A stable local commuting projector and optimal hp approximation estimates in Hcurl

2025-04-30 0 0 558.16KB 34 页 10玖币

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arXiv:2210.09701v2 [math.NA] 5 Dec 2023

A stable local commuting projector and optimal hp

approximation estimates in H(curl)∗

Th´eophile Chaumont-Frelet†Martin Vohral´ık‡§

December 6, 2023

Abstract

We design an operator from the inﬁnite-dimensional Sobolev space H(curl) to its ﬁnite-dimensional

subspace formed by the N´ed´elec piecewise polynomials on a tetrahedral mesh that has the following

properties: 1) it is deﬁned over the entire H(curl), including boundary conditions imposed on a

part of the boundary; 2) it is deﬁned locally in a neighborhood of each mesh element; 3) it is based

on simple piecewise polynomial projections; 4) it is stable in the L2-norm, up to data oscillation;

5) it has optimal (local-best) approximation properties; 6) it satisﬁes the commuting property with

its sibling operator on H(div); 7) it is a projector, i.e., it leaves intact objects that are already in

the N´ed´elec piecewise polynomial space. This operator can be used in various parts of numerical

analysis related to the H(curl) space. We in particular employ it here to establish the two fol-

lowing results: i) equivalence of global-best, tangential-trace- and curl-constrained, and local-best,

unconstrained approximations in H(curl) including data oscillation terms; and ii) fully h- and p-

(mesh-size- and polynomial-degree-) optimal approximation bounds valid under the minimal Sobolev

regularity only requested elementwise. As a result of independent interest, we also prove a p-robust

equivalence of curl-constrained and unconstrained best-approximations on a single tetrahedron in the

H(curl)-setting, including hp data oscillation terms.

Key words: Sobolev space H(curl), N´ed´elec ﬁnite element space, stable local commuting projec-

tor, constrained–unconstrained equivalence, local-best approximation, hp approximation, minimal local

Sobolev regularity

1 Introduction

Let Ω ⊂R3be a Lipschitz polyhedral domain (open, bounded, and connected set) and ΓNa Lipschitz

polygonal relatively open subset of its boundary ∂Ω (details on setting and notation are given in Section 2

below). A central concept in numerical analysis of partial diﬀerential equations including the grad, curl,

and div operators, connected with the Sobolev spaces H1

0,N(Ω), H0,N(curl,Ω), and H0,N(div,Ω), is the

following commuting de Rham complex:

0,N(Ω) ∇

−−→ H0,N(curl,Ω) ∇×

−−−→ H0,N(div,Ω) ∇·

−−→ L2

∗(Ω)



yPp+1,grad

h



yPp,curl

h



yPp,div

h



yΠp

Pp(Th)∩H1

0,N(Ω) ∇

−−→ Np(Th)∩H0,N(curl,Ω) ∇×

−−−→ RTp(Th)∩H0,N(div,Ω) ∇·

−−→ Pp(Th)∩L2

∗(Ω).

(1.1)

Assuming for simplicity in the introduction that Ω is homotopic to a ball (contractible) and either

ΓD=∂Ω or ΓN=∂Ω so that the cohomology spaces are trivial, the ﬁrst line of (1.1) is the well-

known exact sequence on the continuous, inﬁnite-dimensional, level, see, e.g., Arnold et al. [3] and the

references therein. It in particular states that 1) each function from the H0,N(curl,Ω) space whose weak

curl vanishes is a weak gradient of a function from H1

0,N(Ω); 2) each function from the H0,N(div,Ω)

∗This project has received funding from the European Research Council (ERC) under the European Union’s Horizon

2020 research and innovation program (grant agreement No 647134 GATIPOR).

†University Cˆote d’Azur, Inria, CNRS, LJAD, 2004 Route des Lucioles, 06902 Valbonne, France

(theophile.chaumont@inria.fr).

‡Inria, 2 rue Simone Iﬀ, 75589 Paris, France (martin.vohralik@inria.fr).

§CERMICS, Ecole des Ponts, 77455 Marne-la-Vall´ee, France

space whose weak divergence vanishes is a weak curl of a function from H0,N(curl,Ω); 3) each function

from the L2

∗(Ω) space is a weak divergence of a function from H0,N(div,Ω). Similarly, the second line is

the counterpart of the ﬁrst one on the discrete, ﬁnite-dimensional, piecewise polynomial, level, see e.g.,

Boﬃ et al. [9] and the references therein. The passage between the ﬁrst and the second line is then the

key interest in this contribution, where the three operators Pp+1,grad

h,Pp,curl

h, and Pp,div

h,p≥0, should:

1. be deﬁned over the entire inﬁnite-dimensional spaces H1

0,N(Ω), H0,N(curl,Ω), and H0,N(div,Ω);

2. be deﬁned locally, in a neighborhood of mesh elements at most;

3. be based on simple piecewise polynomial projections;

4. be stable in L2(Ω) for H0,N(curl,Ω) and H0,N(div,Ω) and in L2(Ω) of the weak gradient for H1

0,N(Ω),

up to data oscillation;

5. have optimal approximation properties, i.e., that of local-best unconstrained L2-orthogonal projec-

tors;

6. satisfy the commuting properties expressed by the arrows in (1.1);

7. be projectors, i.e., leave intact objects that are already in the piecewise polynomial spaces.

There is an immense literature devoted to (1.1). A ﬁrst consideration for the operators Pp+1,grad

Pp,curl

h, and Pp,div

his given by the canonical projectors, see Ciarlet [18], N´ed´elec [42], and Raviart and

Thomas [44], respectively. These satisfy many of the properties above, but, unfortunately, not prop-

erty 1, since their action is not deﬁned on all objects from the entire inﬁnite-dimensional spaces H1

0,N(Ω),

H0,N(curl,Ω), and H0,N(div,Ω). The commuting diagram (1.1) has been addressed at the abstract level

of the ﬁnite element exterior calculus in, e.g., Christiansen and Winther [17], still leading to the loss

of some of the desirable properties, namely the locality. Simultaneous deﬁnition on the entire Sobolev

spaces, locality, commutativity, and the projection property have been achieved in Falk and Winther [31],

though the stability in the L2(Ω) norms (up to data oscillation) and the local-best unconstrained ap-

proximation properties have not been addressed; the stability in the L2(Ω) norms been recently achieved

in Arnold and Guzm´an [2]. Two diﬀerent sets of projectors, satisfying together (but not individually) all

properties 1–7, were then designed in Ern and Guermond [27,28]. Finally an operator Pp,div

hsatisfying

the integrality of the requested properties (all 1–7above) has been recently devised in Ern et al. [26,

Section 3.1].

The ﬁrst goal of the present contribution is to design an operator Pp,curl

hsatisfying the integrality of

the requested properties 1–7. Deﬁnition 3.3 is designed to this purpose, relying on (a slight modiﬁcation

of) Pp,div

hfrom [26], see Deﬁnition 3.1, and using similar building principles as in [26]. The main result

here is Theorem 3.6. The central technical tool allowing to achieve the commuting property is related

to equilibration in H(curl,Ω). A ﬁrst contribution in this direction is that of Braess and Sch¨oberl [11].

Recent extensions to higher polynomial degrees are developed in Gedicke et al. [35,36] as well as in [16],

which we use here. All 1–7come handy namely in a priori error estimates (especially property 5) and

in approximation classes in proofs of convergence and optimality based on posteriori error estimates.

All 1–7also turn useful in localized orthogonal decomposition techniques for multiscale problems, cf. [33]

and the references therein.

Our contribution stands apart from the existing literature namely in the satisfaction of property 5.

This leads to the result of equivalence of global-best (tangential-trace- and curl-constrained) and local-

best (unconstrained) approximations in H(curl,Ω), see Theorem 3.9. This result, not taking into account

data oscillation, has been recently established in [14], building on the seminal contribution by Veeser [46]

in the H1(Ω)-setting and on [26] in the H(div,Ω)-setting, see also the references therein. Here, we present

a direct proof. We take into account data oscillation, which actually turns out quite demanding. A related

result giving rise to an operator with local-best approximation properties in the general framework of

ﬁnite element spaces of diﬀerential forms has recently been obtained in Gawlik et al. [34]; it, however,

does not address the curl constraint.

Yet a separate, and involved, question in numerical analysis is that of deriving hp-approximation

estimates. This has been addressed in the H(div,Ω)- and H(curl,Ω)-settings in particular in Suri [45],

Monk [41], Demkowicz and Buﬀa [22], and Demkowicz [21], see also the references therein. These ref-

erences feature a slight suboptimality in the polynomial degree pon tetrahedral meshes (presence of a

logarithmic factor), which has been removed in Bespalov and Heuer [7,8] and recently in Melenk and Ro-

jik [40]. Unfortunately, none of these references allows for minimal Sobolev regularity. The result in [26,

Theorem 3.6] is equally fully h- and p-optimal, and this, moreover, under the minimal Sobolev regularity,

only requested elementwise. Deriving such estimates in the H(curl,Ω)-setting is the last goal of the

present contribution. Theorem 3.11 in particular presents a fully h- and p-optimal approximation bound

valid under the minimal Sobolev regularity that is only requested separately on each mesh element (the

treatment of the hp data oscillation is restricted to convex patches or presents a slight suboptimality).

The key ingredients are here again linked to (polynomial-degree-robust) ﬂux equilibration in H(curl,Ω)

of [16], with the cornerstones being the results in a single tetrahedron: on the right inverses in the vol-

ume by Costabel and McIntosh [20] and on the polynomial extension operators from the boundary by

Demkowicz et al. [23,24,25].

This contribution is organized as follows: Section 2ﬁxes the setting and notation. The above-described

main results are collected in Section 3. The well-posedness of the central deﬁnition of our stable local

commuting projector is veriﬁed in Section 4, and the proofs of the three principal theorems are then

presented respectively in Sections 5–7. A result of independent interest, stipulating a polynomial-degree-

robust equivalence of constrained and unconstrained best-approximation on a single tetrahedron in the

H(curl,Ω)-setting, including hp data oscillation terms, is presented in Appendix A. Finally, a technical

result on broken regular decomposition in a patch is presented in Appendix B.

2 Setting and notation

Let ω, Ω⊂R3be open, Lipschitz polyhedral domains; Ω will be used to denote the computational

domain, while we reserve the notation ω⊆Ω for its subdomains. We do not make any assumptions

on the topology of Ω; Ω is not necessarily homotopic to a ball (contractible) and nontrivial cohomology

spaces induced by Ω are allowed. The subdomains ω(patches of elements below) will later, in turn, be

supposed homotopic to a ball (contractible). We will use the notation a.bwhen there exists a positive

constant Csuch that a≤Cb; we will always specify the dependencies of C.

2.1 Sobolev spaces H1,H(curl), and H(div)

We let L2(ω) be the space of scalar-valued square-integrable functions deﬁned on ω; we use the notation

L2(ω) := [L2(ω)]3for vector-valued functions with each component in L2(ω). We denote by k·kωthe

L2(ω) or L2(ω) norm and by (·,·)ωthe corresponding scalar product; we drop the index when ω= Ω.

We will extensively work with the following three Sobolev spaces: 1) H1(ω), the space of scalar-valued

L2(ω) functions with weak gradients in L2(ω), H1(ω) := {v∈L2(ω); ∇v∈L2(ω)}; 2) H(curl, ω), the

space of vector-valued L2(ω) functions with weak curls in L2(ω), H(curl, ω) := {v∈L2(ω); ∇×v∈

L2(ω)}; and 3) H(div, ω), the space of vector-valued L2(ω) functions with weak divergences in L2(ω),

H(div, ω) := {v∈L2(ω); ∇·v∈L2(ω)}. We refer the reader to Adams [1] and Girault and Raviart [37]

for an in-depth description of these spaces. Moreover, component-wise H1(ω) functions will be denoted

by H1(ω) := {v∈L2(ω); vi∈H1(ω), i = 1,...,3}. We will employ the notation h·,·iSfor the integral

product on boundary (sub)sets S⊂∂ω.

2.2 Tetrahedral mesh, patches of elements, and the hat functions

Let Thbe a simplicial mesh of the domain Ω, i.e., ∪K∈ThK= Ω, where any element K∈ This a closed

tetrahedron with nonzero measure, and where the intersection of two diﬀerent tetrahedra is either empty

or their common vertex, edge, or face. The shape-regularity parameter of the mesh This the positive real

number κTh:= maxK∈ThhK/ρK, where hKis the diameter of the tetrahedron Kand ρKis the diameter

of the largest ball contained in K. These assumptions are standard, and allow for strongly graded meshes

with local reﬁnements, though not for anisotropic elements.

We denote the set of vertices of the mesh Thby Vh; it is composed of interior vertices lying in Ω and

of vertices lying on the boundary ∂Ω. For an element K∈ Th,FKdenotes the set of its faces and VK

the set of its vertices. Conversely, for a vertex a∈ Vh,Tadenotes the patch of the elements of Ththat

share a, and ωais the corresponding open subdomain with diameter hωa. We suppose these vertex patch

subdomains ωato be homotopic to a ball (contractible).

A particular role below will be played by the continuous, piecewise aﬃne “hat” function ψawhich

takes value 1 at the vertex aand zero at the other vertices. We note that ωacorresponds to the support

of ψaand that the functions ψaform the partition of unity

a∈Vh

ψa= 1.(2.1)

By [[v]], we denote the jump of the function von a face F, i.e., the diﬀerence of the traces of vfrom the

two elements sharing Falong an arbitrary but ﬁxed normal.

2.3 Sobolev spaces with partially vanishing traces on Ωand ωa

Let ΓD, ΓNbe two disjoint, relatively open, and possibly empty subsets of the computational domain

boundary ∂Ω such that ∂Ω = ΓD∪ΓN. We also require that ΓDand ΓNhave polygonal Lipschitz

boundaries, and we assume that each boundary face of the mesh Thlies entirely either in ΓDor in ΓN.

Notice that the assumption that ΓDand ΓNhave Lipschitz boundaries excludes “checkerboard” patterns

of mixed boundary conditions. In particular, the cohomology spaces with the boundary conditions on ωa

introduced below are trivial.

We denote by L2

∗(Ω) the subspace of L2(Ω) formed by functions of mean value 0 if ΓN=∂Ω and set

∗(Ω) = L2(Ω) otherwise. H1

0,D(Ω) is the subspace of H1(Ω) formed by functions vanishing on ΓDin

the sense of traces, H1

0,D(Ω) := {v∈H1(Ω); v= 0 on ΓD}. Let nΩbe the unit normal vector on ∂Ω,

outward to Ω. Then H0,N(curl,Ω), H0,D(curl,Ω) are the subspaces of H(curl,Ω) formed by functions

with vanishing tangential traces respectively on ΓNor ΓD,

H0,N(curl,Ω) := {v∈H(curl,Ω); v×nΩ= 0 on ΓN},(2.2a)

H0,D(curl,Ω) := {v∈H(curl,Ω); v×nΩ= 0 on ΓD},(2.2b)

where

v×nΩ= 0 on ΓN⇐⇒ (∇×v,ϕ)−(v,∇×ϕ) = 0

∀ϕ∈H1(Ω) such that ϕ×nΩ=0on ΓD

(2.3)

and symmetrically for ΓD. Finally, H0,N(div,Ω) is the subspace of H(div,Ω) formed by functions with

vanishing normal trace on ΓN,

H0,N(div,Ω) := {v∈H(div,Ω); v·nΩ= 0 on ΓN},(2.4)

where

v·nΩ= 0 on ΓN⇐⇒ (v,∇ϕ) + (∇·v, ϕ) = 0 ∀ϕ∈H1

0,D(Ω).(2.5)

Fernandes and Gilardi [32] present a rigorous characterization of tangential (resp. normal) traces of

H(curl,Ω) (resp. H(div,Ω)) on a part of the boundary ∂Ω.

We will also need local spaces on the patch subdomains ωa. Let ﬁrst a∈ Vhbe an interior vertex.

Then we set

∗(ωa) := {v∈H1(ωa); (v, 1)ωa= 0},(2.6a)

H0(curl, ωa) := {v∈H(curl, ωa); v×nωa= 0 on ∂ωa},(2.6b)

H†(curl, ωa) := H(curl, ωa),(2.6c)

H0(div, ωa) := {v∈H(div, ωa); v·nωa= 0 on ∂ωa},(2.6d)

where the tangential trace in (2.6b) is understood by duality as above in (2.3), whereas the normal trace

in (2.6d) is understood by duality as above in (2.5). By deﬁnition, H1

∗(ωa) from (2.6a) is the subspace

of those H1(ωa) functions whose mean value vanishes.

The situation is more subtle for boundary vertices. As a ﬁrst possibility, if a∈ΓN(i.e., a∈ Vh

is a boundary vertex such that all the boundary faces sharing the vertex alie in ΓN), then the spaces

∗(ωa), H0(curl, ωa), H†(curl, ωa), and H0(div, ωa) are deﬁned as above in (2.6). Secondly, when

a∈ΓD, then at least one of the faces sharing the vertex alies in ΓD, and we denote by γDthe subset of

ΓDcorresponding to all such faces. Notice that due to our assumptions on ΓDand ΓNto have polygonal

Lipschitz boundaries, γDis always simply connected. In this situation, we let

∗(ωa) := {v∈H1(ωa); v= 0 on γD},(2.7a)

H0(curl, ωa) := {v∈H(curl, ωa); v×nωa= 0 on ∂ωa\γD},(2.7b)

H†(curl, ωa) := {v∈H(curl, ωa); v×nωa= 0 on γD},(2.7c)

H0(div, ωa) := {v∈H(div, ωa); v·nωa= 0 on ∂ωa\γD}.(2.7d)

In all cases, component-wise H1

∗(ωa) functions are denoted by H1

∗(ωa).

2.4 Piecewise polynomial spaces

Let q≥0 be an integer. For a single tetrahedron K∈ Th, we denote by Pq(K) the space of scalar-valued

polynomials on Kof total degree at most q, and by [Pq(K)]3the space of vector-valued polynomials on

Kwith each component in Pq(K). The N´ed´elec [9,42] space of degree qon Kis then given by

Nq(K) := [Pq(K)]3+x×[Pq(K)]3.(2.8)

Similarly, the Raviart–Thomas [9,44] space of degree qon Kis given by

RTq(K) := [Pq(K)]3+Pq(K)x.(2.9)

We note that (2.8) and (2.9) are equivalent to the writing with a direct sum and only homogeneous

polynomials in the second terms. The second term in (2.8) is also equivalently given by homogeneous

(q+ 1)-degree polynomials vhsuch that x·vh(x) = 0 for all x∈K.

We will below extensively use the broken, piecewise polynomial spaces formed from the above element

spaces

Pq(Th) := {vh∈L2(Ω); vh|K∈ Pq(K)∀K∈ Th},

Nq(Th) := {vh∈L2(Ω); vh|K∈Nq(K)∀K∈ Th},

RTq(Th) := {vh∈L2(Ω); vh|K∈RTq(K)∀K∈ Th}.

To form the usual ﬁnite-dimensional Sobolev subspaces, we will write Pq(Th)∩H1(Ω) (for q≥1),

Nq(Th)∩H(curl,Ω), RTq(Th)∩H(div,Ω) (both for q≥0), and similarly for the subspaces reﬂecting

the diﬀerent boundary conditions. The same notation will also be used on the patches Ta.

2.5 L2-orthogonal projectors and elementwise canonical projectors

For q≥0, let Πq

hdenote the L2(K)-orthogonal projector onto Pq(K) or the elementwise L2(Ω)-orthogonal

projector onto the piecewise (broken) polynomials Pq(Th), i.e., for v∈L2(Ω), Πq

h(v)∈ Pq(Th) is, sepa-

rately for all K∈ Th, given by

(Πq

h(v), vh)K= (v, vh)K∀vh∈ Pq(K).(2.10)

Then, Πq

his given componentwise by Πq

We will also use the L2(Ω)-orthogonal projector Πq

RT onto the broken Raviart–Thomas space RTq(Th),

given for v∈L2(Ω) also elementwise as: for all K∈ Th,Πq

RT (v)|K∈RTq(K) is such that

(Πq

RT (v),vh)K= (v,vh)K∀vh∈RTq(K),(2.11a)

or, equivalently,

Πq

RT (v)|K:= arg min

vh∈RTq(K)kv−vhkK.(2.11b)

Finally, we will also need the elementwise (broken) canonical projectors. Let v∈L2(Ω) such that

v|K∈[C1(K)]3for all K∈ Thbe given. Below, we only use piecewise polynomial vwhich satisfy these

assumptions. Separately on each element K∈ Th, following [9,44], the canonical q-degree Raviart–

Thomas projector Iq

RT (v)|K∈RTq(K), q≥0, is given by

hIq

RT (v)·nK, rhiF=hv·nK, rhiF∀rh∈ Pq(F),∀F∈ FK,(2.12a)

(Iq

RT (v),rh)K= (v,rh)K∀rh∈[Pq−1(K)]3,(2.12b)

where nKis the unit outer normal vector of the element K. Similarly, following [9,42], canonical q-degree

N´ed´elec projector Iq

N(v)|K∈Nq(K), q≥0, is given, separately on each element K∈ Th, by

hIq

N(v)·τe, rhie=hv·τe, rhie∀rh∈ Pq(e),∀e∈ EK,(2.13a)

hIq

N(v)×nK,rhiF=hv×nK,rhiF∀rh∈[Pq−1(F)]2,∀F∈ FK,(2.13b)

(Iq

N(v),rh)K= (v,rh)K∀rh∈[Pq−2(K)]3,(2.13c)

where EKstands for the set of edges of K,τeand h·,·ierespectively denote a (unit) tangential vector (the

orientation does not matter) and the L2(e) scalar product of the edge e∈ EK, and where we implicitly

complement rhby a zero component in the normal direction of the face Fin (2.13b). These projectors

crucially satisfy, on each tetrahedron Kand for all v∈[C1(K)]3, the commuting properties

∇·Iq

RT (v) = Πq

h(∇·v),(2.14a)

∇×(Iq

N(v)) = Iq

RT (∇×v).(2.14b)

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arXiv:2210.09701v2[math.NA]5Dec2023AstablelocalcommutingprojectorandoptimalhpapproximationestimatesinH(curl)∗Th´eophileChaumont-Frelet†MartinVohral´ık‡§December6,2023AbstractWedesignanoperatorfromtheinﬁnite-dimensionalSobolevspaceH(curl)toitsﬁnite-dimensionalsubspaceformedbytheN´ed´elecpiecewisepoly...

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A stable local commuting projector and optimal hp approximation estimates in Hcurl

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