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An arbitrary-order discrete de Rham complex on
polyhedral meshes. Part II: Consistency
Daniele Antonio Di Pietro, Jérôme Droniou
To cite this version:
An arbitrary-order discrete de Rham complex on polyhedral
meshes. Part II: Consistency
Daniele A. Di Pietro1and Jérôme Droniou2
1IMAG, Univ Montpellier, CNRS, Montpellier, France, daniele.di-pietro@umontpellier.fr 2School of Mathematics, Monash University, Melbourne, Australia, jerome.droniou@monash.edu
January 8, 2021
Abstract
In this paper we prove a complete panel of consistency results for the discrete de Rham (DDR) complex introduced in the companion paper [10], including primal and adjoint consistency for the discrete vector calculus operators, and consistency of the corresponding potentials. The theoretical results are showcased by performing a full convergence analysis for a DDR approximation of a magnetostatics model. Numerical results on three-dimensional polyhedral meshes complete the exposition.
Key words. Discrete de Rham complex, compatible discretisations, polyhedral methods, mixed
methods
MSC2010. 65N30, 65N99, 78A30
1
Introduction
We prove complete consistency results for the discrete de Rham (DDR) complex introduced in the companion paper [10]. Specifically, the first set of results concerns primal consistency of the local discrete vector calculus operators introduced in [10, Section 3.3] and of the corresponding potentials defined in Section 3.1 below. The second set of results concerns adjoint consistency, that relates to the ability to approximate formal adjoint operators, and therefore requires to estimate the residuals of global integration by parts formulas.
For specific space dimensions, polynomial degrees, and operators, consistency results that bear relations to ours can be found in the literature on polytopal methods.
Starting from low-order methods, consistency results for Compatible Discrete Operator approxima-tions of the Poisson problem based on nodal unknowns can be found in [5]; see in particular the proof of Theorem 3.3 therein, which contains an adjoint consistency result for a gradient reconstructed from vertex values. In the same framework, an adjoint consistency estimate for a discrete curl constructed from edge values can be found in [6, Lemma 2.3]. A rather complete set of consistency results for Mimetic Finite Difference operators can be found in [4], where they appear as intermediate steps in the error analyses of Chapters 5–7. A notable exception is provided by the adjoint consistency of the curl operator, which is not needed in the error estimate of [4, Theorem 7.3] since the authors consider an approximation of the current density based on the knowledge of a vector potential.
a different analysis is proposed based on the third Strang lemma. The estimate of the consistency error in [9, Theorem 19] involves, in particular, the adjoint consistency of a discrete gradient reconstructed as the gradient of a scalar polynomial rather than a vector-valued polynomial. We note, in passing, that the concept of adjoint consistency for (discrete) gradients is directly related to the notion of limit-conformity in the Gradient Discretisation Method [15], a generic framework which encompasses several polytopal methods. Primal and dual consistency estimates for a discrete divergence and the corresponding vector potential similar (but not identical) to the ones considered here have been established in [14] in the framework of Mixed High-Order methods. Note that these methods, the H1-conforming Virtual Element method, and the Mixed High-Order method, do not lead to a discrete de Rham complex. In the framework of arbitrary-order compatible discretisations, on the other hand, primal consistency results for the curl appear as intermediate results in [3], where an error analysis for a Virtual Element approximation of magnetostatics is carried out assuming interpolation estimates for three-dimensional vector valued virtual spaces; see Remark 4.4 therein. However, [3] does not establish any adjoint consistency property of the discrete curl (the formulation of magnetostatics considered in this reference does not require this).
The results presented in this paper are, to the best of our knowledge, the first ones to span the full set of discrete vector calculus operators for an arbitrary-order discrete de Rham complex on polyhedral meshes. The key ingredients to establish primal consistency are the polynomial consistency of discrete vector calculus operators along with the corresponding potentials, and their boundedness when applied to the interpolates of smooth functions. The proofs of adjoint consistency, on the other hand, rely on operator-specific techniques, all grounded in discrete integration by parts formulas for the corresponding potential reconstructions (see (3.1) along with Remark 3 for the gradient, (3.6) for the curl, and (3.10) for the divergence). Specifically, the key point for the adjoint consistency of the gradient are estimates for local H1-like seminorms of the scalar potentials. The adjoint consistency of curl requires, on the other hand, the construction of liftings of the discrete face potentials that satisfy an orthogonality and a boundedness condition. These reconstructions are inspired by the minimal reconstruction operators of [4, Chapter 3], with a key novelty provided by a curl correction which ensures the well-posedness of the reconstruction inside mesh elements.
In order to showcase the theoretical results derived here and in the companion paper [10], we carry out a full convergence analysis for a DDR approximation of magnetostatics. This is, to the best of our knowledge, the first full theoretical result of this kind for arbitrary-order polytopal methods.
The rest of this paper is organised as follows. In Section 2 we briefly recall the key elements of the setting introduced in [10]. Section 3 contains the statement of the primal and adjoint consistency results, whose proofs are given in Section 4. The application of the theoretical tools to the error analysis of a DDR approximation of magnetostatics is considered in Section 5, where numerical evidence supporting the error estimates is also provided. Finally, Appendix A contains an in-depth and novel study of the div–curl problems defining the curl liftings on polytopal elements: well-posedness, orthogonality and boundedness properties.
2
Setting
We briefly recall here the setting introduced in the companion paper [10], to which we refer for a more detailed description of the following notions.
2.1 Mesh and orientation
Let H ⊂ R∗+ be a countable set with 0 as its unique accumulation point. Let Ω ⊂ R3 denote an
open connected polyhedral set and (Mℎ)ℎ∈H a family of meshes indexed by their size ℎ. We write
Mℎ ≔ Tℎ∪ Fℎ∪ Eℎ∪ Vℎ with Tℎthe set of elements 𝑇 , Fℎ the set of faces 𝐹, Eℎthe set of edges 𝐸 ,
and Vℎ the set of vertices 𝑉 . We additionally denote by Fℎbthe subset of Fℎ collecting the faces that
Definition 1.9] (with 𝜌 ∈ (0, 1) denoting the mesh regularity parameter), and that elements and faces are simply connected with Lipschitz continuous boundary. For 𝑇 ∈ Tℎ, we set F𝑇 ≔ {𝐹 ∈ Fℎ : 𝐹 ⊂ 𝜕𝑇 }
and, for 𝑌 ∈ Tℎ∪ Fℎ, E𝑌 ≔ {𝐸 ∈ Eℎ : 𝐸 ⊂ 𝜕𝑌 }. The real number ℎ𝑌 denotes the diameter of a mesh
element, face, or edge 𝑌 ∈ Tℎ∪ Fℎ∪ Eℎ.
Each face 𝐹 ∈ Fℎ is equipped with a unit normal vector 𝒏𝐹, and each edge 𝐸 ∈ Eℎ with a
unit tangent vector 𝒕𝐸. Given 𝐹 ∈ Fℎ and 𝐸 ∈ E𝐹, we also denote by 𝒏𝐹 𝐸 the unit vector
nor-mal to 𝐸 lying in the plane of 𝐹. The families of numbers {𝜔𝑇 𝐹 ∈ {−1, 1} : 𝑇 ∈ Tℎ, 𝐹 ∈ F𝑇} and
{𝜔𝐹 𝐸 ∈ {−1, 1} : 𝐹 ∈ Fℎ, 𝐸 ∈ E𝐹} collect relative orientations selected so that: for all 𝑇 ∈ Tℎand all
𝐹 ∈ F𝑇, 𝜔𝑇 𝐹𝒏𝐹 points out of 𝑇 and, for all 𝐹 ∈ Fℎand all 𝐸 ∈ E𝐹, 𝜔𝐹 𝐸𝒏𝐹 𝐸 points out of 𝐹. Given 𝐹 ∈ Fℎ, the tangent gradient, divergence, two-dimensional vector and scalar curl operators are denoted
by grad𝐹, div𝐹, rot𝐹 and rot𝐹, respectively. 2.2 Polynomial spaces
Let ℓ ≥ −1 be an integer. For 𝑌 ∈ Tℎ∪ Fℎ∪ Eℎ, with 𝑛 the dimension of 𝑌 , we denote by P ℓ
(𝑌 ) the space of polynomial functions over 𝑌 of total degree ≤ ℓ, and we set Pℓ(𝑌 ) = Pℓ(𝑌 )𝑛. The L2-orthogonal projector on Pℓ(𝑌 ) is 𝜋ℓP,𝑌, and 𝝅
ℓ
P,𝑌 : L2(𝑌 ) → P ℓ
(𝑌 ) is its vector-valued counterpart. The set Pℓ
c(Eℎ) is made of all continuous functions over the mesh skeleton
Ð
𝐸∈ Eℎ
𝐸 that are polynomial of total degree ≤ ℓ on each 𝐸 ∈ Eℎ.
For all 𝑌 ∈ Tℎ∪ Fℎ, denote by 𝒙𝑌 a point inside 𝑌 such that 𝑌 contains a ball centered at 𝒙𝑌 and of
diameter 𝜌ℎ𝑌. For any mesh face 𝐹 ∈ Fℎ, any mesh element 𝑇 ∈ Tℎ, and any integer ℓ ≥ −1, we define
Gℓ (𝐹) ≔ grad𝐹 P ℓ+1 (𝐹), Gc,ℓ(𝐹) ≔ (𝒙 − 𝒙 𝐹) ⊥Pℓ−1 (𝐹), Rℓ (𝐹) ≔ rot𝐹P ℓ+1 (𝐹), Rc,ℓ(𝐹) ≔ (𝒙 − 𝒙 𝐹)P ℓ−1 (𝐹), Gℓ (𝑇 ) ≔ grad Pℓ+1 (𝑇 ), Gc,ℓ(𝑇 ) ≔ (𝒙 − 𝒙 𝑇) × P ℓ−1 (𝑇 ), Rℓ (𝑇 ) ≔ curl Pℓ+1 (𝑇 ), Rc,ℓ(𝑇 ) ≔ (𝒙 − 𝒙 𝑇)P ℓ−1 (𝑇 ) (2.1) where (𝒙 − 𝒙𝐹) ⊥
denotes the vector 𝒙 − 𝒙𝐹 rotated by an angle −𝜋/2 in the plane spanned by 𝐹 and
oriented by 𝒏𝐹. If 𝑌 = 𝐹 or 𝑌 = 𝑇 , the following direct (but not necessarily orthogonal) decompositions
hold: Pℓ (𝑌 ) = Gℓ (𝑌 ) ⊕ Gc,ℓ(𝑌 ) = Rℓ (𝑌 ) ⊕ Rc,ℓ(𝑌 ). (2.2) With obvious notations, the L2-orthogonal projectors on the subspaces appearing in these decomposi-tions are denoted by 𝝅ℓG,𝑌, 𝝅
c,ℓ G,𝑌, 𝝅
ℓ
R,𝑌, and 𝝅 c,ℓ
R,𝑌. The local Nédélec and Raviart–Thomas spaces over
𝑌 are denoted by Nℓ (𝑌 ) ≔ Gℓ−1 (𝑌 ) ⊕ Gc,ℓ(𝑌 ), RTℓ (𝑌 ) ≔ Rℓ−1 (𝑌 ) ⊕ Rc,ℓ(𝑌 ). (2.3) As detailed in [10, Lemma 4], the knowledge of the L2-projections of a polynomial 𝒛 ∈ Pℓ(𝑌 ) on each element of the space pairs (Gℓ(𝑌 ), Gc,ℓ(𝑌 )) or (Rℓ(𝑌 ), Rc,ℓ(𝑌 )) appearing in (2.2) enables the recovery of 𝒛. Specifically, for 𝑌 ∈ Tℎ∪ Fℎ, X ∈ {G, R}, and (𝒗, 𝒘) ∈ X
Above, writing 𝑎 . 𝑏 in place of 𝑎 ≤ 𝐶 𝑏 with 𝐶 depending only on Ω, the mesh regularity parameter 𝜌 of [12, Definition 1.9], and the considered polynomial degree, we have used 𝑎 ' 𝑏 with the meaning of “𝑎 . 𝑏 and 𝑏 . 𝑎”. Both shorthand notations . and ' will be used throughout the paper.
2.3 Discrete spaces
The discrete counterpart of the space H1(Ω) in the DDR sequence is
𝑋𝑘 grad,ℎ ≔ n 𝑞 ℎ = (𝑞𝑇)𝑇∈ Tℎ ,(𝑞𝐹)𝐹∈ F ℎ , 𝑞E ℎ : 𝑞𝑇 ∈ P 𝑘−1 (𝑇 ) for all 𝑇 ∈ Tℎ, 𝑞𝐹 ∈ P 𝑘−1
(𝐹) for all 𝐹 ∈ Fℎ, and 𝑞Eℎ ∈ P
𝑘+1 c (Eℎ)
o ,
and the corresponding interpolator 𝐼𝑘grad,ℎ: C0(Ω) → 𝑋
𝑘
grad,ℎis such that, for all 𝑞 ∈ C0(Ω),
𝐼𝑘 grad,ℎ𝑞≔ (𝜋 𝑘−1 P,𝑇𝑞|𝑇)𝑇∈ T ℎ ,(𝜋𝑘−1 P,𝐹𝑞|𝐹)𝐹∈ Fℎ , 𝑞E ℎ ∈ 𝑋 𝑘 grad,ℎ, where 𝜋𝑘P,𝐸−1(𝑞Eℎ)|𝐸 = 𝜋 𝑘−1
P,𝐸𝑞|𝐸for all 𝐸 ∈ Eℎand 𝑞Eℎ(𝒙𝑉) = 𝑞 (𝒙𝑉) for all 𝑉 ∈ Vℎ,
(2.8)
with 𝒙𝑉 denoting the coordinates vector of the vertex 𝑉 . The discrete H(curl; Ω) space is
𝑿𝑘 curl,ℎ≔ n 𝒗𝑇 = (𝒗R,𝑇 ,𝒗c R,𝑇)𝑇∈ Tℎ ,(𝒗R,𝐹,𝒗c R,𝐹)𝐹∈ Fℎ ,(𝑣𝐸)𝐸∈ E ℎ : (𝒗R,𝑇,𝒗c R,𝑇) ∈ R 𝑘−1 (𝑇 ) × Rc, 𝑘(𝑇 ) for all 𝑇 ∈ T ℎ, (𝒗R,𝐹,𝒗c R,𝐹) ∈ R 𝑘−1 (𝐹) × Rc, 𝑘(𝐹) for all 𝐹 ∈ F ℎ, and 𝑣𝐸 ∈ P 𝑘 (𝐸) for all 𝐸 ∈ Eℎ o ,
with interpolator 𝑰𝑘curl,ℎ: C0(Ω) → 𝑿
𝑘
curl,ℎsuch that, for all 𝒗 ∈ C0(Ω),
𝑰𝑘 curl,ℎ𝒗 ≔ (𝝅 𝑘−1 R,𝑇𝒗|𝑇,𝝅c, 𝑘 R,𝑇𝒗|𝑇)𝑇∈ Tℎ ,(𝝅𝑘−1 R,𝐹𝒗t,𝐹,𝝅c, 𝑘R,𝐹𝒗t,𝐹)𝐹∈ Fℎ ,(𝜋𝑘 P,𝐸(𝒗|𝐸·𝒕𝐸))𝐸∈ Eℎ ,
where, for all 𝐹 ∈ Fℎ, 𝒗t,𝐹 ≔ 𝒏𝐹 × (𝒗|𝐹 × 𝒏𝐹) denotes the orthogonal projection of 𝒗 on the plane
spanned by 𝐹. The role of H(div; Ω) is played, at the discrete level, by 𝑿𝑘 div,ℎ ≔ n 𝒗𝑇 = (𝒗G,𝑇,𝒗 c G,𝑇)𝑇∈ Tℎ,(𝑣𝐹)𝐹∈ Fℎ : (𝒗G,𝑇,𝒗c G,𝑇) ∈ G 𝑘−1 (𝑇 ) × Gc, 𝑘(𝑇 ) for all 𝑇 ∈ T ℎand 𝑣𝐹 ∈ P 𝑘 (𝐹) for all 𝐹 ∈ Fℎ o ,
with interpolator 𝑰𝑘div,ℎ : H1(Ω) → 𝑿𝑘div,ℎsuch that, for all 𝒗 ∈ H1(Ω), 𝑰𝑘 div,ℎ𝒗 ≔ (𝝅 𝑘−1 G,𝑇𝒗|𝑇,𝝅c, 𝑘G,𝑇𝒗|𝑇)𝑇∈ T ℎ ,(𝜋𝑘 P,𝐹(𝒗|𝐹 ·𝒏𝐹))𝐹∈ F𝑇 . (2.9)
Finally, the discrete counterpart of L2(Ω) in the DDR sequence is P𝑘 (Tℎ) ≔ 𝑞ℎ ∈ L2(Ω) : (𝑞ℎ)|𝑇 ∈ P 𝑘 (𝑇 ) for all 𝑇 ∈ Tℎ ,
equipped with the global L2-orthogonal projector 𝜋𝑘P,ℎ : L2(Ω) → P 𝑘
(Tℎ) such that, for all 𝑞 ∈ L2(Ω),
(𝜋𝑘
P,ℎ𝑞)|𝑇 ≔ 𝜋 𝑘
P,𝑇𝑞|𝑇 for all 𝑇 ∈ Tℎ.
2.4 Local discrete vector calculus operators
Given • ∈ {grad, curl, div} and a mesh entity 𝑌 appearing in the definition of 𝑋•,ℎ𝑘 , we denote by 𝑋 𝑘 •,𝑌
2.4.1 Gradients
Throughout the rest of the paper, for 𝐸 ∈ Eℎ and 𝑞 ℎ ∈ 𝑋𝑘 grad,ℎ we set 𝑞𝐸 ≔ (𝑞Eℎ)|𝐸 ∈ 𝑋 𝑘 grad,𝐸 = P𝑘+1
(𝐸). For any 𝐸 ∈ Eℎ, the edge gradient 𝐺 𝑘 𝐸 : 𝑋
𝑘
grad,𝐸 → P
𝑘
(𝐸) is such that, for all 𝑞𝐸 ∈ 𝑋 𝑘 grad,𝐸, 𝐺𝑘 𝐸𝑞𝐸 ≔ 𝑞 0 𝐸,
where the derivative is taken along 𝐸 according to the orientation of 𝒕𝐸. For all 𝐹 ∈ Fℎ, the face
gradientG𝑘𝐹 : 𝑋 𝑘
grad,𝐹 → P
𝑘
(𝐹) is such that, for all 𝑞𝐹 = (𝑞
𝐹, 𝑞E𝐹) ∈ 𝑋 𝑘 grad,𝐹, ∫ 𝐹 G𝑘 𝐹𝑞𝐹 ·𝒘𝐹 = − ∫ 𝐹 𝑞𝐹div𝐹𝒘𝐹 + Õ 𝐸∈ E𝐹 𝜔𝐹 𝐸 ∫ 𝐸 𝑞E 𝐹(𝒘𝐹 ·𝒏𝐹 𝐸) ∀𝒘𝐹 ∈ P 𝑘 (𝐹).
The scalar trace 𝛾𝐹𝑘+1: 𝑋 𝑘
grad,𝐹 → P
𝑘+1
(𝐹) is such that, for all 𝑞𝐹 ∈ 𝑋𝑘
grad,𝐹, ∫ 𝐹 𝛾𝑘+1 𝐹 𝑞𝐹div𝐹𝒗𝐹 = − ∫ 𝐹 G𝑘 𝐹𝑞𝐹·𝒗𝐹+ Õ 𝐸∈ E𝐹 𝜔𝐹 𝐸 ∫ 𝐸 𝑞E 𝐹(𝒗𝐹·𝒏𝐹 𝐸) ∀𝒗𝐹 ∈ R c, 𝑘+2(𝐹). (2.10)
Remark 1 (Validity of (2.10)). Relation (2.10) also holds for all 𝒗𝐹 ∈ RT 𝑘+1
(𝐹); see [10, Remark 9]. Finally, for all 𝑇 ∈ Tℎ, the element gradient G
𝑘 𝑇 : 𝑋
𝑘
grad,𝑇 → P
𝑘
(𝑇 ) is such that, for all 𝑞𝑇 = (𝑞𝑇,(𝑞𝐹)𝐹∈ F𝑇 , 𝑞E 𝑇) ∈ 𝑋 𝑘 grad,𝑇, ∫ 𝑇 G𝑘 𝑇𝑞𝑇 ·𝒘𝑇 = − ∫ 𝑇 𝑞𝑇 div 𝒘𝑇 + Õ 𝐹∈ F𝑇 𝜔𝑇 𝐹 ∫ 𝐹 𝛾𝑘+1 𝐹 𝑞𝐹(𝒘𝑇 ·𝒏𝐹) ∀𝒘𝑇 ∈ P 𝑘 (𝑇 ). (2.11) 2.4.2 Curls
For all 𝐹 ∈ Fℎ, the face curl 𝐶 𝑘 𝐹 : 𝑿
𝑘
curl,𝐹 → P
𝑘
(𝐹) is such that, for all 𝒗𝐹 = 𝒗R,𝐹
,𝒗c R,𝐹,(𝑣𝐸)𝐸∈ E𝐹 ∈ 𝑿𝑘 curl,𝐹, ∫ 𝐹 𝐶𝑘 𝐹𝒗𝐹 𝑟𝐹 = ∫ 𝐹 𝒗R,𝐹 · rot𝐹𝑟𝐹 − Õ 𝐸∈ E𝐹 𝜔𝐹 𝐸 ∫ 𝐸 𝑣𝐸𝑟𝐹 ∀𝑟𝐹 ∈ P 𝑘 (𝐹). (2.12)
The tangential trace 𝜸t,𝐹𝑘 : 𝑿𝑘curl,𝐹 → P
𝑘
(𝐹) is such that, for all 𝒗𝐹 ∈ 𝑿 𝑘
curl,𝐹, recalling the definition
(2.4) of the recovery operator with (X, 𝑌 ) = (R, 𝐹), 𝜸𝑘 t,𝐹𝒗𝐹 ≔ ℜ 𝑘 R,𝐹(𝜸 𝑘 t, R,𝐹𝒗𝐹,𝒗 c R,𝐹), where 𝜸t, R,𝐹𝑘 𝒗𝐹 ∈ R 𝑘 (𝐹) is defined by ∫ 𝐹 𝜸𝑘 t, R,𝐹𝒗𝐹 · rot𝐹𝑟𝐹 = ∫ 𝐹 𝐶𝑘 𝐹𝒗𝐹 𝑟𝐹 + Õ 𝐸∈ E𝐹 𝜔𝐹 𝐸 ∫ 𝐸 𝑣𝐸𝑟𝐹 ∀𝑟𝐹 ∈ P0, 𝑘+1(𝐹). (2.13)
Remark 2 (Validity of (2.13)). We note that this relation actually holds for all 𝑟𝐹 ∈ P 𝑘+1
(𝐹) and also with 𝜸t,𝐹𝑘 instead of 𝜸t, R,𝐹𝑘 ; see [10, Remark 14].
Finally, for all 𝑇 ∈ Tℎ, the element curl C 𝑘 𝑇 : 𝑿
𝑘
curl,𝑇 → P
𝑘
(𝑇 ) is such that, for all 𝒗𝑇 =
2.4.3 Divergence
For all 𝑇 ∈ Tℎ, the element divergence 𝐷 𝑘 𝑇 : 𝑿
𝑘
div,𝑇 → P 𝑘
(𝑇 ) is defined by: For all 𝒗𝑇 =
𝒗G,𝑇,𝒗c G,𝑇,(𝑣𝐹)𝐹∈ F𝑇 ∈ 𝑿 𝑘 div,𝑇, ∫ 𝑇 𝐷𝑘 𝑇𝒗𝑇 𝑟𝑇 = − ∫ 𝑇 𝒗G,𝑇 · grad 𝑟𝑇 + Õ 𝐹∈ F𝑇 𝜔𝑇 𝐹 ∫ 𝐹 𝑣𝐹𝑟𝑇 ∀𝑟𝑇 ∈ P 𝑘 (𝑇 ). (2.15) 2.5 Global sequence
The global discrete gradient 𝑮𝑘ℎ : 𝑋 𝑘 grad,ℎ → 𝑿 𝑘 curl,ℎ, curl 𝑪 𝑘 ℎ : 𝑿 𝑘 curl,ℎ → 𝑿 𝑘
div,ℎ, and divergence
𝐷𝑘
ℎ : 𝑿 𝑘
div,ℎ → P 𝑘
(Tℎ) are obtained by projecting the local operators onto the corresponding spaces:
For all (𝑞ℎ ,𝒗 ℎ,𝒘ℎ) ∈ 𝑋 𝑘 grad,ℎ× 𝑿 𝑘 curl,ℎ×𝑿 𝑘 div,ℎ, 𝑮𝑘 ℎ𝑞 ℎ ≔ (𝝅 𝑘−1 R,𝑇(G 𝑘 𝑇𝑞 𝑇 ),𝝅c, 𝑘 R,𝑇(G 𝑘 𝑇𝑞 𝑇 ))𝑇∈ T ℎ ,(𝝅𝑘−1 R,𝐹(G 𝑘 𝐹𝑞 𝐹 ),𝝅c, 𝑘 R,𝐹(G 𝑘 𝐹𝑞 𝐹 ))𝐹∈ Fℎ ,(𝐺𝑘 𝐸𝑞𝐸)𝐸∈ Eℎ , 𝑪𝑘 ℎ𝒗ℎ ≔ (𝝅 𝑘−1 G,𝑇(C 𝑘 𝑇𝒗𝑇),𝝅 c, 𝑘 G,𝑇(C 𝑘 𝑇𝒗𝑇))𝑇∈ Tℎ ,(𝐶𝑘 𝐹𝒗𝐹)𝐹∈ Fℎ , (𝐷𝑘 ℎ𝒘ℎ)|𝑇 ≔ 𝐷 𝑘 𝑇𝒘𝑇 ∀𝑇 ∈ Tℎ .
Following our previous notation for local spaces and interpolator, we will use the following notations for the restrictions of these discrete gradients and curl operators to mesh elements and faces:
𝑮𝑘 𝐹𝑞𝐹 = 𝝅 𝑘−1 R,𝐹(G 𝑘 𝐹𝑞𝐹),𝝅 c, 𝑘 R,𝐹(G 𝑘 𝐹𝑞𝐹), (𝐺 𝑘 𝐸𝑞𝐸)𝐸∈ F𝐸 , 𝑮𝑘 𝑇𝑞 𝑇 = 𝝅𝑘−1 R,𝑇(G 𝑘 𝑇𝑞 𝑇 ),𝝅c, 𝑘 R,𝑇(G 𝑘 𝑇𝑞 𝑇), (𝝅 𝑘−1 R,𝐹(G 𝑘 𝐹𝑞 𝐹 ),𝝅c, 𝑘 R,𝐹(G 𝑘 𝐹𝑞 𝐹 ))𝐹∈ F𝑇 ,(𝐺𝑘 𝐸𝑞𝐸)𝐸∈ E𝑇 , 𝑪𝑘 𝑇𝒗𝑇 = 𝝅 𝑘−1 G,𝑇(C 𝑘 𝑇𝒗𝑇),𝝅 c, 𝑘 G,𝑇(C 𝑘 𝑇𝒗𝑇), (𝐶 𝑘 𝐹𝒗𝐹)𝐹∈ F𝑇 .
The global sequence reads:
R 𝑋𝑘 grad,ℎ 𝑿 𝑘 curl,ℎ 𝑿 𝑘 div,ℎ P 𝑘 (Tℎ) {0}. 𝐼𝑘 grad,ℎ 𝑮 𝑘 ℎ 𝑪 𝑘 ℎ 𝐷𝑘 ℎ 0 (2.16) It is proved in [10] that this sequence has exactness properties (the specific nature of which depends on the topology of Ω, as for the continuous de Rham sequence), and that the discrete operators satisfy Poincaré inequalities.
3
Consistency results
3.1 Potential reconstructions and L2-products on discrete spaces
Let 𝑇 ∈ Tℎ. In this section, we define polynomial potential reconstructions on the discrete spaces 𝑋•,𝑇
with • ∈ {grad, curl, div}. These potentials have polynomial consistency properties, and enable the construction of discrete L2-inner products on DDR spaces that are also polynomially consistent. 3.1.1 Scalar potential on 𝑋𝑘
grad,𝑇
The scalar potential reconstruction 𝑃grad,𝑇𝑘+1 : 𝑋
𝑘
grad,𝑇 → P
𝑘+1
(𝑇 ) is such that, for all 𝑞𝑇 ∈ 𝑋𝑘
grad,𝑇, ∫ 𝑇 𝑃𝑘+1 grad,𝑇𝑞𝑇 div 𝒗 𝑇 = − ∫ 𝑇 G𝑘 𝑇𝑞 𝑇 ·𝒗𝑇 + Õ 𝐹∈ F𝑇 𝜔𝑇 𝐹 ∫ 𝐹 𝛾𝑘+1 𝐹 𝑞 𝐹(𝒗 𝑇 ·𝒏𝐹) ∀𝒗𝑇 ∈ Rc, 𝑘+2(𝑇 ), (3.1)
with 𝛾𝐹𝑘+1defined by (2.10). This relation defines 𝑃 𝑘+1
grad,𝑇𝑞𝑇 uniquely since div : R
c, 𝑘+2(𝑇 ) → P𝑘+1
Remark 3 (Validity of (3.1)). The definition (2.11) ofG𝑘
𝑇 and the identity div curl = 0 show that both
sides of (3.1) vanish when 𝒗𝑇 ∈ R 𝑘
(𝑇 ). Hence, (3.1) actually holds for any 𝒗𝑇 ∈ R 𝑘
(𝑇 ) ⊕ Rc, 𝑘+2(𝑇 ) =
P𝑘
(𝑇 ) + Rc, 𝑘+2(𝑇 ), the second equality following from Rc, 𝑘(𝑇 ) ⊂ Rc, 𝑘+2(𝑇 ) and the decomposition
(2.2).
Using the polynomial consistency propertiesG𝑇𝑘
𝐼𝑘
grad,𝑇𝑞 = grad 𝑞 and 𝛾
𝑘+1 𝐹
𝐼𝑘
grad,𝐹𝑞|𝐹 = 𝑞|𝐹,
valid for all 𝑞 ∈ P𝑘+1(𝑇 ) (see [10, Eqs. (3.13) and (3.11)]), the following polynomial consistency of 𝑃𝑘+1 grad,𝑇 is obtained: 𝑃𝑘+1 grad,𝑇 𝐼 𝑘 grad,𝑇𝑞 = 𝑞 ∀𝑞 ∈ P 𝑘+1 (𝑇 ). (3.2)
Moreover, applying (3.1) to 𝒗𝑇 ∈ Rc, 𝑘(𝑇 ) (which is possible since Rc, 𝑘(𝑇 ) ⊂ Rc, 𝑘+2(𝑇 )), using the
definition (2.11) ofG𝑇𝑘 with 𝒘𝑇 =𝒗𝑇, and recalling that div : R
c, 𝑘(𝑇 ) → P𝑘−1 (𝑇 ) is onto, we obtain 𝜋𝑘−1 P,𝑇 𝑃 𝑘+1 grad,𝑇𝑞𝑇 = 𝑞𝑇 ∀𝑞 𝑇 ∈ 𝑋𝑘 grad,𝑇. (3.3) 3.1.2 Vector potential on 𝑿𝑘 curl,𝑇
The vector potential reconstruction 𝑷𝑘curl,𝑇 : 𝑿
𝑘
curl,𝑇 → P
𝑘
(𝑇 ) is such that, for all 𝒗𝑇 ∈ 𝑿 𝑘 curl,𝑇, 𝑷𝑘 curl,𝑇𝒗𝑇 ≔ ℜ 𝑘 R,𝑇(𝑷 𝑘 curl, R,𝑇𝒗𝑇,𝒗 c R,𝑇), (3.4) where 𝑷𝑘curl, R,𝑇𝒗𝑇 ∈ R 𝑘
(𝑇 ) is defined, using the isomorphism curl : Gc, 𝑘+1(𝑇 ) → R𝑘
(𝑇 ) (see [10, Eq. (2.10)]), by ∫ 𝑇 𝑷𝑘 curl, R,𝑇𝒗𝑇 · curl 𝒘𝑇 = ∫ 𝑇 C𝑘 𝑇𝒗𝑇 ·𝒘𝑇 − Õ 𝐹∈ F𝑇 𝜔𝑇 𝐹 ∫ 𝐹 𝜸𝑘 t,𝐹𝒗𝐹· (𝒘𝑇 ×𝒏𝐹) ∀𝒘𝑇 ∈ G c, 𝑘+1(𝑇 ). (3.5) Remark 4 (Discrete integration by parts formula for 𝑷𝑘
curl,𝑇). Formula (3.5) can be extended to test
functions in the Nédélec space N𝑘+1(𝑇 ) defined by (2.3). To check it, simply notice that both sides vanish whenever 𝒘𝑇 ∈ G
𝑘
(𝑇 ) (use curl grad = 0 and the definition (2.14) ofC𝑘
𝑇). Since 𝝅 𝑘 R,𝑇 𝑷 𝑘 curl,𝑇𝒗𝑇 = 𝑷𝑘
curl, R,𝑇𝒗𝑇 (see (3.4) and (2.5)), we infer that
∫ 𝑇 𝑷𝑘 curl,𝑇𝒗𝑇 · curl 𝒛𝑇 = ∫ 𝑇 C𝑘 𝑇𝒗𝑇 ·𝒛𝑇 − Õ 𝐹∈ F𝑇 𝜔𝑇 𝐹 ∫ 𝐹 𝜸𝑘 t,𝐹𝒗𝐹· (𝒛𝑇 ×𝒏𝐹) ∀𝒛𝑇 ∈ N 𝑘+1 (𝑇 ). (3.6) Applying (3.6) to 𝒗𝑇 = 𝑰 𝑘 curl,𝑇𝒗 with 𝒗 ∈ P 𝑘
(𝑇 ), using the consistency properties 𝜸𝑘 t,𝐹 𝑰 𝑘 curl,𝐹𝒗 = 𝝅𝑘 P,𝐹𝒗t,𝐹 =𝒗t,𝐹 andC 𝑘 𝑇 𝑰 𝑘
curl,𝑇𝒗 = curl 𝒗 (see [10, Eqs. (3.22) and (3.26)]), and integrating by parts,
and since curl : N𝑘+1(𝑇 ) → R𝑘(𝑇 ) is onto (due to the isomorphism property [10, Eq. (2.10)]), we see that 𝝅𝑘R,𝑇 𝑷 𝑘 curl,𝑇 𝑰 𝑘 curl,𝑇𝒗 = 𝝅 𝑘
R,𝑇𝒗. The definition (3.4) and the property (2.5) of the recovery
operator also yield 𝝅c, 𝑘R,𝑇 𝑷 𝑘 curl,𝑇 𝑰 𝑘 curl,𝑇𝒗 = 𝝅 c, 𝑘 R,𝑇𝒗. As a consequence, 𝑷𝑘 curl,𝑇 𝑰 𝑘 curl,𝑇𝒗 = 𝒗 ∀𝒗 ∈ P 𝑘 (𝑇 ). (3.7)
3.1.3 Vector potential on 𝑿𝑘 div,𝑇
The vector potential reconstruction 𝑷𝑘div,𝑇 : 𝑿div,𝑇𝑘 → P𝑘(𝑇 ) is such that, for all 𝒘𝑇 ∈ 𝑿 𝑘 div,𝑇, 𝑷𝑘 div,𝑇𝒘𝑇 = ℜ 𝑘 G,𝑇(𝑷 𝑘 div, G,𝑇𝒘𝑇,𝒘 c G,𝑇), where 𝑷𝑘div, G,𝑇𝒘𝑇 ∈ G 𝑘 (𝑇 ) is defined by ∫ 𝑇 𝑷𝑘 div, G,𝑇𝒘𝑇 · grad 𝑟𝑇 = − ∫ 𝑇 𝐷𝑘 𝑇𝒘𝑇 𝑟𝑇 + Õ 𝐹∈ F𝑇 𝜔𝑇 𝐹 ∫ 𝐹 𝑤𝐹 𝑟𝑇 ∀𝑟𝑇 ∈ P0, 𝑘+1(𝑇 ). (3.9)
Remark 5 (Discrete integration by parts formula for 𝑷𝑘
div,𝑇). Observing that 𝑷 𝑘 div, G,𝑇 =𝝅 𝑘 G,𝑇𝑷 𝑘 div,𝑇 (use
(2.5)) and that (3.9) holds for any 𝑟𝑇 ∈ P 𝑘+1
(𝑇 ) (as can be proved taking 𝑟𝑇 constant in 𝑇 and observing
that both sides of this equation vanish due to the definition (2.15) of 𝐷𝑇𝑘), we infer
∫ 𝑇 𝑷𝑘 div,𝑇𝒘𝑇 · grad 𝑟𝑇 = − ∫ 𝑇 𝐷𝑘 𝑇𝒘𝑇 𝑟𝑇 + Õ 𝐹∈ F𝑇 𝜔𝑇 𝐹 ∫ 𝐹 𝑤𝐹 𝑟𝑇 ∀𝑟𝑇 ∈ P𝑘+1(𝑇 ). (3.10) Writing (3.10) for 𝒘𝑇 = 𝑰 𝑘 div,𝑇𝒘 with 𝒘 ∈ RT 𝑘+1 (𝑇 ), observing that 𝐷𝑘 𝑇 𝑰 𝑘 div,𝑇𝒘 = 𝜋 𝑘 P,𝑇(div 𝒘) =
div 𝒘 by [10, Eq. (3.36)] and 𝜋𝑘P,𝐹(𝒘|𝐹·𝒏𝐹) = 𝒘|𝐹·𝒏𝐹for all 𝐹 ∈ F𝑇 by [10, Eq. (A.4)], and integrating
by parts the right-hand side of the resulting expression, we infer 𝝅𝑘G,𝑇 𝑷 𝑘 div,𝑇 𝑰 𝑘 div,𝑇𝒘 = 𝝅 𝑘 G,𝑇𝒘; since 𝝅c, 𝑘 G,𝑇 𝑷 𝑘 div,𝑇 𝑰 𝑘 div,𝑇𝒘 = 𝝅 c, 𝑘 G,𝑇𝒘 by definition of 𝑷 𝑘 div,𝑇, 𝑰 𝑘
div,𝑇 and (2.5), we deduce that
𝑷𝑘 div,𝑇 𝑰 𝑘 div,𝑇𝒘 = 𝝅 𝑘 P,𝑇𝒘 ∀𝒘 ∈ RT 𝑘+1 (𝑇 ). (3.11)
Moreover, following similar arguments as in [10, Proposition 15] we get 𝝅𝑘−1 G,𝑇 𝑷 𝑘 div,𝑇𝒘𝑇 = 𝒘G,𝑇 and 𝝅 c, 𝑘 G,𝑇 𝑷 𝑘 div,𝑇𝒘𝑇 = 𝒘 c G,𝑇 ∀𝒘𝑇 ∈ 𝑿 𝑘 div,𝑇. (3.12) 3.1.4 Discrete L2-products
We now define discrete L2-inner products on the DDR spaces. These products are all constructed in a similar way: by assembling local contributions composed of a consistent term based on the potential reconstructions and a stabilisation term that provides a control of the polynomial components on the lower dimensional geometrical objects. Specifically, each L2-product (·, ·)grad,ℎ : 𝑋grad,ℎ𝑘 × 𝑋
𝑘
grad,ℎ → R,
(·, ·)curl,ℎ : 𝑿𝑘curl,ℎ×𝑿
𝑘
curl,ℎ → R, and (·, ·)div,ℎ : 𝑿 𝑘
div,ℎ ×𝑿 𝑘
div,ℎ → R is the sum over 𝑇 ∈ Tℎof its
where the symmetric, positive semidefinite stabilisation bilinear forms s•,𝑇, • ∈ {grad, curl, div}, are defined as follows: sgrad,𝑇(𝑟𝑇, 𝑞𝑇) ≔ Õ 𝐹∈ F𝑇 ℎ𝐹 ∫ 𝐹 𝑃𝑘+1 grad,𝑇𝑟𝑇 − 𝛾 𝑘+1 𝐹 𝑟𝐹 𝑃𝑘+1 grad,𝑇𝑞𝑇 − 𝛾𝑘+1 𝐹 𝑞𝐹 + Õ 𝐸∈ E𝑇 ℎ2 𝐸 ∫ 𝐸 𝑃𝑘+1 grad,𝑇𝑟𝐹 − 𝑟𝐸 𝑃𝑘+1 grad,𝑇𝑞𝐹 − 𝑞𝐸 , (3.14) scurl,𝑇(𝒘𝑇,𝒗𝑇) ≔ Õ 𝐹∈ F𝑇 ℎ𝐹 ∫ 𝐹 (𝑷𝑘 curl,𝑇𝒘𝑇)t,𝐹−𝜸 𝑘 t,𝐹𝒘𝐹 ·(𝑷𝑘 curl,𝑇𝒗𝑇)t,𝐹−𝜸 𝑘 t,𝐹𝒗𝐹 + Õ 𝐸∈ E𝑇 ℎ2 𝐸 ∫ 𝐸 𝑷𝑘 curl,𝑇𝒘𝐹 ·𝒕𝐸 − 𝑤𝐸 𝑷𝑘 curl,𝑇𝒗𝐹 ·𝒕𝐸 − 𝑣𝐸 , (3.15)
where we recall that the index t, 𝐹 denotes the tangential trace on 𝐹, and sdiv,𝑇(𝒘𝑇,𝒗𝑇) ≔ Õ 𝐹∈ F𝑇 ℎ𝐹 ∫ 𝐹 𝑷𝑘 div,𝑇𝒘𝑇 ·𝒏𝐹 − 𝑤𝐹 𝑷𝑘 div,𝑇𝒗𝑇 ·𝒏𝐹 − 𝑣𝐹 . (3.16)
These local stabilisation bilinear forms s•,𝑇 are polynomialy consistent, i.e., they vanish whenever one of their arguments is the interpolate of a polynomial of total degree ≤ 𝑘 + 1 for • = grad, or ≤ 𝑘 for • ∈ {curl, div}. Further consistency properties for interpolates of smooth functions are stated in Theorem 8
For • ∈ {grad, curl, div}, we denote by k·k•,𝑇 the norm on 𝑋•,𝑇𝑘 induced by the corresponding
local discrete L2-product (·, ·)•,𝑇, and by k·k•,ℎthe norm on 𝑋𝑘•,ℎcorresponding to the global discrete
L2-product (·, ·)•,ℎ.
3.2 Primal consistency
In this section we state consistency results for the discrete potentials, vector calculus operators, and stabilisation bilinear forms. Because of the nature of the interpolator on 𝑿curl,𝑇𝑘 (which requires higher
regularity of functions), we introduce the following notation: For 𝑇 ∈ Tℎand 𝒗 ∈ Hmax( 𝑘+1,2)(𝑇 ),
|𝒗|H( 𝑘+1,2)(𝑇 ) ≔
|𝒗|H1(𝑇 )+ ℎ𝑇|𝒗|H2(𝑇 ) if 𝑘 = 0,
|𝒗|H𝑘+1
(𝑇 ) if 𝑘 ≥ 1. (3.17)
The corresponding global broken seminorm |·|H( 𝑘+1,2)( Tℎ) is such that, for all 𝒗 ∈ H
( 𝑘+1,2)(T ℎ), |𝒗|H( 𝑘+1,2)( Tℎ) ≔ Í 𝑇∈ Tℎ|𝒗| 2 H( 𝑘+1,2)(𝑇 ) 1/2
. The proofs of the following theorems are postponed to Section 4.3.
Theorem 6 (Consistency of the potential reconstructions). It holds, for all 𝑇 ∈ Tℎ,
k𝑃𝑘+1 grad,𝑇 𝐼 𝑘 grad,𝑇𝑞 − 𝑞kL2(𝑇 ) . ℎ 𝑘+2 𝑇 |𝑞|H𝑘+2(𝑇 ) ∀𝑞 ∈ H 𝑘+2 (𝑇 ), (3.18) k𝑷𝑘 curl,𝑇 𝑰 𝑘 curl,𝑇𝒗 − 𝒗kL2(𝑇 ) . ℎ 𝑘+1 𝑇 |𝒗|H( 𝑘+1,2)(𝑇 ) ∀𝒗 ∈ H max( 𝑘+1,2)(𝑇 ), (3.19) k𝑷𝑘 div,𝑇 𝑰 𝑘 div,𝑇𝒘 − 𝒘kL2(𝑇 ) . ℎ 𝑘+1 𝑇 |𝒘|H𝑘+1(𝑇 ) ∀𝒘 ∈ H 𝑘+1 (𝑇 ). (3.20)
Theorem 7 (Primal consistency of the discrete vector calculus operators). It holds, for all 𝑇 ∈ Tℎ,
Theorem 8 (Consistency of stabilisation forms). For all 𝑇 ∈ Tℎ, the stabilisation forms defined by
(3.14)–(3.16) satisfy the following consistency properties: sgrad,𝑇(𝐼 𝑘 grad,𝑇𝑞, 𝐼 𝑘 grad,𝑇𝑞) 1 2 . ℎ𝑘+2 𝑇 |𝑞|H𝑘+2(𝑇 ) ∀𝑞 ∈ H 𝑘+2 (𝑇 ), (3.24) scurl,𝑇(𝑰 𝑘 curl,𝑇𝒗, 𝑰 𝑘 curl,𝑇𝒗) 1 2 . ℎ 𝑘+1 𝑇 |𝒗|H( 𝑘+1,2)(𝑇 ) ∀𝒗 ∈ H max( 𝑘+1,2)(𝑇 ), (3.25) sdiv,𝑇(𝑰 𝑘 div,𝑇𝒘, 𝑰 𝑘 div,𝑇𝒘) 1 2 . ℎ𝑇𝑘+1|𝒘|H𝑘+1 (𝑇 ) ∀𝒘 ∈ H 𝑘+1 (𝑇 ). (3.26) 3.3 Adjoint consistency
Whenever a (formal) integration by parts is used to write the weak formulation of a PDE problem underpinning its discretisation, a form of adjoint consistency is required in the convergence analysis. We state here the adjoint consistency of the operators in the DDR sequence (2.16). Since this sequence does not incorporate boundary conditions, the corresponding adjoint consistency will be based on essential boundary conditions. The regularity requirements will be expressed in terms of the broken Sobolev spaces and norms such that, for any ℓ ≥ 1,
Hℓ(Tℎ) ≔ 𝑔∈ L2(Ω) : 𝑔|𝑇 ∈ Hℓ(𝑇 ) for all 𝑇 ∈ Tℎ and |𝑔|Hℓ( T ℎ) ≔ Õ 𝑇∈ Tℎ |𝑔|𝑇|2 Hℓ(𝑇 ) !1 2 .
The corresponding seminorms for vector-valued functions is denoted, as usual, using boldface letters. We denote in what follows by H10(Ω), H0(div; Ω), and H0(curl; Ω) the subspaces of H1(Ω), H(div; Ω),
and H(curl; Ω) spanned by functions whose trace, normal trace, and tangential trace vanish on 𝜕Ω, respectively.
Theorem 9 (Adjoint consistency for the gradient). Let ˜Ediv,ℎ : C0(Ω) ∩ H
0(div; Ω) × 𝑋 𝑘
grad,ℎ → R
be such that, for all 𝑞
ℎ ∈ 𝑋𝑘 grad,ℎ, ˜ Ediv,ℎ(𝒗, 𝑞 ℎ ) ≔ Õ 𝑇∈ Tℎ (𝑰𝑘 curl,𝑇𝒗|𝑇,𝑮 𝑘 𝑇𝑞𝑇)curl,𝑇 + ∫ 𝑇 div 𝒗 𝑃𝑘grad,𝑇+1 𝑞𝑇 .
Then, it holds, for all 𝒗 ∈ C0(Ω) ∩ H
0(div; Ω) such that 𝒗 ∈ Hmax( 𝑘+1,2)(Tℎ) and all 𝑞 ℎ ∈ 𝑋 𝑘 grad,ℎ, | ˜Ediv,ℎ(𝒗, 𝑞 ℎ) | . ℎ 𝑘+1 |𝒗|H( 𝑘+1,2)( Tℎ)k𝑮 𝑘 ℎ 𝑞 ℎ kcurl,ℎ, (3.27)
Proof. See Section 4.4.1.
Theorem 10 (Adjoint consistency for the curl). Let ˜Ecurl,ℎ: C0(Ω) ∩ H0(curl; Ω) × 𝑿 𝑘
curl,ℎ → R be
such that, for all (𝒘, 𝒗ℎ) ∈ C0(Ω) ∩ H
0(curl; Ω) × 𝑿 𝑘 curl,ℎ, ˜ Ecurl,ℎ(𝒘, 𝒗ℎ) ≔ Õ 𝑇∈ Tℎ (𝑰𝑘 div,𝑇𝒘|𝑇,𝑪 𝑘 𝑇𝒗𝑇)div,𝑇 − ∫ 𝑇 curl 𝒘 · 𝑷𝑘 curl,𝑇𝒗𝑇 . (3.28)
Then, for all 𝒘 ∈ C0(Ω) ∩ H
0(curl; Ω) such that 𝒘 ∈ H 𝑘+2 (Tℎ) and all 𝒗ℎ ∈ 𝑿 𝑘 curl,ℎ, | ˜Ecurl,ℎ(𝒘, 𝒗ℎ) | . ℎ 𝑘+1 |𝒘|H𝑘+1 ( Tℎ) + |𝒘|H 𝑘+2 ( Tℎ) k𝒗ℎkcurl,ℎ+ k𝑪 𝑘 ℎ𝒗ℎkdiv,ℎ . (3.29)
Theorem 11 (Adjoint consistency for the divergence). Let ˜Egrad,ℎ : C0(Ω) ∩ 𝐻01(Ω) × 𝑿div,ℎ𝑘 → R be such that, for all (𝑞, 𝒗ℎ) ∈ C0(Ω) ∩ 𝐻1
0(Ω) × 𝑿 𝑘 div,ℎ, ˜ Egrad,ℎ(𝑞,𝒗ℎ) ≔ ∫ Ω 𝜋𝑘 P,ℎ𝑞 𝐷 𝑘 ℎ𝒗ℎ+ Õ 𝑇∈ T ℎ ∫ Ω grad 𝑞 · 𝑷𝑘 div,𝑇𝒗𝑇 . (3.30)
Then, for all 𝑞 ∈ C0(Ω) ∩ 𝐻1
0(Ω) such that 𝑞 ∈ H 𝑘+2 (Tℎ) and all 𝒗ℎ ∈𝑿 𝑘 div,ℎ, | ˜Egrad,ℎ(𝑞,𝒗ℎ) | . ℎ 𝑘+1 |𝑞| H𝑘+2( T ℎ)k𝒗ℎkdiv,ℎ . (3.31)
Proof. See Section 4.4.3.
4
Proofs of the consistency results
In this section, after establishing some preliminary results, we prove the primal and adjoint consistency results stated in Section 3.
4.1 Component norms and bounds on potentials
We recall the definition of the component L2-norm on 𝑋grad,𝑇𝑘 , 𝑿
𝑘
curl,𝑇 and 𝑿
𝑘
div,𝑇 introduced in [10,
Section 4.1], and which correspond to the L2-norms of the components of the vectors of polynomials, with scaling appropriate to the dimensions of the geometrical objects on which these components are defined: |||𝑞 𝑇|||grad,𝑇 ≔ k𝑞𝑇k2 L2(𝑇 ) + Õ 𝐹∈ F𝑇 ℎ𝐹|||𝑞 𝐹||| 2 grad,𝐹 1/2 for all 𝑞𝑇 ∈ 𝑋 𝑘 grad,𝑇, where |||𝑞𝐹 |||grad,𝐹 ≔ k𝑞𝐹k2 L2(𝐹 ) + Õ 𝐸∈ E𝐹 ℎ𝐸k𝑞𝐸k2 L2(𝐸) 1/2 for all 𝐹 ∈ F𝑇, |||𝒗𝑇|||curl,𝑇 ≔ k𝒗R,𝑇k2 L2(𝑇 )+ k𝒗 c R,𝑇k2L2(𝑇 )+ Õ 𝐹∈ F𝑇 ℎ𝐹|||𝒗𝐹|||2curl,𝐹 1/2 for all 𝒗𝑇 ∈𝑿 𝑘 curl,𝑇, where |||𝒗𝐹|||curl,𝐹 ≔ k𝒗R,𝐹k2L2(𝐹 ) + k𝒗cR,𝐹k2L2(𝐹 )+ Õ 𝐸∈ E𝐹 ℎ𝐸k𝑣𝐸k2 L2(𝐸) 1/2 for all 𝐹 ∈ F𝑇, (4.1) and |||𝒘𝑇|||div,𝑇 ≔ k𝒘G,𝑇k2L2(𝑇 ) + k𝒘 c G,𝑇k2L2(𝑇 ) + Õ 𝐹∈ F𝑇 ℎ𝐹k𝑤𝐹k2 L2(𝐹 ) 1/2 for all 𝒘𝑇 ∈ 𝑿 𝑘 div,𝑇.
The next proposition follows from (2.7) and [10, Lemma 31], in a similar way as in the proof of [11, Proposition 13].
Proposition 12 (Boundedness of local potentials). It holds, for all 𝑇 ∈ Tℎand all 𝐹 ∈ F𝑇,
k𝛾𝑘+1 𝐹 𝑞 𝐹 k L2(𝐹 ) . |||𝑞𝐹|||grad,𝐹 and k𝑃 𝑘+1 grad,𝑇𝑞𝑇 k L2(𝑇 ) . |||𝑞𝑇|||grad,𝑇 ∀𝑞 𝑇 ∈ 𝑋𝑘 grad,𝑇, (4.2) k𝜸𝑘 t,𝐹𝒗𝐹kL2(𝐹 ) . |||𝒗𝐹|||curl,𝐹 and k𝑷 𝑘 curl,𝑇𝒗𝑇kL2(𝑇 ) . |||𝒗𝑇|||curl,𝑇 ∀𝒗𝑇 ∈ 𝑿 𝑘 curl,𝑇, (4.3) k𝑷𝑘 div,𝑇𝒘𝑇kL2(𝑇 ) . |||𝒘𝑇|||div,𝑇 ∀𝒘𝑇 ∈ 𝑿 𝑘 div,𝑇. (4.4)
For • ∈ {grad, curl, div}, using triangle inequalities as in [11, Proposition 14], invoking the bounds of Proposition 12, the projection properties (3.3), (3.8) (and similar for 𝜸t,𝐹𝑘 , see [10, Proposition 15]) or (3.12), and recalling (2.7), we have the norm equivalence: For all 𝑇 ∈ Tℎ
|||𝑧
𝑇|||•,𝑇 ' k𝑧𝑇k•,𝑇 ∀𝑧𝑇 ∈ 𝑋 𝑘
Lemma 13 (Boundedness of local interpolators). It holds, for all 𝑇 ∈ Tℎ, |||𝐼𝑘 grad,𝑇𝑞|||grad,𝑇 . k𝑞kL2(𝑇 )+ ℎ𝑇|𝑞| H1(𝑇 )+ ℎ2𝑇|𝑞|H2(𝑇 ) ∀𝑞 ∈ H 2(𝑇 ), (4.6) |||𝑰𝑘 curl,𝑇𝒗|||curl,𝑇 . k𝒗kL2(𝑇 )+ ℎ𝑇|𝒗|H1(𝑇 )+ ℎ𝑇2|𝒗|H2(𝑇 ) ∀𝒗 ∈ H 2(𝑇 ), (4.7) |||𝑰𝑘 div,𝑇𝒘|||div,𝑇 . k𝒘kL2(𝑇 ) + ℎ𝑇|𝒘|H1(𝑇 ) ∀𝒘 ∈ H1(𝑇 ). (4.8)
Proof. The definition of 𝐼𝑘
grad,𝑇 (see (2.8)) clearly shows that |||𝐼
𝑘 grad,𝑇𝑞|||grad,𝑇 . |𝑇 | 1/2 max𝑇 |𝑞|. By [12, Eq. (5.110)], it holds max 𝑇 |𝑞| . |𝑇 | −1 2 2 Õ 𝑟=0 ℎ𝑟 𝑇|𝑞|H𝑟(𝑇 ),
which concludes the proof of (4.6). The estimate (4.7) is obtained the same way. As for (4.8), by the continuous trace inequality of [12, Lemma 1.31], we have
k𝜋𝑘 P,𝐹(𝒘 · 𝒏𝐹) k L2(𝐹 ) ≤ k𝒘 kL2(𝐹 ) . ℎ −1 2 𝐹 k𝒘 kL2(𝑇 )+ ℎ 1 2 𝐹|𝒘|H1(𝑇 ) .
Using this bound in the definition (2.9) of 𝑰div,𝑇𝑘 yields (4.8).
4.2 Links between discrete vector potentials and vector calculus operators
Proposition 14 (Link between discrete vector potentials and vector calculus operators). For all 𝑇 ∈ Tℎ,
it holds 𝑷𝑘 curl,𝑇 𝑮 𝑘 𝑇 𝑞 𝑇 =G𝑘 𝑇𝑞 𝑇 ∀𝑞 𝑇 ∈ 𝑋𝑘 grad,𝑇, (4.9) 𝑷𝑘 div,𝑇 𝑪 𝑘 𝑇𝒗𝑇 =C 𝑘 𝑇𝒗𝑇 ∀𝒗𝑇 ∈ 𝑿 𝑘 curl,𝑇. (4.10)
Proof. 1. Proof of (4.9). By the second projection property in (3.8), we have 𝝅c, 𝑘 R,𝑇 𝑷 𝑘 curl,𝑇 𝑮 𝑘 𝑇𝑞𝑇 = 𝝅c, 𝑘 R,𝑇 G 𝑘 𝑇 𝑞 𝑇
. To infer the conclusion, it then suffices to prove that 𝝅𝑘 R,𝑇 𝑷 𝑘 curl,𝑇 𝑮 𝑘 𝑇𝑞𝑇 = 𝝅 𝑘 R,𝑇 G 𝑘 𝑇𝑞𝑇 (4.11) and invoke (2.6). To prove (4.11), we take 𝒛𝑇 ∈ N
𝑘+1
(𝑇 ) and apply (3.6) with 𝒗𝑇 =𝑮 𝑘
𝑇𝑞𝑇. Using the
inclusion Im 𝑮𝑇𝑘 ⊂ KerC 𝑘
𝑇 (see [10, Remark 21]) and the relation 𝜸 𝑘 t,𝐹 𝑮 𝑘 𝐹 𝑞 𝐹 = G𝑘 𝐹 𝑞
𝐹 valid for all
𝐹 ∈ F𝑇 (see [10, Proposition 15]), we obtain
∫ 𝑇 𝑷𝑘 curl,𝑇 𝑮 𝑘 𝑇𝑞𝑇 · curl 𝒛𝑇 = − Õ 𝐹∈ F𝑇 𝜔𝑇 𝐹 ∫ 𝐹 G𝑘 𝐹𝑞𝐹 · (𝒛𝑇 ×𝒏𝐹) = ∫ 𝑇 G𝑘 𝑇𝑞𝑇 · curl 𝒛𝑇,
the conclusion following from the link between element and face gradient, see [10, Proposition 11]. By the isomorphism curl : Gc, 𝑘+1(𝑇 ) → R𝑘(𝑇 ) of [10, Eq. (2.10)] and since Gc, 𝑘+1(𝑇 ) ⊂ N𝑘+1(𝑇 ), this establishes (4.11) and concludes the proof of (4.9).
2. Proof of (4.10). The second projection property in (3.12) ensures that 𝝅c, 𝑘G,𝑇 𝑷 𝑘 div,𝑇 𝑪 𝑘 𝑇𝒗𝑇 = 𝝅c, 𝑘 G,𝑇 C 𝑘 𝑇𝒗𝑇
. As before, it therefore remains to analyse the projections on G𝑘(𝑇 ). Apply (3.10) to 𝒘𝑇 =𝑪
𝑘
𝑇𝒗𝑇 and a generic 𝑟𝑇 ∈ P 𝑘+1
(𝑇 ), and use the inclusion Im 𝑪𝑘
𝑇 ⊂ Ker 𝐷 𝑘 𝑇 (see [10, Proposition 17]) to get ∫ 𝑇 𝑷𝑘 div,𝑇 𝑪 𝑘 𝑇𝒗𝑇 · grad 𝑟𝑇 = Õ 𝐹∈ F𝑇 𝜔𝑇 𝐹 ∫ 𝐹 𝐶𝑘 𝐹𝒗𝐹 𝑟𝑇 = ∫ 𝑇 C𝑘 𝑇𝒗𝑇 · grad 𝑟𝑇 ,
where the conclusion is obtained applying the link between element and face curls of [10, Proposition 16]. This establishes that 𝝅𝑘G,𝑇 𝑷
Corollary 15 (Bound on discrete gradients and curl). For all 𝐹 ∈ Fℎ, it holds kG𝑘 𝐹𝑞 𝐹 k2 L2(𝐹 ) + Õ 𝐸∈ E𝐹 ℎ𝐸k𝐺 𝑘 𝐸𝑞𝐸k 2 L2(𝐸) . |||𝑮 𝑘 𝐹𝑞 𝐹 |||curl,𝐹 ∀𝑞 𝐹 ∈ 𝑋𝑘 grad,𝐹. (4.12)
For all 𝑇 ∈ Tℎ, it holds
kG𝑘 𝑇𝑞𝑇k 2 L2(𝑇 )+ Õ 𝐹∈ F𝑇 ℎ𝐹kG 𝑘 𝐹𝑞𝐹k 2 L2(𝐹 ) + Õ 𝐸∈ E𝑇 ℎ2 𝐸k𝐺 𝑘 𝐸𝑞𝐸k 2 L2(𝐸) . |||𝑮 𝑘 𝑇𝑞𝑇|||curl,𝑇 ∀𝑞 𝑇 ∈ 𝑋𝑘 grad,𝑇, (4.13) kC𝑘 𝑇𝒗𝑇k 2 L2(𝑇 )+ Õ 𝐹∈ F𝑇 ℎ𝐹k𝐶𝑘 𝐹𝒗𝐹k 2 L2(𝐹 ) . |||𝑪 𝑘 𝑇𝒗𝑇|||div,𝑇 ∀𝒗𝑇 ∈ 𝑿 𝑘 curl,𝑇. (4.14)
Proof. The definitions of |||·|||curl,𝐹, |||·|||curl,𝑇, 𝑮𝑘𝐹 and 𝑮 𝑘
𝑇 show that the edge gradient contributions in
the left-hand sides of (4.12) and (4.13) are bounded by the corresponding right-hand sides. To bound the face and element gradient contributions in the left-hand sides of (4.12) and (4.13), simply apply (4.3) to 𝒗𝑇 =𝑮 𝑘 𝑇𝑞𝑇, use 𝜸 𝑘 t,𝐹 ◦𝑮 𝑘 𝐹 =G 𝑘
𝐹 (see [10, Proposition 15]) and (4.9). The estimate (4.14) is
established in a similar way.
4.3 Primal consistency
Proof of Theorem 6. Let us start with (3.18). Since H2(𝑇 ) ⊂ C0(𝑇 ), the mapping 𝑃𝑘+1
grad,𝑇 ◦ 𝐼
𝑘
grad,𝑇 :
H2(𝑇 ) → P𝑘+1(𝑇 ) is well-defined and, owing to (3.2), it is a projector. Moreover, combining (4.6) and (4.2), it satisfies the L2(𝑇 )-boundedness
k𝑃𝑘+1 grad,𝑇 𝐼 𝑘 grad,𝑇𝑞 kL2(𝑇 ) . k𝑞kL2(𝑇 )+ ℎ𝑇|𝑞| H1(𝑇 )+ ℎ2𝑇|𝑞|H2(𝑇 ) ∀𝑞 ∈ H 2(𝑇 ).
The approximation property (3.18) is thus a direct consequence of [12, Lemma 1.43]. The proofs of (3.19) (for 𝑘 ≥ 1) and (3.20) are similar, using the fact that the considered operators are projectors onto P𝑘(𝑇 ) (see (3.7) and (3.11)) and invoking Proposition 12 and Lemma 13 to establish their L2 -boundedness. In the case 𝑘 = 0, since 𝑷0curl,𝑇 ◦𝑰0curl,𝑇 requires the H2-regularity of its argument, with
2 > 𝑘 + 1, (3.19) cannot be deduced directly from [12, Lemma 1.43]. However, using the bounds (4.3) and (4.7) a direct proof can be done by introducing 𝝅0P,𝑇𝒗 = 𝑷0curl,𝑇 𝑰0curl,𝑇𝝅0P,𝑇𝒗
: k𝑷0
curl,𝑇 𝑰0curl,𝑇𝒗 − 𝒗kL2(𝑇 ) ≤ k𝑷0curl,𝑇 𝑰0curl,𝑇(𝒗 − 𝝅0P,𝑇𝒗) kL2(𝑇 )+ k𝝅0P,𝑇𝒗 − 𝒗 kL2(𝑇 )
. k𝒗 − 𝝅0
P,𝑇𝒗 kL2(𝑇 )+ ℎ𝑇|𝒗 − 𝝅0P,𝑇𝒗|H1(𝑇 )+ ℎ2𝑇|𝒗 − 𝝅 0
P,𝑇𝒗|H2(𝑇 ),
and (3.19) follows using the approximation properties of 𝝅0P,𝑇, the fact that the H1(𝑇 )- and H2(𝑇
)-seminorms of 𝝅0P,𝑇𝒗 vanish, and the definition (3.17) of |·|H( 𝑘+1,2)(𝑇 ).
Proof of Theorem 7. Let us prove (3.21). For any 𝑞𝑇
∈ 𝑋𝑘
grad,𝑇, taking 𝒘𝑇 = G 𝑘
𝑇𝑞𝑇 in (2.11) and
using Cauchy–Schwarz inequalities along with discrete inverse and trace inequalities, it is inferred, after simplification, kG𝑘 𝑇𝑞 𝑇 kL2(𝑇 ) . ℎ𝑇−1k𝑞𝑇k L2(𝑇 )+ Õ 𝐹∈ F𝑇 ℎ−1/2 𝐹 k𝛾 𝑘+1 𝐹 𝑞 𝐹 k L2(𝐹 ) . ℎ−1𝑇 |||𝑞|||grad,𝑇,
where the conclusion follows from the estimate on 𝛾𝐹𝑘+1𝑞𝐹 in (4.2) and from the definition of |||·|||grad,𝑇.
As a result, for any 𝑟 ∈ H2(𝑇 ), making 𝑞𝑇 = 𝐼𝑘grad,𝑇𝑟 and invoking (4.6), we infer
kG𝑘 𝑇 𝐼
𝑘
grad,𝑇𝑟 kL2(𝑇 ) . ℎ𝑇−1k𝑟 kL2(𝑇 )+ |𝑟 |H1(𝑇 )+ ℎ𝑇|𝑟 |
Letting now 𝑞 ∈ H𝑘+2(𝑇 ), we use the polynomial consistency [10, Eq. (3.13)] of G𝑇𝑘 followed by a
triangle inequality to write kG𝑘 𝑇 𝐼 𝑘 grad,𝑇𝑞 − grad 𝑞kL2(𝑇 ) ≤ kG 𝑘 𝑇 𝐼𝑘 grad,𝑇 𝑞− 𝜋 𝑘+1 P,𝑇𝑞 kL2(𝑇 )+ k grad 𝜋 𝑘+1 P,𝑇𝑞− 𝑞 kL2(𝑇 )
and conclude using (4.15) with 𝑟 = 𝑞 − 𝜋𝑘P,𝑇+1𝑞for the first term in the right-hand side followed by the
approximation properties of 𝜋𝑘P,𝑇+1 (see [12, Theorem 1.45]).
To prove (3.22), we notice thatC𝑇𝑘 𝑰 𝑘 curl,𝑇𝒗 = 𝑷 𝑘 div,𝑇𝑪 𝑘 𝑇 𝑰 𝑘 curl,𝑇𝒗 = 𝑷 𝑘 div,𝑇 𝑰 𝑘 div,𝑇 curl 𝒗 owing to (4.10) along with the commutation property [10, Eq. (3.35)], and conclude using the approximation properties (3.20) with 𝒘 = curl 𝒗.
Finally, (3.23) is a straightforward consequence of the commutation property 𝐷𝑇𝑘 𝑰 𝑘 div,𝑇𝒘
= 𝜋𝑘
P,𝑇(div 𝒘) stated in [10, Eq. (3.36)] together with [12, Theorem 1.45].
Remark 16 (Alternative proof of (3.21)). When 𝑞 ∈ C1(𝑇 ) is such that grad 𝑞 ∈ Hmax( 𝑘+1,2)(𝑇 ), the
proof of (3.21) can be done following similar arguments as for (3.22), i.e., we writeG𝑇𝑘
𝐼𝑘 grad,𝑇𝑞 = 𝑷𝑘 curl,𝑇𝑮 𝑘 𝑇 𝐼 𝑘 grad,𝑇𝑞 = 𝑷𝑘 curl,𝑇 𝑰 𝑘 curl,𝑇 grad 𝑞
using (4.9) followed by [10, Eq. (3.34)], and con-clude using the approximation properties (3.19) with 𝒗 = grad 𝑞. This argument, however, requires additional regularity on 𝑞 with respect to the one used above.
Proof of Theorem 8. We only prove (3.25), the other consistency properties being established in a similar way. Let 𝒗 ∈ Hmax( 𝑘+1,2)(𝑇 ). By the polynomial consistency [10, Eq. (3.22)] of 𝜸t,𝐹𝑘 and (3.7) of 𝑷𝑘curl,𝑇,
it is easily checked that, for all 𝒛𝑇 ∈ P 𝑘
(𝑇 ) and all 𝒘𝑇 ∈ 𝑿 𝑘
curl,𝑇, it holds scurl,𝑇(𝑰
𝑘
curl,𝑇𝒛𝑇,𝒘𝑇) = 0.
Applying this with 𝒛𝑇 =𝝅 𝑘 P,𝑇𝒗 we infer scurl,𝑇(𝑰 𝑘 curl,𝑇𝒗, 𝑰 𝑘 curl,𝑇𝒗) = scurl,𝑇(𝑰 𝑘 curl,𝑇(𝒗−𝝅 𝑘 P,𝑇𝒗), 𝑰 𝑘 curl,𝑇(𝒗−𝝅 𝑘 P,𝑇𝒗)) . |||𝑰 𝑘 curl,𝑇(𝒗−𝝅 𝑘 P,𝑇𝒗)|||2curl,𝑇,
the conclusion following from the definition of k·kcurl,𝑇 and the norm equivalence (4.5). Invoking then (4.7) we infer scurl,𝑇(𝑰 𝑘 curl,𝑇𝒗, 𝑰 𝑘 curl,𝑇𝒗) 1 2 . k𝒗 − 𝝅𝑘 P,𝑇𝒗 kL2(𝑇 )+ ℎ𝑇|𝒗 − 𝝅 𝑘 P,𝑇𝒗|H1(𝑇 )+ ℎ2𝑇|𝒗 − 𝝅 𝑘 P,𝑇𝒗|H2(𝑇 )
and the estimate (3.25) follows from the approximation properties of 𝝅𝑘P,𝑇, see [12, Theorem 1.45],
and the definition (3.17) of |·|H( 𝑘+1,2)(𝑇 ), using in the case 𝑘 = 0 the same arguments as in the proof of
Theorem 6.
4.4 Adjoint consistency
4.4.1 Adjoint consistency for the gradient Lemma 17 (Estimates on local H1
-seminorms of potentials). For all 𝐹 ∈ Fℎ and all 𝑞 𝐹 ∈ 𝑋𝑘 grad,𝐹, it holds k grad 𝛾𝑘+1 𝐹 𝑞 𝐹 k2 L2(𝐹 ) + Õ 𝐸∈ E𝐹 ℎ−1 𝐸 k𝛾 𝑘+1 𝐹 𝑞 𝐹 − 𝑞𝐸k2 L2(𝐸) . |||𝑮 𝑘 𝐹 𝑞 𝐹 |||2 curl,𝐹. (4.16)
Proof. The proof follows arguments similar to [10, Lemma 29]. 1. Proof of (4.16). Let 𝑞𝐹
∈ 𝑋𝑘
grad,𝐹 and define 𝐴𝑞 , 𝜕𝐹 ∈ R as the average of 𝑞E𝐹 over 𝜕𝐹. Introducing
𝐴𝑞 , 𝜕𝐹 = 𝛾𝑘+1
𝐹
𝐼𝑘
grad,𝐹𝐴𝑞 , 𝜕𝐹
(see [10, Eq. (3.11)]), using ℎ𝐸 ' ℎ𝐹 and card(E𝐹) . 1, and invoking a
discrete trace inequality on 𝛾𝐹𝑘+1
𝑞 𝐹 − 𝐴𝑞 , 𝜕𝐹 , we have Õ 𝐸∈ E𝐹 ℎ−1 𝐸 k𝛾 𝑘+1 𝐹 𝑞𝐹− 𝑞𝐸k 2 L2(𝐸) . Õ 𝐸∈ E𝐹 ℎ−1 𝐸 k𝑞𝐸− 𝐴𝑞 , 𝜕𝐹k2 L2(𝐸)+ ℎ−2𝐹 k𝛾 𝑘+1 𝐹 𝑞𝐹− 𝐴𝑞 , 𝜕𝐹 k 2 L2(𝐹 ). (4.18)
Since 𝑞E𝐹 is continuous, recalling that 𝑞𝐸 = (𝑞Eℎ)|𝐸 for all 𝐸 ∈ E𝐹 and using a Poincaré–Wirtinger
inequality along 𝜕𝐹 followed by the definition (4.1) of |||·|||curl,𝐹 yields Õ 𝐸∈ E𝐹 ℎ−1 𝐸 k𝑞𝐸 − 𝐴𝑞 , 𝜕𝐹k 2 L2(𝐸) . ℎ𝐹 Õ 𝐸∈ E𝐹 k𝐺𝑘 𝐸𝑞𝐸k 2 L2(𝐸) . |||𝑮 𝑘 𝐹𝑞𝐹||| 2 curl,𝐹. (4.19)
We now turn to the second term in (4.18). Select 𝒗𝐹 ∈ Rc, 𝑘+2(𝐹) such that div 𝒗𝐹 = 𝛾 𝑘+1 𝐹 𝑞𝐹 − 𝐼 𝑘 grad,𝐹𝐴𝑞 , 𝜕𝐹 . By the L2-estimate on 𝒗𝐹 coming from [10, Lemma 31], the discrete trace inequality of [12, Lemma
1.32], and the consistency property [10, Eq. (3.10)] ofG𝑘
𝐹, we have k𝒗𝐹k L2(𝐹 ) + Õ 𝐸∈ E𝐹 ℎ𝐸k𝒗𝐹k2 L2(𝐸) !1 2 . ℎ𝐹k𝛾 𝑘+1 𝐹 𝑞𝐹 − 𝐼 𝑘 grad,𝐹𝐴𝑞 , 𝜕𝐹 k L2(𝐹 ), G𝑘 𝐹 𝑞 𝐹 − 𝐼𝑘 grad,𝐹𝐴𝑞 , 𝜕𝐹 =G 𝑘 𝐹𝑞 𝐹 .
Hence, applying the definition (2.10) of 𝛾𝐹𝑘+1to 𝑞𝐹− 𝐼 𝑘
grad,𝐹𝐴𝑞 , 𝜕𝐹 ∈ 𝑋 𝑘
grad,𝐹, taking 𝒗𝐹 above as a test
function, and using Cauchy–Schwarz inequalities, we obtain k𝛾𝑘+1 𝐹 𝑞 𝐹 − 𝐼𝑘 grad,𝐹𝐴𝑞 , 𝜕𝐹 k2 L2(𝐹 ) . ℎ𝐹kG 𝑘 𝐹𝑞 𝐹 kL2(𝐹 )k𝛾 𝑘+1 𝐹 𝑞 𝐹 − 𝐼𝑘 grad,𝐹𝐴𝑞 , 𝜕𝐹 k L2(𝐹 ) + Õ 𝐸∈ E𝐹 ℎ−1 𝐸 k𝑞𝐸 − 𝐴𝑞 , 𝜕𝐹k2 L2(𝐸) !1 2 ℎ𝐹k𝛾𝑘+1 𝐹 𝑞 𝐹 − 𝐼𝑘 grad,𝐹𝐴𝑞 , 𝜕𝐹 k L2(𝐹 ).
Simplifying and recalling (4.12) and (4.19), we infer k𝛾𝐹𝑘+1
𝑞 𝐹 − 𝐴 𝑞 , 𝜕𝐹 k L2(𝐹 ) . ℎ𝐹|||𝑮 𝑘 𝐹𝑞𝐹|||curl,𝐹
which, plugged together with (4.19) into (4.18), gives the following estimate on the second term in the left-hand side of (4.16): Õ 𝐸∈ E𝐹 ℎ−1 𝐸 k𝛾 𝑘+1 𝐹 𝑞𝐹 − 𝑞𝐸k 2 L2(𝐸) . |||𝑮 𝑘 𝐹𝑞𝐹||| 2 curl,𝐹. (4.20)
Integrating by parts the definition (2.10) of 𝛾𝐹𝑘+1applied to 𝒗𝐹 ∈ P 𝑘
(𝐹) (see Remark 1), we have ∫ 𝐹 grad𝐹𝛾 𝑘+1 𝐹 𝑞𝐹 ·𝒗𝐹 = ∫ 𝐹 G𝑘 𝐹𝑞𝐹 ·𝒗𝐹 + Õ 𝐸∈ E𝐹 𝜔𝐹 𝐸 ∫ 𝐸 (𝛾𝑘+1 𝐹 𝑞𝐹 − 𝑞E𝐹) (𝒗𝐹 ·𝒏𝐹 𝐸). Making 𝒗𝐹 = grad𝐹 𝛾 𝑘+1
𝐹 𝑞𝐹, using Cauchy–Schwarz inequalities, (4.12), a discrete trace inequality,
and (4.20) then yields the bound on the first term in the left-hand side of (4.16).
denote the average over 𝜕𝑇 of the piecewise polynomial function defined by (𝛾𝐹𝑘+1𝑞𝐹)𝐹∈ F𝑇. We write,
using triangle and Cauchy–Schwarz inequalities, Õ 𝐹∈ F𝑇 ℎ−1 𝐹 k𝑃 𝑘+1 grad,𝑇𝑞𝑇 − 𝛾𝑘+1 𝐹 𝑞 𝐹 k2 L2(𝐹 ) . Õ 𝐹∈ F𝑇 ℎ−1 𝐹 k𝛾 𝑘+1 𝐹 𝑞 𝐹 − 𝐴𝑞 , 𝐹k2 L2(𝐹 ) + Õ 𝐹∈ F𝑇 ℎ−1 𝐹 k 𝐴𝑞 , 𝐹 − 𝐴𝑞 , 𝜕𝑇k2 L2(𝐹 ) + Õ 𝐹∈ F𝑇 ℎ−1 𝐹 k𝑃 𝑘+1 grad,𝑇𝑞𝑇 − 𝐴𝑞 , 𝜕𝑇k2 L2(𝐹 ) ≕ 𝔗1+ 𝔗2+ 𝔗3. (4.21)
The first term is estimated using a Poincaré–Wirtinger inequality on 𝛾𝐹𝑘+1𝑞𝐹and invoking (4.16) together
with the definition (4.1) of |||·|||curl,𝑇 to get
𝔗1. Õ 𝐹∈ F𝑇 ℎ−1 𝐹 ℎ𝐹k grad𝐹𝛾 𝑘+1 𝐹 𝑞 𝐹 kL2(𝐹 ) 2 . Õ 𝐹∈ F𝑇 ℎ𝐹|||𝑮 𝑘 𝐹𝑞𝐹||| 2 curl,𝐹 . |||𝑮 𝑘 𝑇𝑞𝑇||| 2 curl,𝑇. (4.22)
For the second term in (4.21), we follow the same steps as in [10, Lemma 29], working from face to face through common edges and using (4.16) to get 𝔗2 . |||𝑮
𝑘 𝑇
𝑞
𝑇
|||2
curl,𝑇. Finally, for 𝔗3, we
apply the definition (3.1) of 𝑃grad,𝑇𝑘+1 𝑞𝑇
− 𝐼𝑘
grad,𝑇𝐴𝑞 , 𝜕𝑇
with 𝒗𝑇 ∈ Rc, 𝑘+2(𝑇 ) such that div 𝒗𝑇 =
𝑃𝑘+1 grad,𝑇(𝑞𝑇 − 𝐼𝑘 grad,𝑇𝐴𝑞 , 𝜕𝑇) and k𝒗𝑇kL2(𝑇 ) . ℎ𝑇k𝑃 𝑘+1 grad,𝑇(𝑞𝑇 − 𝐼𝑘 grad,𝑇𝐴𝑞 , 𝜕𝑇) k L2(𝑇 ), see [10, Lemma
31]. Using the consistency properties (3.2) of 𝑃grad,𝑇𝑘+1 , [10, Eq. (3.13)] ofG
𝑘
𝑇 and [10, Eq. (3.11)] of
𝛾𝑘+1
𝐹 , and a discrete trace inequality, this gives
k𝑃𝑘+1 grad,𝑇𝑞𝑇 − 𝐴 𝑞 , 𝜕𝑇k L2(𝑇 ) . ℎ𝑇kG 𝑘 𝑇𝑞𝑇kL2(𝑇 )+ ℎ𝑇 Õ 𝐹∈ F𝑇 ℎ −1 2 𝐹 k𝛾 𝑘+1 𝐹 𝑞𝐹 − 𝐴𝑞 , 𝜕𝑇kL2(𝐹 ) . ℎ𝑇|||𝑮 𝑘 𝑇𝑞𝑇|||curl,𝑇 + ℎ𝑇 𝔗12 1 + 𝔗 1 2 2 , (4.23)
where the second line follows from (4.13), and a triangle inequality to write Õ 𝐹∈ F𝑇 ℎ −1 2 𝐹 k𝛾 𝑘+1 𝐹 𝑞 𝐹 − 𝐴𝑞 , 𝜕𝑇k L2(𝐹 ) ≤ Õ 𝐹∈ F𝑇 ℎ −1 2 𝐹 k𝛾 𝑘+1 𝐹 𝑞 𝐹 − 𝐴𝑞 , 𝐹k L2(𝐹 )+ Õ 𝐹∈ F𝑇 ℎ −1 2 𝐹 k 𝐴𝑞 , 𝐹− 𝐴𝑞 , 𝜕𝑇kL2(𝐹 ) .
Using discrete trace inequalities and the previous estimates on 𝔗1and 𝔗2, (4.23) leads to
𝔗3 . ℎ−2𝑇 k𝑃 𝑘+1 grad,𝑇𝑞𝑇 − 𝐴𝑞 , 𝜕𝑇k2 L2(𝑇 ) . |||𝑮 𝑘 𝑇𝑞𝑇||| 2 curl,𝑇
Plugging this bound together with the estimates on 𝔗1 and 𝔗2 into (4.21) concludes the proof of the
bound on the second term in the right-hand side of (4.17) To bound the first term in the left-hand side of (4.17), we proceed as for grad𝐹𝛾
𝑘+1 𝐹
𝑞
𝐹 in Step 1, using an integration by parts in the definition (3.1)
of 𝑃grad,𝑇𝑘+1 𝑞𝑇 and selecting the test function 𝒗
𝑇 = grad 𝑃 𝑘+1
grad,𝑇𝑞𝑇 (see Remark 3).
Proof of Theorem 9. It holds, by definition (3.13b) of the local discrete L2
-product in 𝑿𝑘curl,ℎand (4.9),
˜ Ediv,ℎ(𝒗, 𝑞 ℎ ) = Õ 𝑇∈ Tℎ ∫ 𝑇 𝑷𝑘 curl,𝑇 𝑰 𝑘 curl,𝑇𝒗 ·G 𝑘 𝑇𝑞 𝑇 + scurl,𝑇 (𝑰𝑘 curl,𝑇𝒗|𝑇,𝑮 𝑘 𝑇𝑞𝑇) + ∫ 𝑇 div 𝒗 𝑃grad,𝑇𝑘+1 𝑞𝑇 . (4.24)
Subtracting this quantity from (4.24), we obtain ˜ Ediv,ℎ(𝒗, 𝑞 ℎ ) = Õ 𝑇∈ T ℎ ∫ 𝑇 𝑷𝑘 curl,𝑇 𝑰 𝑘 curl,𝑇𝒗 − 𝒘𝑇 ·G𝑘 𝑇𝑞 𝑇 + scurl,𝑇 (𝑰𝑘 curl,𝑇𝒗|𝑇,𝑮 𝑘 𝑇 𝑞 𝑇 ) + Õ 𝑇∈ Tℎ ∫ 𝑇 div(𝒗 − 𝒘𝑇)𝑃 𝑘+1 grad,𝑇𝑞𝑇 + Õ 𝐹∈ F𝑇 𝜔𝑇 𝐹 ∫ 𝐹 (𝒘𝑇 −𝒗) · 𝒏𝐹𝛾 𝑘+1 𝐹 𝑞 𝐹 ,
where 𝒗 is introduced in the boundary term by single-valuedness of the discrete trace, and using 𝒗|𝐹 ·𝒏𝐹 = 0 whenever 𝐹 ∈ Fℎb. Integrating by parts the third term in the right-hand side of the above
expression, we obtain ˜ Ediv,ℎ(𝒗, 𝑞 ℎ ) = Õ 𝑇∈ Tℎ ∫ 𝑇 𝑷𝑘 curl,𝑇 𝑰 𝑘 curl,𝑇𝒗 − 𝒘𝑇 ·G𝑘 𝑇𝑞 𝑇 + scurl,𝑇 (𝑰𝑘 curl,𝑇𝒗|𝑇,𝑮 𝑘 𝑇𝑞𝑇) + Õ 𝑇∈ T ℎ − ∫ 𝑇 (𝒗 − 𝒘𝑇) · grad 𝑃 𝑘+1 grad,𝑇𝑞𝑇 + Õ 𝐹∈ F𝑇 𝜔𝑇 𝐹 ∫ 𝐹 (𝒘𝑇 −𝒗) · 𝒏𝐹(𝛾 𝑘+1 𝐹 𝑞 𝐹 − 𝑃𝑘+1 grad,𝑇𝑞𝑇 ) . (4.25) We set 𝒘𝑇 =𝝅 𝑘
P,𝑇𝒗 and use (3.19) and the approximation properties of 𝝅 𝑘 P,𝑇 stated in [12, Theorem 1.45] to see that k𝑷𝑘 curl,𝑇 𝑰 𝑘 curl,𝑇𝒗 − 𝝅 𝑘 P,𝑇𝒗 kL2(𝑇 )+ k𝒗 − 𝝅 𝑘 P,𝑇𝒗 kL2(𝑇 )+ Õ 𝐹∈ F𝑇 ℎ 1 2 𝐹k𝒗 − 𝝅 𝑘 P,𝑇𝒗 kL2(𝐹 ) . ℎ 𝑘+1 𝑇 |𝒗|H( 𝑘+1,2)(𝑇 ).
Using Cauchy–Schwarz inequalities on the integrals and on the stabilisation bilinear form in (4.25), the bound (4.13) together with the norm equivalence (4.5), and the consistency property (3.25) of the stabilisation term, we arrive at
E˜div,ℎ(𝒗, 𝑞ℎ ) ≤ Õ 𝑇∈ Tℎ ℎ𝑘+1 𝑇 |𝒗|H( 𝑘+1,2)(𝑇 )k𝑮 𝑘 𝑇 𝑞 𝑇 kcurl,𝑇 + Õ 𝑇∈ Tℎ ℎ𝑘+1 𝑇 |𝒗|H( 𝑘+1,2)(𝑇 )k grad 𝑃 𝑘+1 grad,𝑇𝑞𝑇 kL2(𝑇 ) + Õ 𝑇∈ Tℎ Õ 𝐹∈ F𝑇 ℎ𝑘+1 𝑇 |𝒗|H( 𝑘+1,2)(𝑇 ) ℎ −1 2 𝐹 k𝛾 𝑘+1 𝐹 𝑞𝐹 − 𝑃 𝑘+1 grad,𝑇𝑞𝑇kL2(𝐹 ) .
The conclusion follows from the estimate (4.17), and Cauchy–Schwarz inequalities on the sums. 4.4.2 Adjoint consistency for the curl
The proof of the adjoint consistency for the curl hinges on liftings defined as solutions of local problems. For any 𝐹 ∈ Fℎ, the face lifting 𝑹curl,𝐹 : 𝑿
𝑘
curl,𝐹 → H(rot; 𝐹) ∩ H(div; 𝐹) is such that, for all
𝒗𝐹 ∈ 𝑿 𝑘
curl,𝐹, 𝑹curl,𝐹𝒗𝐹 =𝝓𝒗𝐹 + grad𝐹
𝜓𝒗
𝐹 with 𝝓𝒗𝐹 ∈ H(rot; 𝐹) ∩ H(div; 𝐹) such that
Let now 𝑇 ∈ Tℎ. The curl correction 𝜹𝑇 : 𝑿 𝑘
curl,𝑇 → H(curl; 𝑇) ∩ H(div; 𝑇) is such that, for all
𝒗𝑇 ∈ 𝑿 𝑘 curl,𝑇, div 𝜹𝑇𝒗𝑇 = − divC 𝑘 𝑇𝒗𝑇 in 𝑇 , (4.28a) curl 𝜹𝑇𝒗𝑇 = 0 in 𝑇 , (4.28b) 𝜹𝑇𝒗𝑇 ·𝒏𝐹 = 𝐶 𝑘 𝐹𝒗𝐹 −C 𝑘 𝑇𝒗𝑇 ·𝒏𝐹 on all 𝐹 ∈ F𝑇 . (4.28c)
The curl correction lifts the difference between the face curl 𝐶𝐹𝑘𝒗𝐹 and the normal component of the
element curlC𝑇𝑘𝒗𝑇 as a function defined over 𝑇 . Its role is to ensure the well-posedness of the problem
defining the element lifting 𝑹curl,𝑇 : 𝑿curl,𝑇𝑘 → H(curl; 𝑇) ∩ H(div; 𝑇) such that, for all 𝒗𝑇 ∈ 𝑿 𝑘 curl,𝑇, curl 𝑹curl,𝑇𝒗𝑇 =C 𝑘 𝑇𝒗𝑇 +𝜹𝑇𝒗𝑇 in 𝑇 , (4.29a) div 𝑹curl,𝑇𝒗𝑇 = 0 in 𝑇 , (4.29b)
(𝑹curl,𝑇𝒗𝑇)t,𝐹 =𝑹curl,𝐹𝒗𝐹 on all 𝐹 ∈ F𝑇
. (4.29c)
In Appendix A we prove that these lifting operators are well-defined, and that they satisfy the following two key properties:
• Orthogonality of the face lifting: For all 𝐹 ∈ Fℎ,
∫ 𝐹 (𝜸𝑘 t,𝐹𝒗𝐹 −𝑹curl,𝐹𝒗𝐹) ·𝒛𝐹 = 0 ∀(𝒗𝐹 ,𝒛𝐹) ∈ 𝑿𝑘 curl,𝐹× RT 𝑘+1 (𝐹); (4.30)
• Boundedness of the element lifting: For all 𝑇 ∈ Tℎ,
k𝑹curl,𝑇𝒗𝑇kL2(𝑇 )+ k curl 𝑹curl,𝑇𝒗𝑇kL2(𝑇 ) . k𝒗𝑇kcurl,𝑇+ k𝑪 𝑘
𝑇𝒗𝑇kdiv,𝑇 ∀𝒗𝑇 ∈ 𝑿 𝑘
curl,𝑇. (4.31)
Lemma 18 (Approximation properties of N𝑘+1
(𝑇 ) on polyhedral elements). For all 𝑇 ∈ Tℎ and all
𝒘 ∈ H𝑘+2 (𝑇 ), there exists 𝒛𝑇 ∈ N 𝑘+1 (𝑇 ) such that k𝒘 − 𝒛𝑇kL2(𝑇 ) . ℎ 𝑘+1 𝑇 |𝒘|H𝑘+1(𝑇 ) + |𝒘|H𝑘+2(𝑇 ) , (4.32) k curl 𝒘 − curl 𝒛𝑇kL2(𝑇 ) . ℎ 𝑘+1 𝑇 |𝒘|H𝑘+2(𝑇 ). (4.33)
Proof. By the mesh regularity assumption, there is a simplex 𝑆 ⊂ 𝑇 whose inradius is & ℎ𝑇. Following
the arguments in the proof of [12, Lemma 1.25], we infer the norm equivalence k𝑞 k
L2(𝑆) ' k𝑞 kL2(𝑇 ) ∀𝑞 ∈ P 𝑘+1
(𝑇 ). (4.34)
Let us take 𝒛𝑇 as the Nédélec interpolant in N 𝑘+1
(𝑆) of 𝒘; 𝒛𝑇 can be uniquely extended as an element
of N𝑘+1(𝑇 ). By the arguments in the proof of [16, Theorem 3.14 and Corollary 3.17], and since 𝑆 ⊂ 𝑇, it holds k𝒘 − 𝒛𝑇kL2(𝑆) . ℎ 𝑘+1 𝑇 |𝒘|H𝑘+1(𝑇 )+ |𝒘|H𝑘+2(𝑇 ) , k curl 𝒘 − curl 𝒛𝑇kL2(𝑆) . ℎ 𝑘+1 𝑇 |𝒘|H𝑘+2(𝑇 ). (4.35) We then write, introducing 𝝅𝑘P,𝑇+1𝒘 and using triangle inequalities,
where we have used the approximation property of 𝝅𝑘P,𝑇+1 together with the norm equivalence (4.34) in
the second equality, and concluded by introducing 𝒘 and invoking (4.35) to write k𝝅𝑘+1 P,𝑇𝒘 − 𝒛𝑇kL2(𝑆) ≤ k𝝅 𝑘+1 P,𝑇𝒘 − 𝒘 kL2(𝑆) + k𝒘 − 𝒛𝑇kL2(𝑆) . ℎ𝑘+1 𝑇 |𝒘|H𝑘+1(𝑇 )+ ℎ 𝑘+1 𝑇 (|𝒘|H𝑘+1(𝑇 )+ |𝒘|H𝑘+2(𝑇 )).
This concludes the proof of (4.32). The proof of (4.33) is done in a similar way, introducing curl(𝝅𝑘P,𝑇+1𝒘)
and using the approximation property k curl 𝒘 − curl(𝝅𝑘P,𝑇+1𝒘) kL2(𝑇 ) . ℎ 𝑘+1
𝑇 |𝒘|H𝑘+2(𝑇 ).
Proof of Theorem 10. For all 𝑇 ∈ Tℎ, select 𝒛𝑇 ∈ N 𝑘+1
(𝑇 ) given by Lemma 18. Using (3.13c) to expand (·, ·)div,ℎ together with (4.10), and recalling (3.6), we see that it holds, for all 𝒗ℎ ∈ 𝑿
𝑘 curl,ℎ, ˜ Ecurl,ℎ(𝒘, 𝒗ℎ) = Õ 𝑇∈ Tℎ ∫ 𝑇 𝑷𝑘 div,𝑇(𝑰 𝑘 div,𝑇𝒘|𝑇) −𝒛𝑇 ·C 𝑘 𝑇𝒗𝑇 + Õ 𝑇∈ Tℎ sdiv,𝑇(𝑰 𝑘 div,𝑇𝒘|𝑇,𝑪 𝑘 𝑇𝒗𝑇) + Õ 𝑇∈ Tℎ ∫ 𝑇 curl(𝒛𝑇 −𝒘) · 𝑷 𝑘 curl,𝑇𝒗𝑇 + Õ 𝑇∈ Tℎ Õ 𝐹∈ F𝑇 𝜔𝑇 𝐹 ∫ 𝐹 (𝒛𝑇 ×𝒏𝐹) ·𝜸 𝑘 t,𝐹𝒗𝐹 ≕ 𝔗1+ 𝔗2+ 𝔗3+ 𝔗4. (4.36)
Using Cauchy–Schwarz and triangle inequalities, it is readily inferred for the first term
|𝔗1| . " Õ 𝑇∈ Tℎ k𝑷𝑘 div,𝑇(𝑰 𝑘 div,𝑇𝒘) − 𝒘 k 2 L2(𝑇 )+ k𝒘 − 𝒛𝑇k2 L2(𝑇 ) #1 2 Õ 𝑇∈ Tℎ kC𝑘 𝑇𝒗𝑇k 2 L2(𝑇 ) !1 2 . ℎ𝑘+1 |𝒘|H𝑘+1 ( Tℎ)+ |𝒘|H 𝑘+2 ( Tℎ) k𝑪𝑘 ℎ𝒗ℎkdiv,ℎ , (4.37)
where the conclusion follows using the approximation properties (3.20) and (4.32) to bound the first factor, and (4.14) along with the norm equivalence (4.5) to bound the second.
For 𝔗2, combining the consistency property (3.26) of sdiv,𝑇 with discrete Cauchy–Schwarz
inequal-ities and the definition of the k·kdiv,ℎ-norm readily gives
|𝔗2| . ℎ𝑘+1
|𝒘|H𝑘+1
( Tℎ)k𝑪
𝑘
ℎ𝒗ℎkdiv,ℎ. (4.38)
For 𝔗3, Cauchy–Schwarz inequalities, the approximation property (4.35), and the definition of the
norm k·kcurl,ℎyield
|𝔗3| ≤ Õ 𝑇∈ Tℎ k curl(𝒛𝑇 −𝒘) k2 L2(𝑇 ) !1 2 Õ 𝑇∈ Tℎ k𝑷𝑘 curl,𝑇𝒗𝑇k 2 L2(𝑇 ) !1 2 . ℎ𝑘+1 |𝒘|H𝑘+2 ( Tℎ)k𝒗ℎkcurl,ℎ. (4.39)
Let us now consider the last term in the right-hand side of (4.36). Since ( 𝒛𝑇)|𝐹 ×𝒏𝐹 ∈ RT 𝑘+1
(𝐹) (see [10, Proposition 30]), by (4.30) we can replace 𝜸𝑘t,𝐹𝒗𝑇 by 𝑹curl,𝐹𝒗𝐹 in the boundary integral.
Using the fact that both 𝑹curl,𝐹𝒗𝐹 and the (rotated) tangential component of 𝒘 are continuous across
interfaces, along with the fact that 𝜔𝑇
1𝐹 + 𝜔𝑇2𝐹 = 0 for all 𝐹 ∈ Fℎ between two elements 𝑇1
