To cite this version: Daniele Di Pietro, Jerome Droniou. W s,p -approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray-Lions problems. Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2017, 27 (5), pp.879–908. �10.1142/S0218202517500191�. �hal-01326818�

HAL Id: hal-01326818 https://hal.archives-ouvertes.fr/hal-01326818 Submitted on 6 Jun 2016

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W s,p-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray–Lions problems∗ Daniele A. Di Pietro†1 and J´erˆome Droniou‡2 1

University of Montpellier, Institut Montpelli´erain Alexander Grothendieck, 34095 Montpellier, France 2 School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia

June 6, 2016 Abstract s,p

In this work we prove optimal W -approximation estimates (with p P r1, `8s) for elliptic projectors on local polynomial spaces. The proof hinges on the classical Dupont–Scott approximation theory together with two novel abstract lemmas: An approximation result for bounded projectors, and an Lp -boundedness result for L2 -orthogonal projectors on polynomial subspaces. The W s,p -approximation results have general applicability to (standard or polytopal) numerical methods based on local polynomial spaces. As an illustration, we use these W s,p -estimates to derive novel error estimates for a Hybrid High-Order discretization of Leray–Lions elliptic problems whose weak formulation is classically set in W 1,p pΩq for some p P p1, `8q. This kind of problems appears, e.g., in the modelling of glacier motion, of incompressible turbulent flows, and in airfoil design. Denoting by h the meshsize, we prove k`1

that the approximation error measured in a W 1,p -like norm scales as h p´1 when p ě 2 and as hpk`1qpp´1q when p ă 2. 2010 Mathematics Subject Classification: 65N08, 65N30, 65N12 Keywords: W s,p -approximation properties of elliptic projector on polynomials, Hybrid HighOrder methods, nonlinear elliptic equations, p-Laplacian, error estimates

1

Introduction

In this work we prove optimal W s,p -approximation properties for elliptic projectors on local polynomial spaces, and use these results to derive novel a priori error estimates for a Hybrid High-Order discretisation of Leray–Lions elliptic equations. Let U Ă Rd , d ě 1, be an open bounded set of diameter hU . For all integers s P N and p P r1, `8s, we denote by W s,p pU q the space of functions having derivatives up to degree s in Lp pU q with associated seminorm ÿ |v|W s,p pU q :“ }B α v}Lp pU q , (1) αPNd ,}α}1 “s α

where }α}1 :“ α1 ` . . . ` αd and B “ treatment of the case p “ `8). ∗ This

B1α1

. . . Bdαd (this choice for the seminorm enables a seamless

work was partially supported by ANR project HHOMM (ANR-15-CE40-0005)

† [email protected] ‡ [email protected]

1

Let a polynomial degree l ě 0 be fixed, and denote by Pl pU q the space of d-variate polynomials 1,l on U . The elliptic projector πU : W 1,1 pU q Ñ Pl pU q is defined as follows: For all v P W 1,1 pU q, 1,l πU v is the unique polynomial in Pl pU q that satisfies ż ż 1,l 1,l ∇pπU v ´ vq¨∇w “ 0 for all w P Pl pU q, and pπU v ´ vq “ 0. (2) U

U

As a result of the Poincar´e–Wirtinger inequality, the quantity have the following characterisation: 1,l πU v“

is well-defined. Moreover, we

}∇pw ´ vq}2L2 pU qd .

argş min

wPPl pU q,

1,l πU v

pw´vq“0 U

The first main result of this work is summarised in the following theorem. 1,l Theorem 1 (W s,p -approximation for πU ). Assume that U is star-shaped with respect to every point in a ball of radius %hU for some % ą 0. Let s P t1, . . . , l ` 1u and p P r1, `8s. Then, there exists a real number C ą 0 depending only on d, %, l, s, and p such that, for all m P t0, . . . , su and all v P W s,p pU q, 1,l |v ´ πU v|W m,p pU q ď Chs´m |v|W s,p pU q . (3) U

The proof of Theorem 1 is based on the classical Dupont–Scott approximation theory [26] (cf. also [7, Chapter 4]) and hinges on two novel abstract lemmas for projectors on polynomial spaces: A W s,p -approximation result for projectors that satisfy a suitable boundedness property, and an Lp -boundedness result for L2 -orthogonal projectors on polynomial subspaces. Both results make use of the reverse Lebesgue and Sobolev embeddings for polynomial functions proved in [13] (cf., in particular Lemma 5.1 and Remark A.2 therein). Following similar arguments as in [26, Section 7], the results of Theorem 1 still hold if U is a finite union of domains that are star-shaped with respect to balls of radius comparable to hU . The second main result concerns the approximation of traces, and therefore requires more assumptions on the domain U . 1,l Theorem 2 (W s,p -approximation of traces for πU ). Assume that U is a polytope which admits a partition SU into disjoint simplices S of diameter hS and inradius rS , and that there exists a real number % ą 0 such that, for all S P SU ,

%2 hU ď %hS ď rS . Let s P t1, . . . , l ` 1u, p P r1, `8s, and denote by FU the set of hyperplanar faces of U . Then, there exists a real number C depending only on d, %, l, s and p such that, for all m P t0, . . . , s ´ 1u and all v P W s,p pU q, 1

1,l hUp |v ´ πU v|W m,p pFU q ď Chs´m |v|W s,p pU q . U

(4)

m,p

pF q for all F P FU , and

m,p

Here, W pFU q denotes the set of functions that belong to W |¨|W m,p pFU q the corresponding broken seminorm.

The proof of Theorem 2 is obtained combining the results of Theorem 1 with a continuous Lp -trace inequality. The approximation results of Theorems 1 and 2 are used to prove novel error estimates for the Hybrid High-Order (HHO) method of [13] for nonlinear Leray–Lions elliptic problems of the form: Find a potential u : Ω Ñ R such that ´ divpapx, ∇uqq “ f u“0

in Ω, on BΩ,

(5)

where Ω is a bounded polytopal subset of Rd with boundary BΩ, while the source term f : Ω Ñ R and the function a : Ω ˆ Rd Ñ Rd satisfy the requirements detailed in Eq. (20) below. This 2

equation, which contains the p-Laplace equation (cf. (21) below), appears in the modelling of glacier motion [30], of incompressible turbulent flows in porous media [20], and in airfoil design [29]. In the context of conforming Finite Element (FE) approximations of problems which can be traced back to the general form (5), a priori error estimates were derived in [4, 30]. For nonconforming (Crouzeix–Raviart) FE approximations, error estimates are proved in [33], with convergence rates consistent with the ones presented in this work (concerning the link between the HHO method and nonconforming FE, cf. [18, Remark 1]). Error estimates for a nodal Mimetic Finite Difference (MFD) method for a particular kind of operator a and with p “ 2 are proved in [2]. Finite volume methods, on the other hand, are considered in [1], where error estimates similar to the ones obtained here are derived under the assumption that the source term f vanishes on the boundary (additional error terms are present when this is not the case). Finally, we also cite here [21], where the convergence study of a Mixed Finite Volume (MFV) scheme inspired by [22] is carried out using a compactness argument under minimal regularity assumptions on the exact solution. The HHO method analysed here is based on meshes composed of general polytopal elements and its formulation hinges on degrees of freedom (DOFs) that are polynomials of degree k ě 0 on mesh elements and faces; cf. [14–17] for an introduction to HHO methods and and [9,13] for applications to nonlinear problems. Based on such DOFs, a gradient reconstruction operator GkT of degree k of degree pk`1q are devised by solving local problems and a potential reconstruction operator pk`1 T inside each mesh element T . By construction, the composition of the potential reconstruction pk`1 T with the interpolator on the DOF space coincides with the elliptic projector πT1,k`1 . The gradient and potential reconstruction operators are then used to formulate a local contribution composed of a consistent and a stabilisation term. The W s,p -approximation properties for πT1,k`1 play a crucial role in estimating the error associated with the latter. Denoting by h the meshsize, we prove in Theorem 7 below that, for smooth enough exact solutions, the approximation error measured in k`1 a discrete W 1,p -like norm converges as h p´1 when p ě 2 and as hpk`1qpp´1q when p ă 2. As noticed in [17], the lowest-order version of the HHO method corresponding to k “ 0 is essentially analogous (up to equivalent stabilisation) to the SUSHI scheme of [27] when face unknowns are not eliminated by interpolation. This method, in turn, has been proved in [24] to be equivalent to the MFV method of [22] and the mixed-hybrid MFD method [8,32] (cf. also [6] for an introduction to MFD methods). As a consequence, our results extend the analysis conducted in [21], by providing in particular error estimates for the MFV scheme applied to Leray–Lions equations. To conclude, it is worth mentioning that the tools of Theorems 1 and 2, alongside the optimum W s,p -estimates of [13] for L2 -projectors on polynomial spaces (see Lemma 13), are potentially of interest also for the study of other polytopal methods. Elliptic projections on polynomial spaces appear, e.g., in the conforming and nonconforming Virtual Element Methods (cf. [5, Eq. (4.18)] and [3, Eqs. (3.18)–(3.20)], respectively). They also play a role in determining the high-order part of some post-processings of the potential used in the context of Hybridizable Discontinuous Galerkin methods; cf., e.g., the variation proposed in [10] of the post-processing considered in [11, 12]. The rest of the paper is organised as follows. In Section 2 we provide the proofs of Theorems 1 and 2 preceeded by the required preliminary results. In Section 3 we use these results to derive error estimates for the Hybrid High-Order discretization of problem 5. Appendix A collects some useful inequalities for Leray–Lions operators.

2

W s,p -approximation properties of the elliptic projector on polynomial spaces

This section contains the proofs of Theorems 1 and 2 preceeded by two abstract lemmas for projectors on polynomials subspaces. Throughout the paper, to alleviate the notation, when 3

writing integrals we omit the dependence on the integration variable x as well as the differential with the exception of those integrals involving the function a (cf. (5)).

2.1

Two abstract results for projectors on polynomial subspaces

Our first lemma is an abstract approximation result valid for any projector on a polynomial space that satisfies a suitable boundedness property. Lemma 3 (W s,p -approximation for W -bounded projectors). Assume that U is star-shaped with respect to every point of a ball of radius %hU for some % ą 0. Let five integers l ě 0, s P q,1 t1, . . . , l ` 1u, p P r1, `8s, and q, m P t0, . . . , su be fixed. Let Πq,l pU q Ñ Pl pU q be a U : W projector such that there exists a real number C ą 0 depending only on d, %, l, q, and p such that for all v P W q,p pU q, If m ă q :

|Πq,l U v|W m,p pU q ď C

q ÿ

r´m hU |v|W r,p pU q ,

(6a)

r“m

If m ě q :

|Πq,l U v|W q,p pU q ď C|v|W q,p pU q ,

(6b)

Then, there exists a real number C ą 0 depending only on d, %, l, q, m, s, and p such that, for all v P W s,p pU q, s´m |v ´ Πq,l |v|W s,p pU q . (7) U v|W m,p pU q ď ChU Proof. Here A À B means A ď M B with real number M ą 0 having the same dependencies as C in (7). Since smooth functions are dense in W s,p pU q, we can assume v P C 8 pU q X W s,p pU q. We consider the following representation of v proposed in [7, Chapter 4]: v “ Qs v ` Rs v,

(8)

where Qs v P Ps´1 pU q Ă Pl pU q is the averaged Taylor polynomial, while the remainder Rs v satisfies, for all r P t0, . . . , su (cf. [7, Lemma 4.3.8]), |Rs v|W r,p pU q À hs´r U |v|W s,p pU q .

(9)

q,l s s Since Πq,l U is a projector, it holds ΠU pQ vq “ Q v so that, taking the projection of (8), it is inferred q,l s s Πq,l U v “ Q v ` ΠU pR vq. q,l s s Subtracting this equation from (8), we arrive at v ´ Πq,l U v “ R v ´ ΠU pR vq. Hence, the triangle inequality yields q,l s s |v ´ Πq,l U v|W m,p pU q ď |R v|W m,p pU q ` |ΠU pR vq|W m,p pU q .

(10)

For the first term in the right-hand side, the estimate (9) with r “ m readily yields |Rs v|W m,p pU q À hs´m |v|W s,p pU q . U

(11)

Let us estimate the second term. If m ă q, using the boundedness assumption (6a) followed by the estimate (9), it is inferred s |Πq,l U pR vq|W m,p pU q À

q ÿ

hr´m |Rs v|W r,p pU q À U

r“m

q ÿ

s´r hr´m hU |v|W s,p pU q À hs´m |v|W s,p pU q . U U

r“m

If, on the other hand, m ě q, using the reverse Sobolev embeddings on polynomial spaces of [13, Remark A.2] followed by assumption (6b) and the estimate (9) with r “ q, it is inferred that q´m q´m s´m s s |Πq,l |Πq,l |Rs v|W q,p pU q À hU |v|W s,p pU q . U pR vq|W m,p pU q À hU U pR vq|W q,p pU q À hU

4

In conclusion we have, in either case m ă q or m ě q, s´m s |Πq,l |v|W s,p pU q . U pR vq|W m,p pU q À hU

(12)

Using (11) and (12) to estimate the first and second term in the right-hand side of (10), respectively, the conclusion follows. Our second technical result concerns the Lp -boundedness of L2 -orthogonal projectors on polynomial subspaces, and will be central to prove property (6) (with q “ 1) for the elliptic projector 1,l πU . This result generalises [13, Lemma 3.2], which corresponds to P “ Pl pU q. Lemma 4 (Lp -boundeness of L2 -orthogonal projectors on polynomial subspaces). Let two integers l ě 0 and n ě 1 be fixed, and let P be a subspace of Pl pU qn . We consider the L2 -orthogonal projector ΠP : L1 pU qn Ñ P such that, for all Φ P L1 pU qn , ż pΠP Φ ´ Φq¨Ψ “ 0 for all Ψ P P. (13) T

Let p P r1, `8s. Let rU be the inradius of U and assume that there is a real number δ such that rU ě δ ą 0. hU Then there exists a real number C ą 0 depending only on n, d, δ, l, and p such that @Φ P Lp pU qn : }ΠP Φ}Lp pU qn ď C}Φ}Lp pU qn .

(14)

Proof. We abridge as A À B the inequality A ď M B with real number M ą 0 having the same dependencies as C. Since ΠP is an L2 -orthogonal projector, (14) trivially holds with C “ 1 if p “ 2. On the other hand, if p ą 2, we have, using the reverse Lebesgue embeddings on polynomial spaces of [13, Lemma 3.2] followed by (14) for p “ 2, 1

}ΠP Φ}Lp pU qn À |U |dp

´ 12

1

}ΠP Φ}L2 pU qn À |U |dp

´ 12

}Φ}L2 pU qn .

Here, |U |d is the d-dimensional measure of U . Using the H¨older inequality to infer }Φ}L2 pU qn À 1

´1

|U |d2 p }Φ}Lp pU qn concludes the proof for p ą 2. It only remains to treat the case p ă 2. We first observe that, using the definition (13) of ΠP twice, for all Φ, Ψ P L1 pU qn , ż ż ż pΠP Φq¨Ψ “ pΠP Φq¨pΠP Ψq “ Φ¨pΠP Ψq. U

U

U

Hence, with p1 such that 1{p ` 1{p1 “ 1, it holds ż }ΠP Φ}Lp pU qn “

sup ΨPLp1 pU qn ,}Ψ}Lp1 pU qn “1 U

pΠP Φq¨Ψ

ż “

sup ΨPLp1 pU qn ,}Ψ}Lp1 pU qn “1 U

ď

sup ΨPLp1 pU qn ,}Ψ}Lp1 pU qn “1

Φ¨pΠP Ψq

(15)

}Φ}Lp pU qn }ΠP Ψ}Lp1 pU qn ,

where we have used the H¨ older inequality to conclude. Using (14) for p1 ą 2, we have }ΠP Ψ}Lp1 pU qn À }Ψ}Lp1 pU qn “ 1. Plugging this bound into (15) concludes the proof for p ă 2.

5

2.2

Proof of the main results

We are now ready to prove Theorems 1 and 2. Inside the proofs, A À B means A ď M B with M having the same dependencies as the real number C in the corresponding statement. Proof of Theorem 1. The proof consists in verifying the boundedness property (6), with q “ 1, for the elliptic projector first with m “ 1 (Step 1) then with m “ 0 (Step 2). The conclusion then 1,l follows applying Lemma 3 to Π1,l U “ πU . Step 1. |¨|W 1,p pU q -boundedness. We start by proving that 1,l @v P W 1,p pU q : |πU v|W 1,p pU q À |v|W 1,p pU q .

(16)

1,l By definition (2) of πU , it holds, for all v P W 1,1 pT q, 1,l ∇πU v “ Π∇Pl pU q ∇v,

(17)

where Π∇Pl pU q denotes the L2 -orthogonal projector on ∇Pl pU q Ă Pl´1 pU qd . Then, (16) is proved observing that, by definition (1) of the |¨|W 1,p pU q -seminorm, and invoking (17) and the pLp qd boundedness of Π∇Pl pU q resulting from (14) with P “ ∇Pl pU q, we have 1,l 1,l |πU v|W 1,p pU q À }∇πU v}Lp pU qd “ }Π∇Pl pU q ∇v}Lp pU qd À }∇v}Lp pU qd À |v|W 1,p pU q .

Step 2. }¨}Lp pU q -boundedness. We next prove that 1,l @v P W 1,p pU q : }πU v}Lp pU q À hU |v|W 1,p pU q ` }v}Lp pU q .

(18)

Let v P W 1,p pU q and denote by v P P0 pU q the L2 -orthogonal projection of v on P0 pU q such that ż ż 1 pv ´ vq “ 0, that is, v “ v. |U |d U U 1,l v. By definition (2) of the elliptic projector, v is also the L2 -orthogonal projection on P0 pU q of πU 1,l The W s,p -approximation of the L2 -projector (63) (applied with m “ 0 and s “ 1 to πU v instead 1,l 1,l v|W 1,p pU q . This yields of v) therefore gives }πU v ´ v}Lp pU q À hU |πU 1,l 1,l v ´ v}Lp pU q ` }v}Lp pU q v}Lp pU q ď }πU }πU 1,l À hU |πU v|W 1,p pU q ` }v}Lp pU q

À hU |v|W 1,p pU q ` }v}Lp pU q , where we have introduced ˘v inside the norm and used the triangle inequality in the first line, and the terms in the second line are have been estimated using (16) for the first one and the Jensen inequality for the second one. Proof of Theorem 2. Under the assumptions on U , we have the following Lp -trace inequality (cf. [13, Lemma 3.6] for a proof): For all w P W 1,p pU q, 1

hUp }w}Lp pBU q À }w}Lp pU q ` hU }∇w}Lp pU q .

(19)

1,l For m ď s´1, by applying (19) to w “ B α pv ´πU vq P W 1,p pU q for all α P Nd such that }α}1 “ m, we find 1 1,l 1,l 1,l hUp |v ´ πU v|W m,p pFU q À |v ´ πU v|W m,p pU q ` hU |v ´ πU v|W m`1,p pU q .

To conclusion follows using (3) for m and m ` 1 to bound the two terms in the right-hand side. 6

3

Error estimates for a Hybrid High-Order discretisation of Leray–Lions problems

In this section we use the approximation results for the elliptic projector to derive new error estimates for the HHO discretisation of Leray–Lions problems introduced in [13] (where convergence to minimal regularity solutions is proved using a compactness argument).

3.1

Continuous model

We consider problem (5) under the following assumptions for a fixed p P p1, `8q with p1 :“ 1

p p´1 :

f P Lp pΩq,

(20a)

a : Ω ˆ Rd Ñ Rd is a Caratheodory function,

(20b)

1

ap¨, 0q P Lp pΩqd and Dβa P p0, `8q : |apx, ξq ´ apx, 0q| ď βa |ξ|p´1 for a.e. x P Ω, for all ξ P Rd ,

(20c)

Dλa P p0, `8q : apx, ξq ¨ ξ ě λa |ξ|p for a.e. x P Ω, for all ξ P Rd , p´2

Dγa P p0, `8q : |apx, ξq ´ apx, ηq| ď γa |ξ ´ η|p|ξ| for a.e. x P Ω, for all pξ, ηq P Rd ˆ Rd ,

` |η|

p´2

(20d)

q

(20e)

Dζa P p0, `8q : rapx, ξq ´ apx, ηqs ¨ rξ ´ ηs ě ζa |ξ ´ η|2 p|ξ| ` |η|qp´2 for a.e. x P Ω, for all pξ, ηq P Rd ˆ Rd ,

(20f)

Assumptions (20b)–(20d) are the pillars of Leray–Lions operators and stipulate, respectively, the regularity for a, its growth, and its coercivity. Assumptions (20e) and (20f) additionally require the Lipschitz continuity and uniform monotonicity of a in an appropriate form. Remark 5 (p-Laplacian). A particularly important example of Leray–Lions problem is the pLaplace equation, which corresponds to the function apx, ξq “ |ξ|p´2 ξ.

(21)

Properties (20b)–(20d) are trivially verified for this choice, which additionally verifies (20e) and (20f); cf. [4] for a proof of the former and [23] for a proof of both. As usual, problem (5) is understood in the following weak sense: ż ż Find u P W01,p pΩq such that, for all v P W01,p pΩq, apx, ∇upxqq ¨ ∇vpxq dx “ f v, Ω

(22)

Ω

where W01,p pΩq is spanned by the elements of W 1,p pΩq that vanish on BΩ in the sense of traces.

3.2

The Hybrid High-Order (HHO) method

We briefly recall here the construction of the HHO method and a few known results that will be needed in the analysis.

7

3.2.1

Mesh and notations

Let us start by the notion of mesh, and some associated notations.ŤA mesh Th is a finite collection of nonempty disjoint open polytopal elements T such that Ω “ T PTh T and h “ maxT PTh hT , with hT standing for the diameter of T . A face F is defined as a hyperplanar closed connected subset of Ω with positive pd´1q-dimensional Hausdorff measure and such that (i) either there exist T1 , T2 P Th such that F Ă BT1 X BT2 and F is called an interface or (ii) there exists T P Th such that F Ă BT XBΩ and F is called a boundary face. Interfaces are collected in the set Fhi , boundary faces in Fhb , and we let Fh :“ Fhi Y Fhb . The diameter of a face F P Fh is denoted by hF . For all T P Th , FT :“ tF P Fh | F Ă BT u denotes the set of faces contained in BT (with BT denoting the boundary of T ) and, for all F P FT , nT F is the unit normal to F pointing out of T . Throughout the rest of the paper, we assume the following regularity for Th . Assumption 6 (Regularity assumption on Th ). The mesh Th admits a matching simplicial submesh Th and there exists a real number % ą 0 such that: (i) For all simplices S P Th of diameter hS and inradius rS , %hS ď rS , and (ii) for all T P Th , and all S P Th such that S Ă T , %hT ď hS . When working on refined mesh sequences, all the (explicit or implicit) constants we consider below remain bounded provided that % remains bounded away from 0 in the refinement process. Additionally, mesh elements satisfy the geometric regularity assumptions that enable the use of both Theorems 1 and 2 (as well as Lemma 13 below). 3.2.2

Degrees of freedom and interpolation operators

Let a polynomial degree k ě 0 and an element T P Th be fixed. The local space of degrees of freedom (DOFs) is ˜ ¸ ą k k k P pF q , (23) UT :“ P pT q ˆ F PFT

where Pk pF q denotes the set of pd ´ 1q-variate polynomials on F . We use the underlined notation vT “ pvT , pvF qF PFT q for a generic element vT P UkT . If U “ T P Th or U “ F P Fh , we define the 0,l 0,l L2 -projector πU : L1 pU q Ñ Pl pU q such that, for any v P L1 pU q, πU v is the unique element of l P pU q satisfying ż @w P Pl pU q : U

0,l pπU v ´ vq w “ 0.

(24)

0,l When applied to vector-valued function, it is understood that πU acts component-wise. The local interpolation operator IkT : W 1,1 pT q Ñ UkT is then given by

@v P W 1,1 pT q : IkT v :“ pπT0,k v, pπF0,k vqF PFT q.

(25)

Local DOFs are collected in the following global space obtained by patching interface values: ˜ ¸ ˜ ¸ ą ą k k k Uh :“ P pT q ˆ P pF q . T PTh

F PFh

A generic element of Ukh is denoted by vh “ ppvT qT PTh , pvF qF PFh q and, for all T P Th , vT “ pvT , pvF qF PFT q is its restriction to T . We also introduce the ( notation vh for the broken polynomial function in Pk pTh q :“ v P L1 pΩq : v |T P Pk pT q @T P Th obtained from element-based DOFs by setting vh|T “ vT for all T P Th . The global interpolation operator Ikh : W 1,1 pΩq Ñ Ukh is such that @v P W 1,1 pΩq : Ikh v :“ ppπT0,k vqT PTh , pπF0,k vqF PFh q.

8

(26)

3.2.3

Gradient and potential reconstructions

For U “ T P Th or U “ F P Fh , we denote henceforth by p¨, ¨qU the L2 - or pL2 qd -inner product on U . The HHO method hinges on the local discrete gradient operator GkT : UkT Ñ Pk pT qd such that, for all vT “ pvT , pvF qF PFT q P UkT , GkT vT is the unique solution of the following problem: For all φ P Pk pT qd , ÿ pGkT vT , φqT :“ ´pvT , div φqT ` pvF , φ¨nT F qF . (27) F PFT

In (27), the right-hand side mimicks an integration by parts formula where the role of the scalar function inside volumetric and boundary integrals is played by element-based and face-based DOFs, respectively. This recipe for the gradient reconstruction is justified observing that, as a consequence of the definitions (25) of IkT and (24) of the L2 -projector, we have the following commuting property: For all v P W 1,1 pT q, GkT IkT v “ πT0,k p∇vq.

(28)

For further use, we note the following formula inferred from (27) integrating by parts the first term in the right-hand side: For all vT P UkT and all φ P Pk pT qd , ÿ pGkT vT , φqT “ p∇vT , φqT ` pvF ´ vT , φ¨nT F qF . (29) F PFT

We also define the local potential reconstruction operator pk`1 : UkT Ñ Pk`1 pT q such that, for all T vT P UkT , ż ż k k`1 ´ G v q¨∇w “ 0 for all w P P pT q and ppk`1 (30) p∇pk`1 v T T T T T vT ´ vT q “ 0. T

T

As already noticed in [17] (cf., in particular, Eq. (17) therein), we have the following relation which establishes a link between the potential reconstruction pk`1 composed with the interpolation T operator IkT defined by (25) and the elliptic projector πT1,k`1 defined by (2): pk`1 ˝ IkT “ πT1,k`1 . T

(31)

The local gradient and potential reconstructions give rise to the global gradient operator Gkh : Ukh Ñ Pk pTh qd and potential reconstruction pk`1 : Ukh Ñ Pk`1 pTh q such that, for all vh P Ukh , h k`1 pGkh vh q|T “ GkT vT and ppk`1 h vh q|T “ pT vT for all T P Th .

3.2.4

(32)

Discrete problem

For all T P Th , we define the local function AT : UkT ˆ UkT Ñ R such that ż AT puT , vT q :“ apx, GkT uT pxqq ¨ GkT vT pxq dx ` sT puT , vT q,

(33a)

T

with sT : UkT ˆ UkT Ñ R stabilisation term such that ÿ 1´p ż ˇ ˇ ˇδTk F uT ˇp´2 δTk F uT δTk F vT , sT puT , vT q :“ hF F PFT

(33b)

F

where the scaling factor h1´p ensures the dimensional homogeneity of the terms composing AT , F and the face-based residual operator δTk F : UkT Ñ Pk pF q is defined such that, for all vT P UkT , 0,k k`1 δTk F vT :“ πF0,k pvF ´ pk`1 T vT q ´ πT pvT ´ pT vT q.

9

(33c)

A global function Ah : Ukh ˆ Ukh Ñ R is assembled element-wise from local contributions setting ÿ AT puT , vT q. Ah puh , vh q :“ (33d) T PTh

Boundary conditions are strongly enforced by considering the following subspace of Ukh : ! ) Ukh,0 :“ vh P Ukh | vF “ 0 @F P Fhb .

(33e)

The HHO approximation of problem (22) reads: ż Find uh P Ukh,0 such that, for all vh P Ukh,0 , Ah puh , vh q “

f vh .

(33f)

Ω

For a discussion on the existence and uniqueness of a solution to (33) we refer the reader to [13, Theorem 4.5 and Remark 4.7].

3.3

Error estimates

We state in this section an error estimate in terms of the following discrete W 1,p -seminorm on Ukh : ¸ p1 ˜ ¯ p1 ´ ÿ p p . (34) }vh }1,p,h :“ }vT }1,p,T , where }vT }1,p,T :“ }∇pk`1 T vT }Lp pT qd ` sT pvT , vT q T PTh

It is a simple matter to realise that the map }¨}1,p,h defines a norm on Ukh,0 . The regularity assumptions on the exact solution are expressed in terms of the broken W s,p -spaces defined by W s,p pTh q :“ tv P Lp pΩq : @T P Th , v P W s,p pT qu, which we endow with the norm ¸ p1

˜ }v}W s,p pTh q :“

ÿ

}v}pW s,p pT q

.

T PTh

Notice that, if v P W s,p pTh q for a certain mesh Th , then }v}W s,p pTh q depends only on v, not on Th . Our main result is summarised in the following theorem, whose proof makes use of the approximation results for the elliptic projector stated in Theorems 1 and 2; cf. Remark 12 for further insight into their role. Theorem 7 (Error estimate). Let the assumptions in (20) hold, and let u solve (22). Let a polynomial degree k ě 0 and a mesh Th be fixed, and let uh solve (33). Assume the additional 1 p regularity u P W k`2,p pTh q and ap¨, ∇uq P W k`1,p pTh qd (with p1 “ p´1 ), and define the quantity Eh puq as follows: • If p ě 2, ´ 1 1 k`1 p´1 p´1 Eh puq :“ hk`1 |u|W k`2,p pTh q ` h p´1 |u|W k`2,p pT q ` |ap¨, ∇uq| W k`1,p1 pT h • If p ă 2,

¯ d hq

Eh puq :“ hpk`1qpp´1q |u|p´1 ` hk`1 |ap¨, ∇uq|W k`1,p1 pTh qd . W k`2,p pTh q

;

(35a)

(35b)

Then, there exists a real number C ą 0 depending only on Ω, k, the mesh regularity parameter % defined in Assumption 6, the coefficients p, βa , λa , γa , ζa defined in (20), and an upper bound of }f }Lp1 pΩq such that (36) }Ikh u ´ uh }1,p,h ď CEh puq.

10

Proof. See Section 3.4. Some remarks are of order. Remark 8 (Order of convergence). From (36), it is inferred that the approximation error in the k`1 discrete W 1,p -norm scales as the dominant terms in Eh , namely h p´1 if p ě 2 and hpk`1qpp´1q if p ă 2. Remark 9 (Role of the various terms). There is a nice parallel between the various error terms in (35) and the error estimate obtained for gradient schemes in [23]. In the gradient schemes framework [25, 28], the accuracy of a scheme is essentially assessed through two quantities: a measure WD of the default of conformity of the scheme, and a measure SD of the consistency of the scheme. In (35), the terms involving |ap¨, ∇uq|W k`1,p1 pTh qd estimate the contribution to the error of the default of conformity of the method, and the terms involving |u|W k`2,p pTh q come from the consistency error of the method. From the convergence result in Theorem 7, we can infer an error estimate on the potential reconstruction pk`1 h uh and on its jumps measured through the stabilisation function sT . Corollary 10 (Convergence of the potential reconstruction). Under the notations and assumptions in Theorem 7, and denoting by ∇h the broken gradient on Th , we have ¸1{p

˜ }∇h pu ´

p pk`1 h uh q}Lp pΩqd

ÿ `

` ˘ ď C Eh puq ` hk`1 |u|W k`2,p pTh q ,

sT puT , uT q

(37)

T PTh

where C has the same dependencies as in Theorem 7. Proof. See Section 3.4. Remark 11 (Variations). Following [13, Remark 4.4], variations of the HHO scheme (33) are obtained replacing the space UkT defined by (23) by ˜ ¸ ą l,k l k P pF q , UT :“ P pT q ˆ F PFh

for k ě 0 and l P tk ´ 1, k, k ` 1u. For the sake of simplicity, we consider the case l “ k ´ 1 only when k ě 1 (technical modifications, not detailed here, are required for k “ 0 and l “ k ´ 1 owing to the absence of element DOFs). The interpolant IkT naturally has to be replaced with 0,l 0,k k k`1 remain formally the Il,k T v :“ pπT v, pπF vqF PFT q. The definitions (27) of GT and (30) of pT same (only the domain of the operators changes), and a close inspection shows that both key properties (28) and (31) remain valid for all the proposed choices for l –replacing, of course, IkT with Il,k T in (31). In the expression (33b) of the penalization bilinear form sT , we replace the k face-based residual δTk F defined by (33c) with a new operator δTl,kF : Ul,k T Ñ P pF q such that, for l,k all vT P UT , ´ ¯ 0,l k`1 δTl,kF vT :“ πF0,k vF ´ pk`1 T vT ´ πT pvT ´ pT vT q .

Up to minor modifications, the proof of Theorem 7 remains valid, and therefore so is the case for the error estimates (36) and (37).

3.4

Proof of the error estimates

In this section, we write A À B for A ď M B with M having the same dependencies as C in Theorem 7. The notation A « B means A À B and B À A.

11

Proof of Theorem 7. The proof is split into several steps. In Step 1 we obtain an initial estimate involving, on the left-hand side, a and sT , and, on the right-hand side, a sum of four terms. In Step 2 we prove that the left-hand side of this estimate provides an upper bound of the approximation error }Ikh u ´ uh }1,p,h . Then, in Steps 3–5, we estimate each of the four terms in the right-hand side of the original estimate. Combined with the result of Step 2, these estimates prove (36). Throughout the proof, to alleviate the notation, we write OpXq for a quantity that satisfies |OpXq| À X, and we abridge Ikh u into p uh . We will need the following equivalence of local seminorms, established in [13, Lemma 5.2]: For all vT P UkT , ˆ }vT }1,p,T «

}∇vT }pLp pT qd

ÿ

h1´p F }vF

`

´ vT }pLp pF q

˙ p1

ˆ «

}GkT vT }pLp pT qd

˙ p1 ` sT pvT , vT q . (38)

F PFT

Step 1. Initial estimate. Let vh be a generic element of Ukh,0 , and denote by vT P UkT its restriction to a generic mesh element T P Th . In this step, we estimate the error made when using p uh , instead of uh , in the scheme, namely ı ÿ ż ” ÿ :“ Eh pvh q apx, GkT p uT q ´ apx, GkT uT q ¨GkT vT ` psT pp uT , vT q ´ sT puT , vT qq. (39) T PTh

T

T PTh

Let T P Th be fixed. Setting T1,T :“ }ap¨, GkT p uT q ´ ap¨, ∇uq}Lp1 pT qd ,

(40)

by the H¨ older inequality we infer ż ż apx, GkT p uT pxqq¨GkT vT pxq dx “ apx, ∇upxqq¨GkT vT pxq dx ` OpT1,T q}GkT vT }Lp pT qd . T

T

To benefit from the definition (29) of GkT vT , we approximate ap¨, ∇uq by its L2 -orthogonal projetion on the polynomial space Pk pT qd . We therefore introduce T2,T :“ }ap¨, ∇uq ´ πT0,k ap¨, ∇uq}Lp1 pT qd ,

(41)

and we have ż apx, GkT p uT pxqq¨GkT vT pxq dx “ T ż πT0,k apx, ∇upxqq¨GkT vT pxq dx ` OpT1,T ` T2,T q}GkT vT }Lp pT qd . (42) T

Using (29) with φ “ πT0,k ap¨, ∇uq, the first term in the right-hand side rewrites ż ÿ πT0,k apx, ∇upxqq¨GkT vT pxq dx “ pπT0,k ap¨, ∇uq, ∇vT qT ` pπT0,k ap¨, ∇uq¨nT F , vF ´ vT qF . T

F PFT

We now want to eliminate the projectors πT0,k , in order to utilise the fact that u is a solution to (5). In the first term, the projector πT0,k can be cancelled simply by observing that ∇vT P Pk´1 pT qd Ă Pk pT qd , whereas for the second term we introduce an error controlled by ¸ 11

˜ T3,T :“

ÿ

hF }ap¨, ∇uq ´

F PFT

12

1 πT0,k ap¨, ∇uq}pLp1 pF q

p

(43)

1

(this quantity is well defined since ap¨, ∇uq P W 1,p pT qd by assumption). We therefore have, using the H¨ older inequality, ż ÿ pap¨, ∇uq¨nT F , vF ´ vT qF πT0,k apx, ∇upxqq¨GkT vT pxq dx “ pap¨, ∇uq, ∇vT qT ` T

F PFT

¸ p1

˜ ÿ

` OpT3,T q

1´p hF }vF

´

vT }pLp pF q

.

F PFT

We plug this expression into (42) and use the equivalence of seminorms (38) to obtain ż ÿ pap¨, ∇uq¨nT F , vF ´ vT qF apx, GkT p uT pxqq¨GkT vT pxq dx “ pap¨, ∇uq, ∇vT qT ` T

F PFT

` OpT1,T ` T2,T ` T3,T q}vT }1,p,T . Integrating by parts the first term in the right-hand side and writing ´ divpap¨, ∇uqq “ f in T , we arrive at ż apx, GkT p uT pxqq¨GkT vT pxq dx “ T ÿ pf, vT qT ` pap¨, ∇uq¨nT F , vF qF ` OpT1,T ` T2,T ` T3,T q}vT }1,p,T . F PFT

We then sum over T P Th , use ap¨, ∇uq¨nT1 F “ ´ap¨, ∇uq¨nT2 F on F whenever F P FT1 X FT2 1 (this is because ´ divpap¨, ∇uqq P Lp pΩq) together with vF “ 0 whenever F P Fhb to infer ÿ ÿ pap¨, ∇uq¨nT F , vF qF “ 0, T PTh F PFT

invoke the scheme (33), and use the H¨older inequality on the O terms to write ı ÿ ÿ ż ” sT puT , vT q apx, GkT p uT pxqq ´ apx, GkT uT pxqq ¨GkT vT pxq dx ´ T PTh

T

T PTh

“ OpT1 ` T2 ` T3 q}vh }1,p,h where, for i P t1, 2, 3u, we have set ¸ 11

˜ Ti :“

ÿ

1 Tpi,T

p

.

(44)

sT pp uT , vT q , }vh }1,p,h

(45)

T PTh

Finally, introducing the last error term ř T4 :“

T PTh

sup vh PUk h ,vh ‰0h

we have Eh pvh q “ OpT1 ` T2 ` T3 ` T4 q}vh }1,p,h .

(46)

Step 2. Lower bound for Eh pp uh ´ uh q. uh ´ uh . The goal of this step is to find a lower bound for Let, for the sake of conciseness, eh :“ p Eh peh q in terms of the error measure }eh }1,p,h . To this end, we let vh “ eh in the definition (39) of Eh and distinguish two cases. 13

Case p ě 2: Using for all T P Th the bound (70) below with ξ “ GkT p uT and η “ GkT uT for the first term in the right-hand side of (39), the definition (33b) of sT and, for all F P FT , the bound (72) below with t “ δTk F p uT and r “ δTk F uT for the second, and concluding by the norm equivalence (38), we have ˜ ¸ ÿ ÿ 1´p p p k k }GT eT }Lp pT qd ` hF }δT F eT }Lp pF q Á }eh }p1,p,h . Eh peh q Á (47) T PTh

F PFT

Case p ă 2: Let an element T P Th be fixed. Applying (69) below to ξ “ GkT p uT and η “ GkT uT , 2 , we get integrating over T and using the H¨ older inequality with exponents p2 and 2´p ˙ p2

ˆż }GkT eT }pLp pT qd

rapx, GkT p uT pxqq

À

´

apx, GkT uT pxqqs¨GkT eT pxq dx

T

´ ¯ 2´p 2 . ˆ }GkT p uT }pLp pT qd ` }GkT uT }pLp pT qd Summing over T P Th and using the discrete H¨older inequality, we obtain ´ ¯ 2´p p 2 }Gkh eh }pLp pΩqd À Eh peh q 2 ˆ }Gkh p uh }pLp pΩqd ` }Gkh uh }pLp pΩqd . 1´p

(48)

1´p

A similar reasoning starting from (71) with t “ hF p δTk F p uT and r “ hF p δTk F uT , integrating over F , summing over F P FT and using the H¨older inequality gives p

sT peT , eT q À psT pp uT , eT q ´ sT puT , eT qq 2 psT pp uT , p uT q ` sT puT , uT qq

2´p 2

.

Summing over T P Th and using the discrete H¨older inequality, we get ¸ 2´p 2

˜ ÿ T PTh

p 2

sT peT , eT q À Eh peh q ˆ

ÿ

uT q ` sT pp uT , p

T PTh

ÿ

sT puT , uT q

.

(49)

T PTh

Combining (48) and (49), and using the seminorm equivalence (38) leads to ´ ¯ 2´p p 2 }eh }p1,p,h À Eh peh q 2 ˆ }p . uh }p1,p,h ` }uh }p1,p,h From the W 1,p -boundedness of IkT and the a priori bound on }uh }1,p,h proved in [13, Proposition 7.1 and Proposition 6.1], respectively, we infer that 1{pp´1q

}p uh }1,p,h À }u}W 1,p pΩq À 1 and }uh }1,p,h À }f }Lp1 pΩq À 1,

(50)

}eh }21,p,h À Eh peh q.

(51)

so that

In conclusion, combining the initial estimate (46) with vh “ eh with the bounds (47) (if p ě 2) and (51) (if p ă 2), we obtain ˆ 1 ˙ 1 1 1 If p ě 2 : }eh }1,p,h À O T1p´1 ` T2p´1 ` T3p´1 ` T4p´1 , (52) If p ă 2 : }eh }1,p,h À O pT1 ` T2 ` T3 ` T4 q .

14

Step 3. Estimate of T1 . Recall that, by (44) and (40), ¸ 11

˜ ÿ

T1 “

}ap¨, GkT p uT q

´

p

1 ap¨, ∇uq}pLp1 pT qd

.

T PTh

Notice also that, by (28), GkT p uT “ GkT IkT u “ πT0,k p∇uq. Thus, using the approximation properties 0,k of πT summarised in Lemma 13 below (with v “ Bi u for i “ 1, . . . , d), we infer }GkT p uT ´ ∇u}Lp pT qd À hk`1 T |u|W k`2,p pT q .

(53)

Case p ě 2: Assume first p ą 2. Recalling (20e), and using the generalised H¨older inequality with p exponents pp1 , p, rq such that p11 “ p1 ` 1r (that is r “ p´2 ) together with (53) yields, for all T P Th , ´ ¯ p´2 }ap¨, GkT p uT q ´ ap¨, ∇uq}Lp1 pT qd À }GkT p uT }p´2 ` }∇u} uT ´ ∇u}Lp pT qd }GkT p p d p d L pT q L pT q ´ ¯ p´2 p´2 k k`1 uT }Lp pT qd ` }∇u}Lp pT qd . À hT |u|W k`2,p pT q }GT p This relation is obviously also valid if p “ 2. We then sum over T P Th and use, as before, the generalised H¨ older inequality, and (50) to infer ¯ ´ p´2 k`1 ` }u} |u|W k`2,p pTh q . T1 À hk`1 |u|W k`2,p pTh q }Gkh p uh }p´2 1,p p d W pΩq À h L pΩq uT ´ ∇u}p´1 Case p ă 2: By (66) below, }ap¨, GkT p uT q ´ ap¨, ∇uq}Lp1 pT qd À }GkT p . Use then (53) Lp pT qd and sum over T P Th to obtain T1 À hpk`1qpp´1q |u|p´1 . W k`2,p pTh q In conclusion, we obtain the following estimates on T1 : If p ě 2 :

T1 À hk`1 |u|W k`2,p pTh q ,

If p ă 2 :

T1 À hpk`1qpp´1q |u|p´1 . W k`2,p pTh q

(54)

Step 4. Estimate of T2 ` T3 . Owing to (44) together with the definitions (41) and (43) of T2,T and T3,T , we have ¸ ˜ ÿ ÿ 1 1 1 1 Tp2 `Tp3 “ }ap¨, ∇uq ´ πT0,k pap¨, ∇uqq}pLp1 pT qd ` hF }ap¨, ∇uq ´ πT0,k pap¨, ∇uqq}pLp1 pF qd . T PTh

F PFT

Using the approximation properties (63) and (64) of πT0,k with v replaced by the components of ap¨, ∇uq, p1 instead of p, and m “ 0, s “ k ` 1, we get 1

1

1

1

Tp2 ` Tp3 À hpk`1qp |ap¨, ∇uq|pW k`1,p1 pT 1

hq

1

d

.

1

1

1

Taking the power 1{p1 of this inequality and using pa ` bq p1 ď 2 p1 a p1 ` 2 p1 b p1 leads to T2 ` T3 À hk`1 |ap¨, ∇uq|W k`1,p1 pTh qd .

15

(55)

Step 5. Estimate of T4 . Recall that T4 is defined by (45). Using the H¨older inequality, we have for all T P Th , 1

1

uT , p uT q p1 sT pvT , vT q p . sT pp uT , vT q À sT pp ř Hence, using again the H¨ older inequality, since T PTh sT pvT , vT q ď }vh }p1,p,h , ¸ 11

˜ T4 À

p

ÿ

uT q sT pp uT , p

.

(56)

T PTh

We proceed in a similar way as in [17, Lemma 4] to estimate sT pp uT , p uT q. Let F P FT . We use the definition (33c) of the face-based residual operator δTk F together with the triangle inequality, the k relation πF0,k πT0,k “ πT0,k , the Lp pF q-boundedness (65) of πF0,k , the equality pk`1 uT “ pk`1 T p T IT u “ 1,k`1 p 1,p πT u (cf. (31)), the trace inequality (19), and the L pT q- and W pT q-boundedness (65) of πT0,k to write }δTk F p uT }Lp pF q ď }πF0,k pu ´ pk`1 uT q}Lp pF q ` }πT0,k pu ´ pk`1 uT q}Lp pF q T p T p ´1

1 1´ p

ď }u ´ πT1,k`1 u}Lp pF q ` hT p }πT0,k pu ´ πT1,k`1 uq}Lp pT q ` hT ´1

1 1´ p

ď }u ´ πT1,k`1 u}Lp pF q ` hT p }u ´ πT1,k`1 u}Lp pT q ` hT

|πT0,k pu ´ πT1,k`1 uq|W 1,p pT q

(57)

|u ´ πT1,k`1 u|W 1,p pT q .

The optimal W s,p -estimates on the elliptic projector (3) and (4) therefore give, for all F P FT , 1 k`2´ p

}δTk F p uT }Lp pF q À hT

|u|W k`2,p pT q .

1´p Raise this inequality to the power p, multiply by h1´p F , use hF hT pk`1qp pk`1qp hF Àh , and sum over F P FT to obtain

pk`2qp´1

1´p`pk`2qp´1

À hF

“

uT q À hpk`1qp |u|pW k`2,p pT q . sT pp uT , p

(58)

T4 À hpk`1qpp´1q |u|p´1 . W k`2,p pTh q

(59)

Substituted into (56), this gives

Conclusion. Use (54), (55) and (59) in (52). Remark 12 (Role of Theorems 1 and 2). Theorems 1 and 2 are used in Step 5 of the proof of Theorem 7 below to derive a bound on the stabilisation term sT when its arguments are the interpolate of the exact solution. Proof of Corollary 10. Let an element T P Th be fixed and set, as in the proof of Theorem 7, p uT :“ IkT u. Recalling the definition (33b) of sT , and using the inequality pa ` bqp ď 2p´1 ap ` 2p´1 bp ,

(60)

it is inferred sT puT , uT q “

ÿ F PFT

ż h1´p F

ÿ

|δTk F uT |p “ F

ż h1´p F

F PFT

À sT pp uT , p uT q ` sT puT ´ p uT , uT ´ p uT q.

16

|δTk F p uT ` δTk F puT ´ p uT q|p F

(61)

On the other hand, inserting pk`1 uT ´ πT1,k`1 u “ 0 (cf. (31)), and using again (60), we have T p p 1,k`1 }∇pu ´ pk`1 uq}pLp pT qd ` }∇pk`1 uT ´ uT q}pLp pT qd . T uT q}Lp pT qd À }∇pu ´ πT T pp

(62)

Summing (61) and (62), and recalling the definition (34) of }¨}1,p,T , we obtain p 1,k`1 }∇pu ´ pk`1 uq}pLp pT qd ` sT pp uT , p uT q ` }p uT ´ uT }p1,p,T . T uT q}Lp pT qd ` sT puT , uT q À }∇pu ´ πT

The result follows by summing this estimate over T P Th and invoking Theorem 1 for the first term in the right-hand side, (58) for the second, and (36) for the third. The following optimal approximation properties for the L2 -orthogonal projector were used in Step 4 of the proof of Theorem 7 with U “ T P Th . 0,l Lemma 13 (W s,p -approximation for πU ). Let U be as in Theorem 2. Let s P t0, . . . , l ` 1u and p P r1, `8s. Then, there exists C depending only on d, %, l, s and p such that, for all v P W s,p pU q, 0,l @m P t0, . . . , su : |v ´ πU v|W m,p pU q ď Chs´m |v|W s,p pU q U

(63)

and, if s ě 1, 1

0,l v|W m,p pFU q ď Chs´m |v|W s,p pU q , @m P t0, . . . , s ´ 1u : hUp |v ´ πU U

(64)

with FU , W m,p pFU q and corresponding seminorm as in Theorem 2. Proof. This result is a combination of [13, Lemmas 3.4 and 3.6]. We give here an alternative proof based on the abstract results of Section 2.1. By Lemma 4 with P “ Pl pU q, we have the following 0,l 0,l boundedness property for πU : For all v P L1 pU q, }πU v}Lp pU q ď C}v}Lp pU q with real number C ą 0 depending only on d, %, and l. The estimate (63) is then an immediate consequence of 0,l Lemma 3 with q “ 0 and Π0,l U “ πU . To prove (64), proceed as in Theorem 2 using (63) in place of (3). 0,l Corollary 14 (W s,p -boundedness of πU ). With the same notation as in Theorem 13, it holds, s,p for all v P W pU q, 0,l |πU v|W s,p pU q ď C|v|W s,p pU q . (65) 0,l 0,l v ´ v|W s,p pU q ` |v|W s,p pU q and Proof. Use the triangle inequality to write |πU v|W s,p pU q ď |πU conclude using (63) with m “ s for the first term.

3.5

Numerical examples

For the sake of completeness, we present here some new numerical examples that demonstrate the orders of convergence achieved by the HHO method in practice. The test were run using the hho software platform1 . We solve on the unit square domain Ω “ p0, 1q2 the homogeneous p-Laplace Dirichlet problem corresponding to the exact solution upxq “ sinpπx1 q sinpπx2 q, with p P t2, 3, 4u and source term inferred from u (cf. (21) for the expression of a in this case). We consider the matching triangular, Cartesian, locally refined, and (predominantly) hexagonal mesh families depicted in Figure 1 and polynomial degrees ranging from 0 to 3. The three former mesh families are taken from the FVCA5 benchmark [31], whereas the latter is taken from [19]. The local refinement in the third mesh family has no specific meaning for the problem considered here: its purpose is to demonstrate the seamless treatment of nonconforming interfaces. 1 Agence

pour la Protection des Programmes deposit number IDDN.FR.001.220005.000.S.P.2016.000.10800

17

Figure 1: Matching triangular, Cartesian, locally refined and hexagonal mesh families used in the numerical examples of Section 3.5. We report in Figure 2 the error }Ikh u ´ uh }1,p,h versus the meshsize h. From the leftmost column, we see that the error estimates are sharp for p “ 2, which confirms the results of [17] (a known superconvergence phenomenon is observed on the Cartesian mesh for k “ 0). For p “ 3, 4, better orders of convergence than the asymptotic ones (cf. Remark 8) are observed in most of the cases. One possible explanation is that the lowest-order terms in the right-hand side of (36) are not yet dominant for the specific problem data and mesh at hand. Another possibility is that compensations occur among lowest-order terms that are separately estimated in the proof of Theorem 7. For k “ 3 and p “ 3, the observed orders of convergence in the last refinement steps are inferior to the predicted value for smooth solutions, which can likely be ascribed to the violation of the regularity assumption on ap¨, ∇uq (cf. Theorem 7), due to the lack of smoothness of a for that p.

A

Inequalities involving the Leray–Lions operator

This section collects inequalities involving the Leray–Lions operator adapted from [23]. Lemma 15. Assume (20c), (20e), and p ď 2. Then, for a.e. x P Ω and all pξ, ηq P Rd ˆ Rd , |apx, ξq ´ apx, ηq| ď p2γa ` 2p´1 βa ` βa q|ξ ´ η|p´1 .

(66)

Proof. Let r ą 0. If |ξ| ě r and |η| ě r then, using (20e) and p ´ 2 ď 0, we have |apx, ξq ´ apx, ηq| ď γa |ξ ´ η|p|ξ|p´2 ` |η|p´2 q ď 2γa rp´2 |ξ ´ η|.

(67)

Otherwise, assume for example that |η| ă r. Then |ξ| ď |ξ ´ η| ` r and thus, owing to (20c), |apx, ξq ´ apx, ηq| ď |apx, ξq ´ apx, 0q| ` |apx, 0q ´ apx, ηq| ď βa p|ξ|p´1 ` |η|p´1 q ď βa p|ξ ´ η| ` rqp´1 ` βa rp´1 .

(68)

Combining (67) and (68) shows that, in either case, |apx, ξq ´ apx, ηq| ď 2γa rp´2 |ξ ´ η| ` βa p|ξ ´ η| ` rqp´1 ` βa rp´1 . Taking r “ |ξ ´ η| concludes the proof of (66). Lemma 16. Under Assumption (20f) we have, for a.e. x P Ω and all pξ, ηq P Rd ˆ Rd , • If p ă 2, ´p

|ξ ´ η|p ď ζa 2 2pp´1q

2´p 2

´

¯ p2 ´ ¯ 2´p 2 rapx, ξq ´ apx, ηqs ¨ rξ ´ ηs |ξ|p ` |η|p ;

18

(69)

k“0

k“1

k“2

k“3

10´1

10´1

10´2

10´2

10´3

10´3

10´2

´4

10

4

10´6

3

2 3/2 1 1/2

10´4

2 1 ´8

10

10´5

1

10´2.5

10´2

10´2.5

(a) Triangular, p “ 2

10´2

10´1

10´2

10´1.5

10´1

10´2 10´2

10´4 10´5

4 3

10´6

2

10´3

1

10´8 10´2.5

10´2

10´4

10´1.5

10´3

1/2

1 1

1

10´2.5

(d) Cartesian, p “ 2

10´2

1

10´1.5

10´2.5

(e) Cartesian, p “ 3

10´1

10´2

10´1.5

(f) Cartesian, p “ 4

10´1

10´1

10´2 10´3

4/3 1 2/3 1/3

3/2

2

10´7

10´2 10´2

10´4 10´5

4

10´6

2

10´3

3

3/2

2

1

4/3 1 2/3 1/3

10´3

1/2

1

10´7 10

10´2.5

(c) Triangular, p “ 4

10´2 10´3

1

10´1.5

(b) Triangular, p “ 3

10´1

4/3 1 2/3 1/3

10´5

1

10´1.5

10´4

10´4 1

´8

10´1.5

10´1

1

10´0.5

10´1.5

(g) Loc. ref., p “ 2

1

10´1

10´1.5

10´0.5

(h) Loc. ref., p “ 3

10´1

10´0.5

(i) Loc. ref., p “ 4

10´1 10´1

10´2 10´3

10´1

10´2 10´2

10´4 4

10

´5

3

10´3

3/2

2

10´6 10´7

2 1

1 1

10´2.4 10´2.2 10´2 10´1.8 10´1.6 10´1.4 10´1.2

(j) Hexagonal, p “ 2

1/2

10´4

1

10´2.4 10´2.2 10´2 10´1.8 10´1.6 10´1.4 10´1.2

(k) Hexagonal, p “ 3

4/3 1 2/3 1/3

10´3 1

10´2.4 10´2.2 10´2 10´1.8 10´1.6 10´1.4 10´1.2

(l) Hexagonal, p “ 4

Figure 2: }Ikh u ´ uh }1,p,h versus h for the mesh families of Figure 1. The slopes represent the orders of convergence expected from Theorem 7, i.e. k`1 p´1 for k P t0, . . . , 3u and p P t2, 3, 4u.

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• If p ě 2, |ξ ´ η|p ď ζa´1 rapx, ξq ´ apx, ηqs ¨ rξ ´ ηs.

(70)

Proof. Estimate (69) is obtained by raising (20f) to the power p{2 and using p|ξ| ` |η|qp ď 2p´1 p|ξ|p ` |η|p q. To prove (70), we simply write |ξ ´ η|p ď |ξ ´ η|2 p|ξ| ` |η|qp´2 . Remark 17. The (real-valued) mapping a : t ÞÑ |t|p´2 t corresponds to the p-Laplace operator in dimension 1, and it therefore satisfies (20f). Hence, by Lemma 16, 2´p ‰ ˘p |t|p´2 t ´ |r|p´2 r rt ´ rs 2 p|t|p ` |r|p q p , “ ‰ If p ě 2: |t ´ r|p ď C |t|p´2 t ´ |r|p´2 r rt ´ rs,

If p ă 2: |t ´ r|p ď C

`“

(71) (72)

where C depends only on p.

References [1] B. Andreianov, F. Boyer, and F. Hubert. On the finite-volume approximation of regular solutions of the p-Laplacian. IMA J. Numer. Anal., 26:472–502, 2006. [2] P. F. Antonietti, N. Bigoni, and M. Verani. Mimetic finite difference approximation of quasilinear elliptic problems. Calcolo, 52:45–67, 2014. [3] B. Ayuso de Dios, K. Lipnikov, and G. Manzini. The nonconforming virtual element method. ESAIM: Math. Model Numer. Anal. (M2AN), 50(3):879–904, 2016. [4] J. W. Barrett and W. B. Liu. Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow. Numer. Math., 68(4):437–456, 1994. [5] L. Beir˜ ao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo. Basic principles of virtual element methods. Math. Models Methods Appl. Sci. (M3AS), 199(23):199–214, 2013. [6] L. Beir˜ ao da Veiga, K. Lipnikov, and G. Manzini. The Mimetic Finite Difference Method for Elliptic Problems, volume 11 of Modeling, Simulation and Applications. Springer, 2014. [7] S. C. Brenner and L. R. Scott. The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics. Springer, New York, third edition, 2008. [8] F. Brezzi, K. Lipnikov, and M. Shashkov. Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal., 43(5):1872–1896, 2005. [9] F. Chave, D. A. Di Pietro, F. Marche, and F. Pigeonneau. A Hybrid High-Order method for the Cahn–Hilliard problem in mixed form. SIAM J. Numer. Anal., 2016. Accepted for publication. Preprint arXiv:1509.07384 [math.NA]. [10] B. Cockburn, D. A. Di Pietro, and A. Ern. Bridging the Hybrid High-Order and Hybridizable Discontinuous Galerkin methods. ESAIM: Math. Model. Numer. Anal. (M2AN), 50(3):635–650, 2016. [11] B. Cockburn, J. Gopalakrishnan, and F.-J. Sayas. A projection-based error analysis of HDG methods. Math. Comp., 79:1351–1367, 2010. [12] B. Cockburn, W. Qiu, and K. Shi. Conditions for superconvergence of HDG methods for second-order elliptic problems. Math. Comp., 81:1327–1353, 2012. [13] D. A. Di Pietro and J. Droniou. A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes. Math. Comp., 2016. Accepted for publication. Preprint arXiv:1508.01918 [math.NA]. [14] D. A. Di Pietro, J. Droniou, and A. Ern. A discontinuous-skeletal method for advection–diffusion–reaction on general meshes. SIAM J. Numer. Anal., 53(5):2135–2157, 2015. [15] D. A. Di Pietro and A. Ern. A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Meth. Appl. Mech. Engrg., 283:1–21, 2015. [16] D. A. Di Pietro and A. Ern. Arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes. IMA J. Numer. Anal., 2016. Published online. DOI 10.1093/imanum/drw003. [17] D. A. Di Pietro, A. Ern, and S. Lemaire. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Meth. Appl. Math., 14(4):461–472, 2014. [18] D. A. Di Pietro, A. Ern, A. Linke, and F. Schieweck. A discontinuous skeletal method for the viscositydependent Stokes problem. Comput. Meth. Appl. Mech. Engrg., 306:175–195, 2016. [19] D. A. Di Pietro and S. Lemaire. An extension of the Crouzeix–Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow. Math. Comp., 84(291):1–31, 2015.

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[20] J. I. Diaz and F. de Thelin. On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal., 25(4):1085–1111, 1994. [21] J. Droniou. Finite volume schemes for fully non-linear elliptic equations in divergence form. ESAIM: Math. Model Numer. Anal. (M2AN), 40:1069–1100, 2006. [22] J. Droniou and R. Eymard. A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math., 105:35–71, 2006. [23] J. Droniou, R. Eymard, T. Gallou¨ et, C. Guichard, and R. Herbin. Gradient schemes for elliptic and parabolic problems. 2016. In preparation. [24] J. Droniou, R. Eymard, T. Gallou¨ et, and R. Herbin. A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. (M3AS), 20(2):1–31, 2010. [25] J. Droniou, R. Eymard, T. Gallouet, and R. Herbin. Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Methods Appl. Sci. (M3AS), 23(13):2395–2432, 2013. [26] T. Dupont and R. Scott. Polynomial approximation of functions in Sobolev spaces. Math. Comp., 34(150):441– 463, 1980. [27] R. Eymard, T. Gallou¨ et, and R. Herbin. Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal., 30(4):1009–1043, 2010. [28] R. Eymard, C. Guichard, and R. Herbin. Small-stencil 3D schemes for diffusive flows in porous media. ESAIM Math. Model. Numer. Anal., 46(2):265–290, 2012. [29] R. Glowinski. Numerical methods for nonlinear variational problems. Springer Series in Computational Physics. Springer-Verlag, New York, 1984. [30] R. Glowinski and J. Rappaz. Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology. ESAIM: Math. Model Numer. Anal. (M2AN), 37(1):175–186, 2003. [31] R. Herbin and F. Hubert. Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In R. Eymard and J.-M. H´ erard, editors, Finite Volumes for Complex Applications V, pages 659–692. John Wiley & Sons, 2008. [32] Y. Kuznetsov, K. Lipnikov, and M. Shashkov. Mimetic finite difference method on polygonal meshes for diffusion-type problems. Comput. Geosci., 8:301–324, 2004. [33] W. Liu and N. Yan. Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian. Numer. Math., 89:341–378, 2001.

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