free vectorfields, see [25, Prop. 2.1] for the simple proof, and [13, Sect. 2.13] for ageneral proof based on differential forms. Lemma 2.2. If div u = 0 , then, for any a ∈ B , curl R a u = u for all u ∈ C (Ω) . Unfortunately, the mapping R a cannot be extended to a continuous mapping L ( D ) H ( D ), cf. [25, Thm. 2.1]. As discovered in the breakthrough paper [16]based on earlier work of Bogovskiˇi [10], it takes a smoothed version to accomplishthis: we introduce the smoothed Poincar´e lifting R ( u ) := Z B Φ( a ) R a ( u ) d a , (2.3)where Φ ∈ C ∞ ( R ) , supp Φ ⊂ B , Z B Φ( a ) d a = 1 . (2.4)The substitution y := a + t ( x − a ) , τ := 11 − t , (2.5)transforms the integral (2.4) into R ( u )( x ) = Z R ∞ Z τ (1 − τ ) u ( y ) × ( x − y )Φ( y + τ ( y − x )) d τ d y = Z R k ( x , y − x ) × u ( y ) d y , (2.6) Bold symbols will generally be used to tag vector valued functions and spaces of such. The dependence of R on Φ is dropped from the notation. R. Hiptmair that is, R is a convolution-type integral operator with kernel k ( x , z ) = Z ∞ τ (1 + τ )Φ( x + τ z ) z d τ = z | z | Z ∞ ζ Φ( x + ζ z | z | ) d ζ + z | z | Z ∞ ζ Φ( x + ζ z | z | ) d ζ . (2.7)The kernel can be bounded by | k ( x , z ) | ≤ K ( x ) | z | − , where K ∈ C ∞ ( R ) dependsonly on Φ and is locally uniformly bounded. As a consequence, (2.6) exists as animproper integral.The intricate but elementary analysis of [16, Sect. 3.3] further shows, that k be-longs to the H¨ormander symbol class S − , ( R ), see [41, Ch. 7]. Invoking the theory ofpseudo-differential operators [41, Prop. 5.5] we obtain the following following conti-nuity result, which is a special case of [16, Cor. 3.4] Theorem 2.3.
The mapping R can be extended to a continuous linear operator L ( D ) H ( D ) , which is still denoted by R . It satisfies curl R u = u ∀ u ∈ H (div 0 , D ) . (2.8)The smoothed Poincar´e lifting shares this continuity property with many othermappings, see [29, Sect. 2.4]. Yet, it enjoys another essential feature, which is imme-diate from its definition (2.2): R maps polynomials of degree p to other polynomialsof degree ≤ p + 1. The next section will highlight the significance of this observation.
3. Tetrahedral edge elements.
In [38] Ned´el´ec introduced a family of H ( curl , Ω)-conforming, that is, tangentially continuous, finite element spaces. Ona tetrahedral triangulation M of Ω, the corresponding finite element spaces of degree p are given by W p ( M ) := { v ∈ H ( curl , Ω) : v | T ∈ W p ( T ) ∀ T ∈ M} , W p ( T ) := { v ∈ C ∞ ( T ) : v ( x ) = p ( x ) + q ( x ) × x , p , q ∈ P p ( R ) , x ∈ T } . We wrote P p ( R ) for the space of 3-variate polynomials of total degree ≤ p , p ∈ N ,and the bold symbol P p ( R ) for vectorfields with three components in P p ( R ). Toemphasize that polynomials on a tetrahedron T are being considered, we may use thenotations P p ( T )/ P p ( T ) instead of P p ( R )/ P ( R ). We also adopt the convention that P p ( R ) = { } , if p <
0. Another relevant polynomial space is P p (div 0 , R ) := { q ∈ P p ( R ) : div q = 0 } . (3.1)Deep insights can be gained by regarding edge elements as discrete 1-forms. This pro-vides a very elegant construction of higher order edge element spaces and immediatelyreveals their relationships with standard Lagrangian finite elements and H (div , Ω)-conforming face elements (see below). In particular, the Poincar´e lifting becomes apowerful tool for building discrete differential forms of high polynomial degree. This isexplored in [27, 28], [29, Sect. 3.4], and [4, Sect. 1.4] in arbitrary dimension, using thecalculus of differential forms. In this article we prefer to stick to the classical calculusof vector analysis, because we are only concerned with 3D. We hope, that, thus, thepresentation will be more accessible to an audience of numerical analysts. Yet, thedifferential forms background has inspired our notations: integer superscripts label iscrete compactness W p ( M ) can be readas a space of discrete 1-forms.According to [27, Sect. 3], for any T ∈ M , a ∈ T , we can obtain the local spaceas W p ( T ) = P p ( R ) + R a (cid:0) P p (div 0 , R ) (cid:1) . (3.2)Independence of a is discussed in [27, Sect. 3]. The representation (3.2) can be estab-lished by dimensional arguments: from the formula (2.2) for the Poincar´e lifting weimmediately see that P p ( R )+ R a ( P p ( R )) ⊂ W p ( T ). In addition, from [38, Lemma 4]and [27, Thm. 6, case l = 1, n = 3] we learn that the dimensions of both spaces agreeand are equal to dim W p ( T ) = (1 + p )(3 + p )(4 + p ) . (3.3)As a consequence, the two finite dimensional spaces must agree.For the remainder of this section, which focuses on local spaces, we single out atetrahedron T ∈ M . On T we can introduce a smoothed Poincar´e lifting R T accordingto (2.3) with B = T and a suitable Φ ∈ C ∞ ( T ) complying with (2.4). An immediateconsequence of (3.2) is that R T (cid:0) P p (div 0 , R ) (cid:1) ⊂ W p ( T ) . (3.4)We introduce the notation F m ( T ) for the set of all m -dimensional facets of T , m =0 , , ,
3. Hence, F ( T ) contains the vertices of T , F ( T ) the edges, F ( T ) the faces,and F ( T ) = { T } . Moreover, for some F ∈ F m ( T ), m = 1 , , P p ( F ) denotes thespace of m -variate polynomials of total degree ≤ p in a local coordinate system of thefacet F , and P p ( F ) will designate corresponding tangential polynomial vectorfields.Further, we write W p ( e ) = W p ( T ) · t e , t e the unit tangent vector of e, e ∈ F ( T ) , (3.5) W p ( f ) = W p ( T ) × n f , n f the unit normal vector of f, f ∈ F ( T ) , (3.6)for the tangential traces of local edge element vectorfields onto edges and faces. Simplevector analytic manipulations permit us to deduce from (3.2) that W p ( e ) = P p ( e ) , e ∈ F ( T ) , (3.7) W p ( f ) = P p ( f ) + R D a ( P p ( f )) , a ∈ f , f ∈ F ( T ) , (3.8)where the projection R D a of the Poincar´e lifting in the plane reads R D a ( u )( x ) := Z tu ( a + t ( x − a )]( x − a ) d t , a ∈ R . (3.9)It satisfies div Γ R D a ( u ) = u for all u ∈ C ∞ ( R ). We point out that, along with (3.2),the formulas (3.7) and (3.8) are special versions of the general representation formulafor discrete 1-forms, see [27, Formula (16)]. Special facet tangential trace spaces willalso be needed: ◦ W p ( e ) := { u ∈ W p ( e ) : Z e u d l = 0 } , e ∈ F ( T ) , (3.10) ◦ W p ( f ) := { u ∈ W p ( f ) : u · n e,f ≡ ∀ e ∈ F ( T ) , e ⊂ ∂f } , f ∈ F ( T ) , (3.11) ◦ W p ( T ) := { u ∈ W p ( T ) : u × n f ≡ ∀ f ∈ F ( T ) } . (3.12) R. Hiptmair
Here n f represents an exterior face unit normal of T , n e,f the in plane normal of aface w.r.t. an edge e ⊂ ∂f .According to [38, Sect. 1.2] and [27, Sect. 4], the local degrees of freedom for W p ( T ) are given by the first p − M , the first p − M and the first p tangential moments along the edges of T , see (3.14) for concrete formulas. Then theset dof p ( T ) can be partitioned asdof p ( T ) = [ e ∈F ( T ) ldf p ( e ) ∪ [ f ∈F ( T ) ldf p ( f ) ∪ ldf p ( T ) , (3.13)where the functionals in ldf p ( e ), ldf p ( f ), and ldf p ( T ) are supported on an edge, face,and T , respectively, and read κ ∈ ldf p ( e ) ⇒ κ ( u ) = R e p ξ · t e d l for e ∈ F ( T ) , suitable p ∈ P p ( e ) ,κ ∈ ldf p ( f ) ⇒ κ ( u ) = R f p · ( ξ × n ) d S for f ∈ F ( T ) , suitable p ∈ P p − ( f ) ,κ ∈ ldf p ( T ) ⇒ κ ( u ) := R T p · ξ d x for suitable p ∈ P p − ( T ) . (3.14)These functionals are unisolvent on W p ( T ) and locally fix the tangential trace of u ∈ W p ( T ). There is a splitting of W p ( T ) dual to (3.13): Defining Y p ( F ) := { v ∈ W ( T ) : κ ( v ) = 0 ∀ κ ∈ dof p ( T ) \ ldf p ( F ) } (3.15)for F ∈ F m ( T ), m = 1 , ,
3, we find the direct sum decomposition W p ( T ) = X m =1 X F ∈F m ( T ) Y p ( F ) . (3.16)In addition, note that the tangential trace of u ∈ X p ( F ) vanishes on all facets = F ,whose dimension is smaller or equal the dimension of F . By the unisolvence of dof p ( T ),there are bijective linear extension operators E e,p : W p ( e )
7→ Y p ( e ) , e ∈ F ( T ) , (3.17) E f,p : ◦ W p ( f )
7→ Y p ( f ) , f ∈ F ( T ) . (3.18)The curl connects the edge element spaces W p ( M ) and the so-called face elementspaces of discrete 2-forms [38, Sect. 1.3] W p ( M ) := { v ∈ H (div , Ω) : v | T ∈ W p ( T ) ∀ T ∈ M} , W p ( T ) := { v ∈ C ∞ ( T ) : v ( x ) = p ( x ) + q ( x ) x , p ∈ P p ( T ) , q ∈ P p ( T ) } . An alternative representation of the local face element space is [27, Formula (16) for l = 2, n = 3] W p ( T ) = P p ( T ) + D a ( P p ( T )) , (3.19)where the appropriate version of the Poincar´e lifting reads( D a u )( x ) := Z t u ( a + t ( x − a ))( x − a ) d t , a ∈ T . (3.20) iscrete compactness D a u = u , see [25, Prop. 1.2].The normal trace space of W p ( T ) onto a face is W p ( f ) := W p ( T ) · n f = P p ( f ) , f ∈ F ( T ) , (3.21)and as relevant space “with zero trace” we are going to need ◦ W p ( f ) := { u ∈ W p ( f ) : Z f u d S = 0 } , f ∈ F ( T ) , (3.22) ◦ W p ( T ) := { u ∈ W p ( T ) : u · n ∂T = 0 } . (3.23)The connection between the local spaces W p ( T ), W p ( T ) and full polynomialspaces is established through a local discrete DeRham exact sequence: To elucidatethe relationship between differential operators and various traces onto faces and edges,we also include those in the statement of the following theorem. There n f stands foran exterior face unit normal of T , n e,f for the in plane normal of a face w.r.t. an edge e ⊂ ∂f , and ddl is the differentiation w.r.t. arclength on an edge. Theorem 3.1.
For f ∈ F ( T ) , e ∈ F ( T ) , e ⊂ ∂f , all the sequences in const Id −−−−→ P p +1 ( T ) grad −−−−→ W p ( T ) curl −−−−→ W p ( T ) div −−−−→ P p ( T ) Id −−−−→ { } . | f y . × n f | f y y . · n f | f const Id −−−−→ P p +1 ( f ) curl Γ −−−−→ W p ( f ) div Γ −−−−→ P p ( f ) Id −−−−→ { } . | e y . · n e,f | e y const Id −−−−→ P p +1 ( e ) ddl −−−−→ P p ( e ) Id −−−−→ { } are exact and the diagram commutes.Proof . The assertion about the top exact sequence is an immediate consequenceof representations (3.2) and (3.19) and the relationships curl R a ( u ) = u ∀ u ∈ P p (div 0 , T ) , div D a ( u ) = u ∀ u ∈ P p ( T ) . For further discussions and the proof of the other exact sequence properties see [27,Sect. 5 for n = 3].
4. Projection based interpolation.
The degrees of freedom introduced abovedefine local finite element projectors onto W p ( T ). In conjunction with suitably definedinterpolation operators for degree p Lagrangian finite elements, they possess a verydesirable commuting diagram property [27, Thm. 13], which will be explained below.However, they do not enjoy favorable continuity properties with increasing p . Thus,L. Demkowicz [19–21], taking the cue from the theory of p -version Lagrangian finiteelements, invented an alternative in the form of local projection based interpolation. Again, consider a single tetra-hedron T ∈ M and fix the polynomial degree p ∈ N . Following the developmentsof [29, Sect. 3.5], projection based interpolation requires building blocks in the form R. Hiptmair of local orthogonal projections P l ∗ and liftings L l ∗ . Some operators will depend on aregularity parameter 0 < ǫ < , which is considered fixed below and will be specifiedin Sect. 5. To begin with, we define for every e ∈ F ( T ) P e,p : H − ǫ ( e ) ddl ◦ P p +1 ( e ) = ◦ W p ( e ) (4.1)as the H − ǫ ( e )-orthogonal projection. Here, ◦ P p ( F ) denotes the space of degree p polynomials on a facet F that vanish on ∂F .Similarly, for every face f ∈ F ( T ) introduce P f,p : H − + ǫ ( f ) curl Γ ◦ P p +1 ( f ) = { v ∈ ◦ W p ( f ) : div Γ v = 0 } , (4.2) P f,p : H − + ǫ ( f ) div Γ ◦ W p ( f ) = ◦ W p ( f ) , (4.3)as the corresponding H − + ǫ ( f )-orthogonal projections. Eventually, let P T,p : L ( T ) grad ◦ P p +1 ( T ) = { v ∈ ◦ W p ( T ) : curl v = 0 } , (4.4) P T,p : L ( T ) curl ◦ W p ( T ) = { v ∈ ◦ W p ( T ) : div v = 0 } , (4.5) P T,p : L ( T ) div ◦ W p ( T ) = { v ∈ P p ( T ) : Z T v ( x ) d x = 0 } , (4.6)stand for the respective L ( T )-orthogonal projections.The lifting operators L e,p : ◦ W p ( e ) ◦ P p +1 ( e ) , e ∈ F ( T ) , (4.7) L f,p : { v ∈ ◦ W p ( f ) : div Γ v = 0 } 7→ ◦ P p +1 ( f ) , f ∈ F ( T ) , (4.8) L T,p : { v ∈ ◦ W p ( T ) : curl v = 0 } 7→ ◦ P p +1 ( T ) , (4.9)are uniquely defined by requiring ddl L e,p u = u ∀ u ∈ ◦ W p ( e ) , (4.10) curl Γ L f,p u = u ∀ u ∈ { ◦ W p ( f ) : div Γ v = 0 } , (4.11) grad L T,p u = u ∀ u ∈ { v ∈ ◦ W p ( T ) : curl v = 0 } . (4.12)Another class of liftings provides right inverses for curl and div Γ : Pick a face f ∈F ( T ), and, without loss of generality, assume the vertex opposite to the edge e e tocoincide with 0. Then define L f,p : (cid:26) div Γ ◦ W p ( f ) ◦ W p ( f ) u R D u − curl Γ E e e,p L e e,p ( R D u · n e e,f ) . (4.13)This is a valid definition, since, by virtue of definition (3.9), the normal componentsof R D u will vanish on ∂f \ e e . Moreover, div Γ R D u = u ensures that the normalcomponent of R D u has zero average on e e . We infer (cid:16) curl Γ E e e,p L e e,p (cid:0) ( R D u · n e e,f ) | e e (cid:1) · n e e,f (cid:17) | e e = ddl L e e,p (cid:0) ( R D u ) · n e e,f (cid:17) | e e = R D u · n e e,f on e e , The parameter l in the notations for the extension operators E l ∗ , the projections P l ∗ , and theliftings L l ∗ refers to the degree of the discrete differential form they operate on. This is explainedmore clearly in [29, Sect. 3.5].iscrete compactness ∂f is satisfied. The same idea underlies thedefinition of L T,p : ( curl ◦ W p ( T ) ◦ W p ( T ) u R u − grad E e f,p L e f,p (cid:0) (( R u ) × n e f ) | e f (cid:1) , (4.14)where e f is the face opposite to vertex 0, and the definition of L T,p : ( div ◦ W p ( T ) ◦ W p ( T ) u D u − curl E e f,p L e f,p (( D u · n e f ) | e f ) . (4.15)The relationships between the various facet function spaces with vanishing traces canbe summarized in the following exact sequences: { } Id −−−−→ ◦ P p +1 ( T ) grad −−−−→ L T,p ◦ W p ( T ) curl −−−−→ L T,p ◦ W p ( T ) div −−−−→ L T,p P p ( T ) Id −−−−→ { } , { } Id −−−−→ ◦ P p +1 ( f ) curl Γ −−−−→ L f,p ◦ W p ( f ) div Γ −−−−→ L f,p P p ( f ) Id −−−−→ { } , { } Id −−−−→ ◦ P p +1 ( e ) ddl −−−−→ L e,p P p ( e ) Id −−−−→ { } , (4.16)where P p ( F ) designates degree p polynomial spaces on F with vanishing mean. Theserelationships and the lifting mappings are studied in [29, Sect. 3.4].Finally we need polynomial extension operators E e,p : ◦ P p +1 ( e )
7→ P p +1 ( T ) , (4.17) E f,p : ◦ P p +1 ( f )
7→ P p +1 ( T ) (4.18)that satisfy E e,p u | e ′ = 0 ∀ e ′ ∈ F ( T ) \ { e } , (4.19) E f,p u | f ′ = 0 ∀ f ′ ∈ F ( T ) \ { f } . (4.20)Such extension operators can be constructed relying on a representation of a polyno-mial on F , F ∈ F m ( T ), m = 1 ,
2, as a homogeneous polynomial in the barycentriccoordinates of F , see [29, Lemma 3.4]. As an alternative, one may use the polynomialpreserving extension operators proposed in [22, 37] and [1]. We stress that continuityproperties of the extensions E lF , l = 0 , F ∈ F m ( T ), are immaterial. Now, we are in a position to define the projec-tion based interpolation operators locally on a generic tetrahedron T with vertices a i , i = 1 , , , < ǫ < , which is usually suppressed to keep notations manageable)Π T,p (= Π T,p ( ǫ )) : C ∞ ( T )
7→ P p +1 ( T ) (4.21)0 R. Hiptmair for degree p Lagrangian H (Ω)-conforming finite elements. For u ∈ C ( T ) define ( λ i is the barycentric coordinate function belonging to vertex a i of T ) u (0) := u − X i =1 u ( a i ) λ i | {z } := w (0) , (4.22) u (1) := u (0) − X e ∈F ( T ) E e,p L e,p P e,p dds u (0) | e | {z } := w (1) , (4.23) u (2) := u (1) − X f ∈F ( T ) E f,p L f,p P f,p curl Γ ( u (1) | f ) | {z } := w (2) , (4.24)Π T,p u := L T,p P T,p grad u (2) + w (2) + w (1) + w (0) . (4.25)Observe that w ( i ) | F = 0 for all F ∈ F m ( T ), 0 ≤ m < i ≤
3. We point out that w (0) isthe standard linear interpolant of u . Lemma 4.1.
The linear mapping Π T,p , p ∈ N , is a projection onto Cp p +1 ( T ) Proof . Assume u ∈ P p +1 ( T ), which will carry over to all intermediate functions.Since u (0) ( z i ) = 0, i = 1 , . . . ,
4, we conclude from the projection property of P e,p that L e P e dds u (0) | e = u (0) | e for any edge e ∈ F ( T ). As a consequence u (1) = u (0) − X e ∈F ( T ) E e,p u (0) | e ⇒ u (1) | e = 0 ∀ e ∈ F ( T ) . (4.26)We infer L f,p P f curl Γ ( u (1) | f ) = u (1) | f on each face f ∈ F ( T ), which implies u (2) = u (1) − X f ∈F ( T ) E f,p ( u (1) | f ) ⇒ u (2) | f = 0 ∀ f ∈ F ( T ) . (4.27)This means that L T,p P T,p grad u (2) = u (2) and finishes the proof.A similar stage by stage construction applies to edge elements and gives a pro-jection Π T,p (= Π T,p ( ǫ )) : C ∞ ( T )
7→ W ( T ) : (4.28)for a directed edge e := [ a i , a j ] we introduce the Whitney-1-form basis function b e = λ i grad λ j − λ j grad λ i . (4.29) iscrete compactness W ( T ). Next, for u ∈ C ( T ) define u (0) := u − (cid:16) X e ∈F ( T ) Z e u · d ~s (cid:17) b e | {z } := w (0) , (4.30) u (1) := u (0) − X e ∈F ( T ) grad E e,p L e,p P e,p (( u (0) · t e ) | e ) | {z } := w (1) , (4.31) u (2) := u (1) − X f ∈F ( T ) E f,p L f,p P f,p div Γ (( u (1) × n f ) | f ) | {z } := w (2) , (4.32) u (3) := u (2) − X f ∈F ( T ) grad E f,p L f,p P f,p (( u (2) × n f ) | f ) | {z } := w (3) , (4.33) u (4) := u (3) − L T,p P T,p curl u (3) | {z } := w (4) , (4.34)Π T,p u := grad L T,p P T,p u (4) + w (4) + w (3) + w (2) + w (1) + w (0) . (4.35)The contribution w (0) is the standard interpolant Π T, of u onto the local space ofWhitney-1-forms (lowest order edge elements, see [35, Sect. 5.5.1]). The extensionoperators were chosen in a way that guarantees that w (2) · t e = 0 and w (3) · t e = 0for all e ∈ F ( T ). Lemma 4.2.
The linear mapping Π T,p , p ∈ N , is a projection onto W p ( T ) andsatisfies the commuting diagram propertyΠ T,p ◦ grad = grad ◦ Π T,p on C ∞ ( T ) . (4.36) Proof . The proof of the projection property runs parallel to that of Lemma 4.1.Assuming u ∈ W p ( T ) it is obvious that the same will hold for all u ( i ) and w ( i ) from(4.30)-(4.35). In order to confirm that all projections can be discarded, we have tocheck that their arguments satisfy conditions of zero trace on the facet boundariesand, in some cases, belong to the kernel of differential operators.First, recalling the properties of the interpolation operator Π for Whitney-1-forms, we find ( u (0) · t e ) | e ∈ ◦ W p ( e ). This implies grad E e,p L e,p P e,p (( u (0) · t e ) | e ) = ( u (0) · t e ) | e ∀ e ∈ F ( T ) , (4.37)and ( u (1) · t e ) | e ≡ ∀ e ∈ F ( T ) . (4.38)2 R. Hiptmair
We see that ( u (1) × n f ) | f ∈ ◦ W p ( f ) for any f ∈ F ( T ), so that P f,p div Γ (( u (1) × n f ) | f ) = div Γ (( u (1) × n f ) | f ) (4.39) ⇒ div Γ L f,p P f,p div Γ (( u (1) × n f ) | f ) = div Γ (( u (1) × n f ) | f ) (4.40) ⇒ div Γ (( u (2) × n f ) | f ) = 0 ∀ f ∈ F ( T ) , ( u (2) · t e ) | e ≡ ∀ e ∈ F ( T ) (4.41) ⇒ P f,p (( u (2) × n f ) | f ) = ( u (2) × n f ) | f ∀ f ∈ F ( T ) (4.42) ⇒ grad E f,p L f,p P f,p (( u (2) × n f ) | f ) × n f = ( u (2) × n f ) | f ∀ f ∈ F ( T ) (4.43) ⇒ ( u (3) × n f ) | f = 0 ∀ f ∈ F ( T ) (4.44) ⇒ P T,p curl u (3) = curl u (3) (4.45) ⇒ curl L T,p P T,p curl u (3) = curl u (3) (4.46) ⇒ curl u (4) = 0 ⇒ P T u (4) = u (4) (4.47) ⇒ grad L T P T u (4) = u (4) , (4.48)which confirms the projector property.Now assume u = grad u for some u ∈ C ∞ ( T ). The commuting diagram propertywill follow, if we manage to show grad u (0) = u (0) , grad u (1) = u (1) , grad u (2) = u (3) ,etc., for the intermediate functions in (4.22)-(4.25) and (4.30)-(4.35), respectively.By the commuting diagram property for the standard local interpolation operatorsonto the spaces of Whitney-0-forms (linear polynomials) and Whitney-1-forms, weconclude grad u (0) = u (0) ⇒ dds u (0) | e = ( u (0) · t e ) | e ∀ e ∈ F ( T ) (4.49) ⇒ u (1) = grad u (1) ⇒ div Γ (( u (1) × n f ) | f ) = 0 ∀ f ∈ F ( T ) (4.50) ⇒ u (2) = u (1) (4.51) ⇒ ( u (2) × n f ) | f = curl Γ u (1) f ∀ f ∈ F ( T ) ⇒ u (3) = grad u (2) (4.52) ⇒ u (4) = u (3) . (4.53)Of course, analogous relationships for the functions w ( i ) and w ( i ) hold, which yieldsΠ T,p u = grad Π T,p u .Following [29, Sect. 3.5], a projection based interpolation onto W p ( T ), the oper-ator Π T,p (= Π T,p ( ǫ )) : C ∞ ( T )
7→ W p ( T ), involves the stages u (0) := u − (cid:16) X f ∈F ( T ) Z f u · n f d S (cid:17) b f | {z } := w (0) , (4.54) u (1) := u (0) − X f ∈F ( T ) curl E f,p L f,p P f,p (cid:0) ( u (0) · n f ) | f (cid:1)| {z } := w (1) (4.55) u (2) := u (1) − L T,p P T,p div u (1) | {z } := w (2) (4.56)Π T,p u := curl L T,p P T,p u (2) + w (0) + w (1) + w (2) . (4.57) iscrete compactness b f refers to the local basis functions for Whitney-2-forms [29, Sect. 3.2]: b f = λ i grad λ j × grad λ k + λ j grad λ k × λ i + λ k grad λ i × λ j . (4.58)Analogous to Lemma 4.2 one proves the following result. Lemma 4.3.
The linear operator Π T,p , p ∈ N , is a projection onto W p ( T ) andsatisfies the commuting diagram propertyΠ T,p ◦ curl = curl ◦ Π T,p on C ∞ ( T ) . (4.59)The next lemma makes it possible to patch together the local projection basedinterpolation operator to obtain global interpolation operatorsΠ lp : C ∞ (Ω)
7→ W lp ( M ) , l = 1 , . (4.60) Lemma 4.4.
For any F ∈ F m ( T ) , m = 0 , , , and u ∈ C ∞ ( T ) the restriction Π T,p u | F depends only on u | F .For any F ∈ F m ( T ) , m = 1 , , and u ∈ C ∞ ( T ) the tangential trace of Π T,p u onto F depends only on the tangential trace of u on F .For any face f ∈ F ( T ) and u ∈ C ∞ ( T ) the normal trace of Π T,p u onto f dependsonly on the normal component of u on f .Proof . The assertion is immediate from the construction, in particular, the prop-erties of the extension operators used therein.It goes without saying that density arguments permit us to extend Π lp , l = 0 , , p : H s (Ω)
7→ { v ∈ H (Ω) : v | T ∈ P p +1 ( T ) ∀ T ∈ M} , (4.61)Π p : H + s (Ω)
7→ W p ( M ) , (4.62)Π p : H s (Ω)
7→ W p ( M ) , (4.63)for any s > . In addition, by virtue of Lemma 4.4, zero pointwise/tangential/normaltrace on ∂ Ω of the argument function will be preserved by Π lp , l = 0 , ,
2, for instance,Π p ( H + s (Ω) ∩ H ( curl , Ω)) = W p ( M ) ∩ H ( curl , Ω) . (4.64)
5. Interpolation error estimates.
Closely following the ingenious approachin [21, Section 6] we first examine the interpolation error for Π T,p . Please notice thatΠ T,p still depends on the fixed regularity parameter 0 < ǫ < . The argument functionof Π T,p is assumed to lie in H s ( T ) for some s > , cf. (4.61). The continuousembedding H s ( T ) ֒ → C ( T ) plus trace theorems for Sobolev spaces render alloperators well defined in this case.We start with an observation related to the local best approximation propertiesof the projection based interpolant. Lemma 5.1.
For any u ∈ H s ( T ) holds (cid:0) grad ( u − Π T,p u ) , grad v (cid:1) L ( T ) = 0 ∀ v ∈ ◦ P p +1 ( T ) , (5.1) (cid:16) curl Γ ( u − Π T,p u ) | f , curl Γ v (cid:17) H −
12 + ǫ ( f ) = 0 ∀ v ∈ ◦ P p +1 ( f ) , f ∈ F ( T ) , (5.2) (cid:18) ddl ( u − Π T,p u ) | e , ddl v (cid:19) H − ǫ ( e ) = 0 ∀ v ∈ ◦ P p +1 ( e ) , e ∈ F ( T ) . (5.3)4 R. Hiptmair
Proof . We use the notations of (4.22)-(4.25). Setting w := w (0) + w (1) + w (2) , wefind Π T,p u = L T,p P T,p grad ( u − w ) + w , (5.4)which implies, because L T,p is a right inverse of ddl , grad Π T,p u = P T,p grad u + ( Id − P T,p ) grad w . (5.5)This means that u − grad Π T,p u belongs to the range of Id − P T,p and (5.1) follows from(4.4) and the properties of orthogonal projections. Similar manipulations establish(5.2): curl Γ Π T,p u | f = curl Γ w | f (5.6)= curl Γ L f,p | {z } = Id P f,p curl Γ u (1) + curl Γ ( w (0) + w (1) ) | f (5.7)= P f,p curl Γ u | f + ( Id − P f,p ) curl Γ ( w (0) + w (1) ) ∀ f ∈ F ( T ) . (5.8)The same arguments as above verify (5.3).From this we can conclude the result of [21, Section 6, Corollary 1]. To state itwe now assume a dependence0 < ǫ = ǫ ( p ) := 110 log( p + 1) < , p ∈ N , (5.9)of the parameter ǫ in the definition of the local projection based interpolation opera-tors. Below, all parameters ǫ are linked to p via (5.9). Please note that we retain thenotation (cid:0) Π lT,p (cid:1) p ∈ N , l = 0 , ,
2, for these new families of operators.
Theorem 5.2 (
Spectral interpolation error estimate for Π T,p ). There is a con-stant C T > depending only on T and < s ≤ , and, in particular, independent of p , such that (cid:12)(cid:12) ( Id − Π T,p ) v (cid:12)(cid:12) H ( T ) ≤ C T log / pp s | v | H s ( T ) ∀ v ∈ H s ( T ) , p ≥ . (5.10)Stable polynomial extensions are instrumental for the proof, which will be post-poned until Page 16. First, we recall the results of [37, Thm. 1] and [1, Thm. 1]: Theorem 5.3 (
Stable polynomial extension for tetrahedra ). For a tetrahedron T there is linear operator S T : H ( ∂T ) H ( T ) such that S T u | ∂T = u ∀ u ∈ H ( ∂T ) , (5.11) | S T u | H ( T ) ≤ C | u | H ( ∂T ) ∀ u ∈ H ( ∂T ) , (5.12) S T w ∈ P p +1 ( T ) ∀ w ∈ P p +1 ( T ) | ∂T , (5.13) where C > only depends on the shape regularity measure of T . Theorem 5.4 (
Stable polynomial extension for triangles ). Given a triangle F , The shape regularity measure of a tetrahedron is the ratio of the radii of its circumscribed sphereand the largest inscribed sphere.iscrete compactness there is a continuous linear mapping S F : L ( ∂F ) H ( T ) such that | S F u | H ( F ) ≤ C | u | H ( ∂F ) ∀ u ∈ H ( ∂F ) , (5.14) S F w ∈ P p +1 ( F ) ∀ w ∈ P p +1 ( F ) | ∂F , (5.15) where C > depends only on the shape regularity measure of T . By interpolation in Sobolev scale from the last theorem we can conclude ∃ C > | S F u | H s ( F ) ≤ C | u | H s − ( ∂F ) ∀ u ∈ H s − ( ∂F ) , ≤ s ≤ . (5.16)We also need to deal with the awkward property of the H ( ∂T )-norm that itcannot be localized to faces. To that end we resort to a result from [34, Proof ofLemma 3.31], see also [21, Lemma 13]. Lemma 5.5 (
Splitting of H ( ∂T )-norm ). The exists
C > depending only onthe shape regularity of the tetrahedron T such that | u | H
12 + ǫ ( ∂T ) ≤ Cǫ X f ∈F ( T ) | u | H
12 + s ( f ) ∀ u ∈ H + ǫ ( ∂T ) , < ǫ ≤ . (5.17)Another natural ingredient for the proof are polynomial best approximation esti-mates, see [40] or [37, Sect. 3]. Lemma 5.6.
Let F be either a tetrahedron or a triangle. Then, there is a constant C > depending only on F such that for all p ≥ v p ∈P p +1 ( F ) | u − v p | H r ( F ) ≤ Cp r − − s | u | H s ( F ) ∀ u ∈ H s ( F ) , ≤ r ≤ . (5.18)Define a semi-norm projection Q T,p : H ( T )
7→ P p +1 ( T ) on the tetrahedron T by Z T grad ( u − Q T,p u ) · grad v p d x = 0 ∀ v p ∈ P p +1 ( T ) , Z T u − Q T,p u d x = 0 , (5.19)and semi-norm projections Q f,p : H s + ( f )
7→ P p +1 ( f ), f ∈ F ( T ), by( curl Γ ( u − Q f,p u ) , curl Γ v p ) H ǫ − ( f ) = 0 ∀ v p ∈ P p +1 ( T ) , Z f u − Q f,p u d x = 0 . (5.20)These definitions involve best approximation properties of Q T,p u and Q f,p u . Thus, welearn from Lemma 5.6 that with constants independent of 0 < ǫ < < s ≤ | u − Q T,p u | H ( T ) ≤ C ( p + 1) − s | u | H s ( T ) ∀ u ∈ H s ( T ) , (5.21) | u − Q f,p u | H
12 + ǫ ( f ) ≤ C ( p + 1) ǫ − s | u | H
12 + s ( T ) ∀ u ∈ H + s ( f ) . (5.22)The latter estimate follows from the fact that |·| H
12 + ǫ ( f ) and k curl Γ ·k H −
12 + ǫ ( f ) areequivalent semi-norms, uniformly in ǫ .6 R. Hiptmair
We also need error estimates for the L ( e )-orthogonal projections, Q ∗ e,p : L ( e ) ◦ P p +1 ( e ) , e ∈ F ( T ) . (5.23) Lemma 5.7 ( see [21, Lemma 18] ). With a constant
C > independent of p , ≤ ǫ ≤ , and ǫ ≤ r ≤ ǫ (cid:12)(cid:12) e − Q ∗ e,p u (cid:12)(cid:12) H ǫ ( e ) ≤ C ( p + 1) ǫ − r | u | H r ( e ) ∀ u ∈ H r ( e ) ∩ H ( e ) . Proof . Write I e,p : H ( e ) ◦ P p +1 for the interpolation operator( I e,p u )( ξ ) = u (0) + Z ξ (cid:0) Q e,p dudξ (cid:1) ( τ ) d τ , ≤ ξ ≤ | e | , where ξ is the arclength parameter for the edge e and Q e,p : L (Ω)
7→ P p ( e ) is the L ( e )-orthogonal projection. From [40, Sect. 3.3.1, Thm. 3.17] we learn that | u − I e,p u | H ( e ) ≤ C ( p + 1) − | u | H ( e ) ∀ u ∈ H ( e ) , (5.24) k u − I e,p u k L ( e ) ≤ C ( p + 1) − m | u | H m ( e ) ∀ u ∈ H m ( e ) , m = 1 , . (5.25)Here and the in the remainder of the proof, all constants may depend only on thelength of e . As I e,p u ∈ ◦ P p +1 ( e ) for u ∈ H ( e ), this permits us to conclude (cid:13)(cid:13) u − Q ∗ e,p u (cid:13)(cid:13) L ( e ) ≤ k u − I e,p u k L ( e ) ≤ C ( p + 1) − k u k H ( e ) , (5.26)which yields, by interpolation between H ( e ) and L ( e ), (cid:13)(cid:13) u − Q ∗ e,p u (cid:13)(cid:13) L ( e ) ≤ C ( p + 1) − q k u k H q ( e ) , ≤ q ≤ , (5.27)where C > q . On the other hand, using the inverse inequality [6,Lemma 1] k u k H ( e ) ≤ C ( p + 1) k u k L ( e ) ∀ u ∈ P p +1 ( e ) (5.28)and (5.24), (5.25) we find the estimate (cid:12)(cid:12) u − Q ∗ e,p u (cid:12)(cid:12) H ( e ) ≤ | u − I e,p u | H ( e ) + (cid:12)(cid:12) Q ∗ e,p u − I e,p u (cid:12)(cid:12) H ( e ) ≤ | u − I e,p u | H ( e ) + ( p + 1) (cid:13)(cid:13) Q ∗ e,p u − I e,p u (cid:13)(cid:13) L ( e ) ≤ | u − I e,p u | H ( e ) + C ( p + 1) k u − I e,p u k L ( e ) ≤ C k u k H ( e ) . (5.29)Interpolation between (5.27) with q = r − ǫ − ǫ and (5.29) finishes the proof. Proof . [of Thm. 5.2, borrowed from [21, Sect. 6]] Orthogonality (5.1) of Lemma 5.1combined with the definition of Q T,p involves Z T grad ((Π T,p − Q T,p ) u ) · grad v p d x = 0 ∀ v p ∈ ◦ P p +1 ( T ) . (5.30) iscrete compactness T,p − Q T,p ) u turns out to be the |·| H ( T ) -minimal degree p + 1 polynomialextension of (Π T,p − Q T,p ) u | ∂T , which,thanks to Thm. 5.3, implies (cid:12)(cid:12) (Π T,p − Q T,p ) u (cid:12)(cid:12) H ( T ) ≤ (cid:12)(cid:12)(cid:12) S T ((Π T,p u − Q T,p u ) | ∂T ) (cid:12)(cid:12)(cid:12) H ( T ) ≤ C (cid:12)(cid:12)(cid:12) (Π T,p u − Q T,p u ) | ∂T (cid:12)(cid:12)(cid:12) H ( ∂T ) . (5.31)Thus, by the continuity of the trace operator H ( T ) H ( ∂T ), (cid:12)(cid:12) u − Π T,p u (cid:12)(cid:12) H ( T ) ≤ | u − Q T,p u | H ( T ) + C (cid:18)(cid:12)(cid:12)(cid:12) ( u − Π T,p u ) | ∂T (cid:12)(cid:12)(cid:12) H ( ∂T ) + (cid:12)(cid:12)(cid:12) ( u − Q T,p u ) | ∂T (cid:12)(cid:12)(cid:12) H ( ∂T ) (cid:19) ≤ C (cid:18) | u − Q T,p u | H ( T ) + (cid:12)(cid:12)(cid:12) ( u − Π T,p u ) | ∂T (cid:12)(cid:12)(cid:12) H ( ∂T ) (cid:19) . (5.32)To estimate (cid:12)(cid:12)(cid:12) ( u − Π T,p u ) | ∂T (cid:12)(cid:12)(cid:12) H ( ∂T ) we appeal to Lemma 5.5 and get (cid:12)(cid:12)(cid:12) ( u − Π T,p u ) | ∂T (cid:12)(cid:12)(cid:12) H ( ∂T ) ≤ (cid:12)(cid:12)(cid:12) ( u − Π T,p u ) | ∂T (cid:12)(cid:12)(cid:12) H
12 + ǫ ( ∂T ) ≤ Cǫ X f ∈F ( T ) (cid:12)(cid:12)(cid:12) ( u − Π T,p u ) | f (cid:12)(cid:12)(cid:12) H
12 + ǫ ( f ) . (5.33)Next, we use (5.2) from Lemma 5.1 together with (5.20), which confirms that(Π T,p u ) | f − Q f,p u is the minimum |·| H
12 + ǫ ( f ) -seminorm polynomial extension of(Π T,p u ) | ∂f − Q f,p ( u ) | ∂f . Hence, based on arguments parallel to the derivation of (5.32),this time using Thm. 5.4, we can bound (cid:12)(cid:12)(cid:12) ( u − Π T,p u ) | f (cid:12)(cid:12)(cid:12) H
12 + ǫ ( f ) ≤ (cid:12)(cid:12) u | f − Q f,p u (cid:12)(cid:12) H
12 + ǫ ( f ) + C (cid:12)(cid:12)(cid:12) (Π T,p u − Q f,p u ) | ∂f (cid:12)(cid:12)(cid:12) H ǫ ( ∂f ) , (5.34)where the ( ǫ -independent !) continuity constant of the trace mapping S f enters theconstant C >
0. Also recall the continuity of the trace mapping H + ǫ ( f ) H ǫ ( ∂f )[34, Proof of Lemma 3.35]: with C > ǫ , (cid:13)(cid:13) u | ∂f (cid:13)(cid:13) H ǫ ( ∂f ) ≤ C √ ǫ k u k H
12 + ǫ ( f ) ∀ u ∈ H + ǫ ( f ) . (5.35)Use this to continue the estimate (5.34) (cid:12)(cid:12)(cid:12) ( u − Π T,p u ) | f (cid:12)(cid:12)(cid:12) H
12 + ǫ ( f ) ≤ C (cid:18) √ ǫ (cid:12)(cid:12) u | f − Q f,p u (cid:12)(cid:12) H
12 + ǫ ( f ) + (cid:12)(cid:12)(cid:12) ( u − Π T,p u ) | ∂f (cid:12)(cid:12)(cid:12) H ǫ ( ∂f ) (cid:19) . (5.36)As ǫ < , we can localize the norm (cid:12)(cid:12)(cid:12) ( u − Π T,p u ) | ∂f (cid:12)(cid:12)(cid:12) H ǫ ( ∂f ) to the edges of f : (cid:12)(cid:12)(cid:12) ( u − Π T,p u ) | ∂f (cid:12)(cid:12)(cid:12) H ǫ ( ∂f ) ≤ C − ǫ X e ∈F ( T ) ,e ⊂ ∂f (cid:12)(cid:12)(cid:12) ( u − Π T,p u ) | e (cid:12)(cid:12)(cid:12) H ǫ ( e ) . (5.37)8 R. Hiptmair
Recall the ǫ -uniform equivalence of the norms |·| H ǫ ( e ) and (cid:13)(cid:13) ddl · (cid:13)(cid:13) H − ǫ ( e ) . Hence, owingto (5.3), we have from Lemma 5.7 with r = s : (cid:12)(cid:12)(cid:12) ( u − Π T,p u ) | e (cid:12)(cid:12)(cid:12) H ǫ ( e ) ≤ C inf v p ∈ ◦ P p +1 (cid:12)(cid:12)(cid:12) ( u − Π T, u ) | e − v p (cid:12)(cid:12)(cid:12) H ǫ ( e ) ≤ C (cid:12)(cid:12)(cid:12) ( u − Π T, u ) | e − Q ∗ e,p (( u − Π T, u ) | e ) (cid:12)(cid:12)(cid:12) H ǫ ( e ) ≤ C ( p + 1) ǫ − s (cid:12)(cid:12)(cid:12) ( u − Π T, u ) | e (cid:12)(cid:12)(cid:12) H s ( e ) . (5.38)Moreover, H s ( T ) is continuously embedded into C ( T ). Consequently, applyingtrace theorems twice and appealing to the equivalence of all norms on the finitedimensional space P ( T ), (cid:12)(cid:12)(cid:12) ( u − Π T, u ) | e (cid:12)(cid:12)(cid:12) H s ( e ) ≤ (cid:12)(cid:12) u | e (cid:12)(cid:12) H s ( e ) + (cid:12)(cid:12)(cid:12) (Π T, u ) | e (cid:12)(cid:12)(cid:12) H s ( e ) ≤ C | u | H s ( T ) , (5.39)where C > s and T , but not on p . Combining the estimates (5.32),(5.33), (5.36), and (5.37), (5.38) with (5.39), we find (cid:12)(cid:12) u − Π T,p u (cid:12)(cid:12) H ( T ) ≤ C (cid:16) | u − Q T,p u | H ( T ) + 1 ǫ / X f ∈F ( T ) (cid:12)(cid:12) u | f − Q f,p ( u | f ) (cid:12)(cid:12) H
12 + ǫ ( f ) +( p + 1) ǫ − s ǫ ( − ǫ ) X e ∈F ( T ) | u | H s ( T ) (cid:17) , (5.40)with C > p . Finally, we plug in the projection error estimates (5.21),(5.22), and arrive at ( C > u , ǫ , p , s ) (cid:12)(cid:12) u − Π T,p ( ǫ ) u (cid:12)(cid:12) H ( T ) ≤ C (cid:16) ( p + 1) − s | u | H s ( T ) + ( p + 1) − s + ǫ ǫ / X f ∈F ( T ) | u | H
12 + s ( f ) +( p + 1) − s +2 ǫ ǫ ( − ǫ ) X e ∈F ( T ) | u | H s ( e ) (cid:17) . (5.41)The choice (5.9) of ǫ together with an application of trace theorems then finishes theproof.The next lemma plays the role of [8, Lemma 9] and makes it possible to adaptthe approach of [8, Sect. 4.4] to 3D edge elements. Lemma 5.8. If < s ≤ and u ∈ H s (Ω) satisfies curl u | T ∈ P p ( T ) for all T ∈ M , then (cid:13)(cid:13) ( Id − Π p ) u (cid:13)(cid:13) L (Ω) ≤ C log / pp s (cid:0) k u k H s (Ω) + k curl u k L ( T ) (cid:1) , (5.42) with a constant C > depending only on M and s .Proof . Pick any u complying with the assumptions of the lemma. The locality ofthe projector allows purely local considerations. Single out one tetrahedron T ∈ M ,still write u = u | T , and split on T u = ( u − R T curl u ) + R T curl u , (5.43)Note that the properties of the smoothed Poincar´e lifting R T stated in Thm. 2.3 imply iscrete compactness curl ( u − R T curl u ) = 0 on T , as a consequence of (2.8), and2. R T curl u ∈ H ( T ) and the bound k R T curl u k H ( T ) ≤ C k curl u k L (Ω) , (5.44)where here and below no constant may depend on u or p .Hence, as u ∈ H s ( T ), there exists v ∈ H s ( T ) such that u = grad v + R T curl u . (5.45)The continuity of R T reveals that, with a constant C > T , | v | H s ( T ) ≤ k u k H s ( T ) + | R T curl u | H ( T ) ≤ k u k H s ( T ) + C k curl u k L ( T ) . (5.46)By the assumptions of the lemma and (3.4) we know that R T curl u ∈ W p ( T ) . (5.47)By the commuting diagram property from Lemma 4.2 and the projector property ofΠ T,p the task is reduced to an interpolation estimate for Π T,p :( Id − Π T,p ) u (5.45) = grad ( Id − Π T,p ) v + ( Id − Π T,p ) R T curl u | {z } =0 (5.48)As a consequence, invoking Thm. 5.2, (cid:13)(cid:13) ( Id − Π T,p ) u (cid:13)(cid:13) L ( T ) (5.48) = (cid:12)(cid:12) ( Id − Π T,p ) v (cid:12)(cid:12) H ( T ) ≤ C log / pp s | v | H s ( T ) (5.46) ≤ C log / pp s (cid:0) k u k H s ( T ) + k curl u k L ( T ) (cid:1) , (5.49)which furnishes a local version of the estimate. Squaring (5.49) and summing over alltetrahedra finishes the proof.
6. Discrete compactness.
Smoothness of the solenoidal part of the Helmholtzdecomposition of H ( curl , Ω) and H ( curl , Ω), respectively, plays a pivotal role. Itcan be deduced from elliptic lifting theorems for 2nd-order elliptic boundary valueproblems [18, Ch. 6]. Proofs of the following lemma can be found in [29, Sect. 4.1]and [2, Sect. 3].
Lemma 6.1.
For any Lipschitz polyhedron Ω ⊂ R there is a < s ≤ such that X := H (div , Ω) ∩ H ( curl , Ω) and X := H (div , Ω) ∩ H ( curl , Ω) are continuouslyembedded into H s (Ω) , that is, ∃ C = C ( s, Ω) > k u k H s (Ω) ≤ C (cid:16) k u k H ( curl , Ω) + k u k H (div , Ω) (cid:17) ∀ u ∈ X . (6.1)We first verify the discrete compactness property of Def. 1.1 for ǫ ≡
1: consider asequence ( u p ) p ∈ N , which satisfies(i) u p ∈ W p ( M ) , (6.2)(ii) ( u p , z p ) L (Ω) = 0 ∀ z p ∈ { v ∈ W p ( M ) : curl v = 0 } , (6.3)(iii) k u p k L (Ω) + k curl u p k L (Ω) ≤ ∀ p ∈ N . (6.4)0 R. Hiptmair
Theorem 6.2.
A sequence ( u p ) p ∈ N compliant with (6.2) – (6.4) possesses a subse-quence that converges in L (Ω) .Proof . The proof resorts to the “standard policy” for tackling the problem ofdiscrete compactness, introduced by Kikuchi [32,33] for analyzing the h -version of edgeelements. It forms the core of most papers tackling the issue of discrete compactness,see [9, Thm. 2], [8, Thm. 11], [29, Thm. 4.9], [23, Thm. 2], [8, Thm. 11], etc.We start with the continuous Helmholtz decomposition of u p : let e u p be the uniquevector field in H ( curl , Ω) with curl e u p = curl u p , (6.5)( e u p , z ) L (Ω) = 0 ∀ z ∈ Ker( curl ) ∩ H ( curl , Ω) . (6.6)The inclusion grad H (Ω) ⊂ Ker( curl ) enforcesdiv e u p = 0 in Ω , e u p · n = 0 on ∂ Ω . (6.7)Hence, by virtue of Lemma 6.1, e u p satisfies e u p ∈ H s (Ω) , k e u p k H s (Ω) ≤ C k u p k H ( curl , Ω) , (6.8)where C > < s ≤ e u p is L (Ω)-orthogonal to Ker( curl ) ∩ H ( curl , Ω), see (6.6). Thus,using Ned´el´ec’s trick [38], we have k e u p − u p k L (Ω) = (cid:0)e u p − u p , e u p − Π p e u p + Π p e u p − u p (cid:1) L (Ω) = (cid:0)e u p − u p , e u p − Π p e u p (cid:1) L (Ω) , (6.9)because, thanks to Lemma 4.3 and (6.5), curl (Π p e u p − u p ) = (Π p − Id ) curl u p | {z } ∈W p ( M ) = 0 . (6.10) ⇒ Π p e u p − u p ∈ { v ∈ W p ( M ) : curl v = 0 } . (6.11)Hence, appealing to Lemma 5.8, with C > p , k e u p − u p k L (Ω) ≤ (cid:13)(cid:13)e u p − Π p e u p (cid:13)(cid:13) L (Ω) ≤ C log / pp s (cid:0) | e u p | H s (Ω) + k curl e u p k L (Ω) (cid:1) ≤ C log / pp s k u p k H ( curl , Ω) → p → ∞ . (6.12)Since bounded in H s (Ω), by Rellich’s theorem ( e u p ) p ∈ N has a convergent subsequencein L (Ω): owing to (6.12), the same subsequence of ( u p ) p ∈ N will converge in L (Ω). Theorem 6.3.
Replacing W p ( M ) with W p ( M ) ∩ H ( curl , Ω) in (6.2) – (6.4) , theassertion of Thm. 6.2 remains true.Proof . Since the projection based interpolation operators respect homogeneousDirichlet boundary conditions, cf. (4.64), the proof runs parallel to that of Thm. 6.2.We point out that now e u p × n = 0 on ∂ Ω instead of e u p · n = 0, but Lemma 6.1 canstill be applied. iscrete compactness ǫ ≡ Proof . [of Thm. 1.2 in the Introduction] We adapt the proof of [29, Thm. 4.9].Consider a H ( curl , Ω)-bounded sequence ( w p ) p ∈ N , w p ∈ W p ( M ), in the L ǫ (Ω)-orthogonal complement X p ( M ) (see (1.2)) of the discrete kernel of curl , i.e. ,( ǫ w p , z p ) L (Ω) = 0 ∀ z p ∈ Ker( curl ) ∩ W p ( M ) . (6.13)We continue with the L (Ω)-orthogonal discrete Helmholtz decomposition w p = u p ⊕ L w p , u p ∈ W p ( M ) , w p ∈ Ker( curl ) ∩ W p ( M ) . (6.14)As ( u p , z p ) L (Ω) = 0 for all z p ∈ Ker( curl ) ∩ W p ( M ), by Thm. 6.2 we can find asubsequence, again denoted by ( u p ) p ∈ N , with u p p →∞ −−−−→ q in L (Ω) . (6.15)Since w p satisfies (5.41), we conclude (cid:0) ǫ w p , z p (cid:1) L (Ω) = − ( ǫ u p , z p ) L (Ω) ∀ z p ∈ Ker( curl ) ∩ W p ( M ) . (6.16)This can be regarded as a perturbed spectral Galerkin approximation of the followingcontinuous variational problem: seek y ∈ H ( curl , Ω) := Ker( curl ) ∩ H ( curl , Ω)such that ( ǫ y , z ) L (Ω) = − ( ǫ q , z ) L (Ω) ∀ z ∈ H ( curl , Ω) , (6.17)which, obviously, has unique solution y . From Strang’s lemma [14, Thm. 4.4.1] weinfer (cid:13)(cid:13) y − w p (cid:13)(cid:13) L (Ω) ≤ C (cid:0) inf z p ∈W p ( M ) ∩ Ker( curl ) k y − z p k L (Ω) + k u p − q k L (Ω) | {z } → p →∞ (cid:1) , (6.18)where C > ǫ . Next recall, that there is a representation y = grad v + h , v ∈ H (Ω) , h ∈ H (Ω) , (6.19)with H (Ω) standing for the finite dimensional first co-homology space H (Ω) of har-monic vector fields, which is contained in H ( curl , Ω) ∩ H (div 0 , Ω), [29, Lemma 2.2]and [2, Prop. 3.14, Prop. 3.18]. Owing to Lemma 6.1 it belongs to H s (Ω) for some s > , which implies, thanks to Lemma 5.8, (cid:13)(cid:13) h − Π p h (cid:13)(cid:13) L (Ω) ≤ C log / pp s | h | H s (Ω) → p → ∞ . (6.20)Further, the commuting diagram property of Lemma 4.3 confirms that curl Π p h = 0.Besides, asymptotic density of the spectral family of Lagrangian finite element spacesin H (Ω) means that also the first term on the right hand side of (6.18) tends to zeroas p → ∞ .Thus, selecting the same subsequence of ( w p ) p ∈ N (and keeping the notation), itis immediate that w p p →∞ −−−−→ q + y in L (Ω) . (6.21)The case of w p ∈ W p ( M ) ∩ H ( curl , Ω) is amenable to almost the same proof: theboundary conditions are imposed on all fields and in the counterpart of (6.19) thesecond co-homology space H (Ω) has to be considered [29, Lemma 2.2].2 R. Hiptmair
7. Acknowledgment.
The author would like to thank his colleague C. Schwabfor pointing out the crucial reference [16]. He is grateful to A. Buffa, M. Costabel,M. Dauge, and L. Demkowicz, and all the other mathematicians who have laid thefoundations for this work.
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