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
WW s,p -approximation properties of elliptic projectors onpolynomial spaces, with application to the error analysis of aHybrid High-Order discretisation of Leray–Lions problems ˚ Daniele A. Di Pietro : and J´erˆome Droniou ; University of Montpellier, Institut Montpelli´erain Alexander Grothendieck, 34095 Montpellier, France School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia
September 28, 2018
Abstract
In this work we prove optimal W s,p -approximation estimates (with p P r , `8s ) for el-liptic projectors on local polynomial spaces. The proof hinges on the classical Dupont–Scottapproximation theory together with two novel abstract lemmas: An approximation result forbounded projectors, and an L p -boundedness result for L -orthogonal projectors on polyno-mial subspaces. The W s,p -approximation results have general applicability to (standard orpolytopal) numerical methods based on local polynomial spaces. As an illustration, we usethese W s,p -estimates to derive novel error estimates for a Hybrid High-Order discretization ofLeray–Lions elliptic problems whose weak formulation is classically set in W ,p p Ω q for some p P p , `8q . This kind of problems appears, e.g., in the modelling of glacier motion, of in-compressible turbulent flows, and in airfoil design. Denoting by h the meshsize, we prove thatthe approximation error measured in a W ,p -like discrete norm scales as h k ` p ´ when p ě h p k ` qp p ´ q when p ă Keywords: W s,p -approximation properties of elliptic projector on polynomials, Hybrid High-Order methods, nonlinear elliptic equations, p -Laplacian, error estimates In this work we prove optimal W s,p -approximation properties for elliptic projectors on local poly-nomial spaces, and use these results to derive novel a priori error estimates for a Hybrid High-Order(HHO) discretisation of Leray–Lions elliptic equations.Let U Ă R d , d ě
1, be an open bounded connected set of diameter h U . For all integers s P N andall reals p P r , `8s , we denote by W s,p p U q the space of functions having derivatives up to degree s in L p p U q with associated seminorm | v | W s,p p U q : “ ÿ α P N d , } α } “ s }B α v } L p p U q , (1)where } α } : “ α ` . . . ` α d and B α “ B α . . . B α d d (this choice for the seminorm enables a seamlesstreatment of the case p “ `8 ). ˚ This work was partially supported by ANR project HHOMM (ANR-15-CE40-0005) : [email protected] ; [email protected] a r X i v : . [ m a t h . NA ] J a n et a polynomial degree l ě P l p U q the space of d -variate polynomialson U . The elliptic projector π ,lU : W , p U q Ñ P l p U q maps a generic function v P W , p U q on theunique polynomial π ,lU v P P l p U q obtained in the following way: We start by imposing ż U ∇ π ,lU v ¨ ∇ w “ ż U ∇ v ¨ ∇ w @ w P P l p U q . (2a)By the Riesz representation theorem in ∇ P l p U q for the L p U q d -inner product, this relation definesa unique element ∇ π ,lU v , and thus a polynomial π ,lU v up to an additive constant. This constantis then fixed by writing ż U p π ,lU v ´ v q “ . (2b)We have the following characterisation: π ,lU v “ arg min w P P l p U q , ş U p w ´ v q“ } ∇ p w ´ v q} L p U q d . The first main result of this work is summarised in the following theorem.
Theorem 1 ( W s,p -approximation for π ,lU ) . Assume that U is star-shaped with respect to everypoint in a ball of radius (cid:37)h U for some (cid:37) ą . Let s P t , . . . , l ` u and p P r , `8s . Then, thereexists a real number C ą depending only on d , (cid:37) , l , s , and p such that, for all m P t , . . . , s u and all v P W s,p p U q , | v ´ π ,lU v | W m,p p U q ď Ch s ´ mU | v | W s,p p U q . (3)The proof of Theorem 1, given in Section 2.2.1, is based on the classical Dupont–Scott approxi-mation theory [30] (cf. also Chapter 4 in Ref. [9]) and hinges on two novel abstract lemmas forprojectors on polynomial spaces: A W s,p -approximation result for projectors that satisfy a suitableboundedness property, and an L p -boundedness result for L -orthogonal projectors on polynomialsubspaces. Both results make use of the reverse Lebesgue and Sobolev embeddings for polynomialfunctions proved in Ref. [16] (cf., in particular, Lemma 5.1 and Remark A.2 therein). Followingsimilar arguments as in Section 7 of Ref. [30], the results of Theorem 1 still hold if U is a finiteunion of domains that are star-shaped with respect to balls of radius comparable to h U .The second main result concerns the approximation of traces, and therefore requires more assump-tions on the domain U . Theorem 2 ( W s,p -approximation of traces for π ,lU ) . Assume that U is a polytope which admits apartition S U into disjoint simplices S of diameter h S and inradius r S , and that there exists a realnumber (cid:37) ą such that, for all S P S U , (cid:37) h U ď (cid:37)h S ď r S . Let s P t , . . . , l ` u , p P r , `8s , and denote by F U the set of hyperplanar faces of U . Then,there exists a real number C depending only on d , (cid:37) , l , s and p such that, for all m P t , . . . , s ´ u and all v P W s,p p U q , h p U | v ´ π ,lU v | W m,p p F U q ď Ch s ´ mU | v | W s,p p U q . (4) Here, W m,p p F U q denotes the set of functions that belong to W m,p p F q for all F P F U , and |¨| W m,p p F U q the corresponding broken seminorm. The proof of Theorem 2, given in Section 2.2.2, is obtained combining the results of Theorem 1with a continuous L p -trace inequality.The approximation results of Theorems 1 and 2 are used to prove novel error estimates for theHHO method of Ref. [16] for nonlinear Leray–Lions elliptic problems of the form: Find a potential u : Ω Ñ R such that ´ div p a p x , ∇ u qq “ f in Ω ,u “ B Ω , (5)2here Ω is a bounded polytopal subset of R d with boundary B Ω, while the source term f : Ω Ñ R and the function a : Ω ˆ R d Ñ R d satisfy the requirements detailed in Eq. (20) below. Throughoutthe paper, it is assumed that Ω does not have cracks, that is, Ω lies on one side of its boundary.The family of problems (5), which contains the p -Laplace equation as a special case (cf. (21)below), appears in the modelling of glacier motion [35], of incompressible turbulent flows in porousmedia [24], and in airfoil design [33].In the context of conforming Finite Element (FE) approximations of problems which can be tracedback to the general form (5), a priori error estimates were derived in Ref. [5,35]. For nonconforming(Crouzeix–Raviart) FE approximations, error estimates are proved in Ref. [38], with convergencerates consistent with the ones presented in this work (concerning the link between the HHOmethod and nonconforming FE, cf. Remark 1 in Ref. [22] and also Ref. [8]). Error estimates fora nodal Mimetic Finite Difference (MFD) method for a particular kind of operator a and with p “ f vanishes on the boundary (additional error terms are present when this is not thecase). Finally, we also cite here Ref. [25], where the convergence study of a Mixed Finite Volume(MFV) scheme inspired by Ref. [26] is carried out using a compactness argument under minimalregularity assumptions on the exact solution.The HHO method analysed here is based on meshes composed of general polytopal elements andits formulation hinges on degrees of freedom (DOFs) that are polynomials of degree k ě G kT of degree k and a potential reconstruction operator p k ` T of degree p k ` q are devised bysolving local problems inside each mesh element T . By construction, the composition of thepotential reconstruction p k ` T with the interpolator on the DOF space coincides with the ellipticprojector π ,k ` T . The gradient and potential reconstruction operators are then used to formulatea local contribution composed of a consistent and a stabilisation term. The W s,p -approximationproperties for π ,k ` T play a crucial role in estimating the error associated with the latter. Denotingby h the meshsize, we prove in Theorem 12 below that, for smooth enough exact solutions, theapproximation error measured in a discrete W ,p -like norm converges as h k ` p ´ when p ě h p k ` qp p ´ q when 1 ă p ă
2. A detailed comparison with the literature is provided in Remark 13.As noticed in Ref. [21], the lowest-order version of the HHO method corresponding to k “ W s,p -estimates of Ref. [16] for L -projectors on polynomial spaces (see Lemma 18), are potentiallyof interest also for the study of other polytopal methods. Elliptic projections on polynomial spacesappear, e.g., in the conforming and nonconforming Virtual Element Methods (cf. Eq. (4.18) inRef. [6] and Eqs. (3.18)–(3.20) in Ref. [4], respectively). They also play a role in determiningthe high-order part of some post-processings of the potential used in the context of HybridizableDiscontinuous Galerkin methods; cf., e.g., the variation proposed in Ref. [13] of the post-processingconsidered in Refs. [14, 15].The rest of the paper is organised as follows. In Section 2 we provide the proofs of Theorems 1and 2 preceeded by the required preliminary results. In Section 3 we use these results to deriveerror estimates for the HHO discretization of problem 5. A collects some useful inequalities forLeray–Lions operators. 3 W s,p -approximation properties of the elliptic projector onpolynomial spaces This section contains the proofs of Theorems 1 and 2 preceeded by two abstract lemmas forprojectors on polynomials subspaces. Throughout the paper, to alleviate the notation, whenwriting integrals we omit the dependence on the integration variable x as well as the differentialwith the exception of those integrals involving the function a (cf. (5)). Our first lemma is an abstract approximation result valid for any projector on a polynomial spacethat satisfies a suitable boundedness property.
Lemma 3 ( W s,p -approximation for W -bounded projectors) . Assume that U is star-shaped withrespect to every point of a ball of radius (cid:37)h U for some (cid:37) ą . Let a real number p P r , `8s andfour integers l ě , s P t , . . . , l ` u , and q, m P t , . . . , s u be fixed. Let Π q,lU : W q, p U q Ñ P l p U q be a projector such that there exists a real number C ą depending only on d , (cid:37) , l , q , and p suchthat for all v P W q,p p U q ,If m ă q : | Π q,lU v | W m,p p U q ď C q ÿ r “ m h r ´ mU | v | W r,p p U q , (6a) If m ě q : | Π q,lU v | W q,p p U q ď C | v | W q,p p U q , (6b) Then, there exists a real number C ą depending only on d , (cid:37) , l , q , m , s , and p such that, forall v P W s,p p U q , | v ´ Π q,lU v | W m,p p U q ď Ch s ´ mU | v | W s,p p U q . (7) Proof.
Here A À B means A ď M B with real number M ą C in (7). Since smooth functions are dense in W s,p p U q , we can assume v P C p U q X W s,p p U q . Weconsider the following representation of v , proposed in Chapter 4 of Ref. [9]: v “ Q s v ` R s v, (8)where Q s v P P s ´ p U q Ă P l p U q is the averaged Taylor polynomial, while the remainder R s v satisfies, for all r P t , . . . , s u (cf. Lemma 4.3.8 in Ref. [9]), | R s v | W r,p p U q À h s ´ rU | v | W s,p p U q . (9)Since Π q,lU is a projector, it holds Π q,lU p Q s v q “ Q s v so that, taking the projection of (8), it isinferred Π q,lU v “ Q s v ` Π q,lU p R s v q . Subtracting this equation from (8), we arrive at v ´ Π q,lU v “ R s v ´ Π q,lU p R s v q . Hence, the triangleinequality yields | v ´ Π q,lU v | W m,p p U q ď | R s v | W m,p p U q ` | Π q,lU p R s v q| W m,p p U q . (10)For the first term in the right-hand side, the estimate (9) with r “ m readily yields | R s v | W m,p p U q À h s ´ mU | v | W s,p p U q . (11)4et us estimate the second term. If m ă q , using the boundedness assumption (6a) followed bythe estimate (9), it is inferred | Π q,lU p R s v q| W m,p p U q À q ÿ r “ m h r ´ mU | R s v | W r,p p U q À q ÿ r “ m h r ´ mU h s ´ rU | v | W s,p p U q À h s ´ mU | v | W s,p p U q . If, on the other hand, m ě q , using the reverse Sobolev embeddings on polynomial spaces ofRemark A.2 in Ref. [16] followed by assumption (6b) and the estimate (9) with r “ q , it is inferredthat | Π q,lU p R s v q| W m,p p U q À h q ´ mU | Π q,lU p R s v q| W q,p p U q À h q ´ mU | R s v | W q,p p U q À h s ´ mU | v | W s,p p U q . In conclusion we have, in either case m ă q or m ě q , | Π q,lU p R s v q| W m,p p U q À h s ´ mU | v | W s,p p U q . (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 L p -boundedness of L -orthogonal projectors on polyno-mial subspaces, and will be central to prove property (6) (with q “
1) for the elliptic projector π ,lU . This result generalises Lemma 3.2 in Ref. [16], which corresponds to P “ P l p U q . Lemma 4 ( L p -boundeness of L -orthogonal projectors on polynomial subspaces) . Let two integers l ě and n ě be fixed, and let P be a subspace of P l p U q n . We consider the L -orthogonalprojector Π P : L p U q n Ñ P such that, for all Φ P L p U q n , ż T p Π P Φ ´ Φ q¨ Ψ “ for all Ψ P P . (13) Let p P r , `8s . Let r U be the inradius of U and assume that there is a real number δ such that r U h U ě δ ą . Then, there exists a real number C ą depending only on n , d , δ , l , and p such that @ Φ P L p p U q n : } Π P Φ } L p p U q n ď C } Φ } L p p U q n . (14) Remark C in (14)) . At least on selected geometries, inequality (14) holds withconstant C independent of δ . Whether this is true in general remains an open question, whichpossibly requires different techniques than the ones used here to answer. In any case, this doesnot change the fact that the constants appearing in Theorems 1 and 2 do depend on (cid:37) . Proof.
We abridge as A À B the inequality A ď M B with real number M ą C . Since Π P is an L -orthogonal projector, (14) trivially holds with C “ p “
2. On the other hand, if p ą
2, we have, using the reverse Lebesgue embeddings on polynomialspaces of Lemma 3.2 in Ref. [16] followed by (14) for p “ } Π P Φ } L p p U q n À | U | p ´ d } Π P Φ } L p U q n À | U | p ´ d } Φ } L p U q n . Here, | U | d is the d -dimensional measure of U . Using the H¨older inequality to infer } Φ } L p U q n À| U | ´ p d } Φ } L p p U q n concludes the proof for p ą
2. It only remains to treat the case p ă
2. We firstobserve that, using the definition (13) of Π P twice, for all Φ , Ψ P L p U q n , ż U p Π P Φ q¨ Ψ “ ż U p Π P Φ q¨p Π P Ψ q “ ż U Φ ¨p Π P Ψ q . p such that p ` p “
1, it holds } Π P Φ } L p p U q n “ sup Ψ P L p p U q n , } Ψ } Lp U q n “ ż U p Π P Φ q¨ Ψ “ sup Ψ P L p p U q n , } Ψ } Lp U q n “ ż U Φ ¨p Π P Ψ qď sup Ψ P L p p U q n , } Ψ } Lp U q n “ } Φ } L p p U q n } Π P Ψ } L p p U q n , (15)where we have used the H¨older inequality to conclude. Using (14) for p ą
2, we have } Π P Ψ } L p p U q n À} Ψ } L p p U q n “
1. Plugging this bound into (15) concludes the proof for p ă 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. The proof of (3) is obtained applying Lemma 3 with q “ ,lU “ π ,lU . To prove that thecondition (6) holds, we distinguish two cases: m ě
1, treated in
Step 1 , and m “
0, treated in
Step 2 . Step 1.
The case m ě . We need to show that (6b) holds, i.e., @ v P W ,p p U q : | π ,lU v | W ,p p U q À | v | W ,p p U q . (16)By definition (2) of π ,lU , it holds, for all v P W , p U q , ∇ π ,lU v “ Π ∇ P l p U q ∇ v, (17)where Π ∇ P l p U q denotes the L -orthogonal projector on ∇ P l p U q Ă P l ´ p U q d . Then, (16) is provedobserving that, by definition (1) of the |¨| W ,p p U q -seminorm, and invoking (17) and the p L p q d -boundedness of Π ∇ P l p U q resulting from (14) with P “ ∇ P l p U q , we have | π ,lU v | W ,p p U q À } ∇ π ,lU v } L p p U q d “ } Π ∇ P l p U q ∇ v } L p p U q d À } ∇ v } L p p U q d À | v | W ,p p U q . Step 2.
The case m “ . We need to prove that (6a) holds, i.e., @ v P W ,p p U q : } π ,lU v } L p p U q À h U | v | W ,p p U q ` } v } L p p U q . (18)Let v P W ,p p U q and denote by v P P p U q the L -orthogonal projection of v on P p U q such that ż U p v ´ v q “
0, that is, v “ | U | d ż U v .By definition (2) of the elliptic projector, v is also the L -orthogonal projection on P p U q of π ,lU v .The W s,p -approximation of the L -projector (62) (applied with m “ s “ π ,lU v insteadof v ) therefore gives } π ,lU v ´ v } L p p U q À h U | π ,lU v | W ,p p U q . This yields } π ,lU v } L p p U q ď } π ,lU v ´ v } L p p U q ` } v } L p p U q À h U | π ,lU v | W ,p p U q ` } v } L p p U q À h U | v | W ,p p U q ` } v } L p p U q , ˘ v inside the norm and used the triangle inequality in the first line,and the terms in the third line are have been estimated using (16) for the first one and the Jenseninequality for the second one. Under the assumptions on U , we have the following L p -trace inequality (cf. Lemma 3.6 in Ref. [16]for a proof): For all w P W ,p p U q , h p U } w } L p pB U q À } w } L p p U q ` h U } ∇ w } L p p U q . (19)For m ď s ´
1, by applying (19) to w “ B α p v ´ π ,lU v q P W ,p p U q for all α P N d such that } α } “ m ,we find h p U | v ´ π ,lU v | W m,p p F U q À | v ´ π ,lU v | W m,p p U q ` h U | v ´ π ,lU v | W m ` ,p p U q . To conclusion follows using (3) for m and m ` In this section we use the approximation results for the elliptic projector to derive new errorestimates for the HHO discretisation of Leray–Lions problems introduced in Ref. [16] (whereconvergence to minimal regularity solutions is proved using a compactness argument).
We consider problem (5) under the following assumptions for a fixed p P p , `8q with p : “ pp ´ : f P L p p Ω q , (20a) a : Ω ˆ R d Ñ R d is a Caratheodory function , (20b) a p¨ , q P L p p Ω q d and D β a P p , `8q : | a p x , ξ q ´ a p x , q| ď β a | ξ | p ´ for a.e. x P Ω, for all ξ P R d , (20c) D λ a P p , `8q : a p x , ξ q ¨ ξ ě λ a | ξ | p for a.e. x P Ω, for all ξ P R d , (20d) D γ a P p , `8q : | a p x , ξ q ´ a p x , η q| ď γ a | ξ ´ η |p| ξ | p ´ ` | η | p ´ q for a.e. x P Ω, for all p ξ , η q P R d ˆ R d , (20e) D ζ a P p , `8q : r a p x , ξ q ´ a p x , η qs ¨ r ξ ´ η s ě ζ a | ξ ´ η | p| ξ | ` | η |q p ´ for a.e. x P Ω, for all p ξ , η q P R d ˆ R d , (20f)Assumptions (20b)–(20d) are the pillars of Leray–Lions operators and stipulate, respectively, theregularity for a , its growth, and its coercivity. Assumptions (20e) and (20f) additionally requirethe Lipschitz continuity and uniform monotonicity of a in an appropriate form. Remark p -Laplacian) . A particularly important example of Leray–Lions problem is the p -Laplace equation, which corresponds to the function a p x , ξ q “ | ξ | p ´ ξ . (21)Properties (20b)–(20d) are trivially verified for this choice, which additionally verifies (20e) and(20f); cf. Ref. [5] for a proof of the former and Ref. [27] for a proof of both.7s usual, problem (5) is understood in the following weak sense:Find u P W ,p p Ω q such that, for all v P W ,p p Ω q , ż Ω a p x , ∇ u p x qq ¨ ∇ v p x q d x “ ż Ω f v, (22)where W ,p p Ω q is spanned by the elements of W ,p p Ω q that vanish on B Ω in the sense of traces.
We briefly recall here the construction of the HHO method and a few known results that will beneeded in the analysis.
Let us start by the notion of mesh, inspired from Definition 7.2 in Ref. [27], and some associatednotations.
Definition 7 (Mesh and set of faces) . A mesh T h of the domain Ω is a finite collection of nonemptydisjoint open polytopal elements T with boundary B T and diameter h T such that Ω “ Ť T P T h T and h “ max T P T h h T .The set of faces F h is a finite family of disjoint subsets of Ω such that, for any F P F h , F isan open subset of a hyperplane of R d , the p d ´ q -dimensional Hausdorff measure of F is strictlypositive, and the p d ´ q -dimensional Hausdorff measure of its relative interior F z F is zero. Thediameter of F is denoted by h F . Additionally,(i) For each F P F h , either (a) there exist distinct mesh elements T , T P T h such that F ĂB T X B T and F is called an interface or (b) there exists a mesh element T P T h (whichis unique since Ω is assumed to have no cracks) such that F Ă B T X B
Ω and F is called aboundary face.(ii) The set of faces is a partition of the mesh skeleton: Ť T P T h B T “ Ť F P F h F .For any mesh element T P T h , F T : “ t F P F h | F Ă B T u denotes the set of faces contained in B T .For all F P F T , n T F is the unit normal to F pointing out of T .Interfaces are collected in the set F i h , boundary faces in F b h , and F h “ F i h Y F b h . Remark . As a result of Definition 7, above, it holds that B T “ Ť F P F T F for all T P T h , and that B Ω “ Ť F P F b h F .Throughout the rest of the paper, we assume the following regularity for T h inspired by Chapter1 in Ref. [18]. Assumption 9 (Regularity assumption on T h ) . The mesh T h admits a matching simplicial sub-mesh T h and there exists a real number (cid:37) ą S P T h of diameter h S and inradius r S , (cid:37)h S ď r S , and (ii) for all T P T h , and all S P T h such that S Ă T , (cid:37)h T ď h S .When working on refined mesh sequences, all the (explicit or implicit) constants we considerbelow remain bounded provided that (cid:37) remains bounded away from 0 in the refinement process.Additionally, mesh elements satisfy the geometric regularity assumptions that enable the use ofboth Theorems 1 and 2 (as well as Lemma 18 below).8 .2.2 Degrees of freedom and interpolation operators Let a polynomial degree k ě T P T h be fixed. The local space of degrees offreedom (DOFs) is U kT : “ P k p T q ˆ ˜ ą F P F T P k p F q ¸ , (23)where P k p F q denotes the space spanned by the restriction to F of d -variate polynomials. We usethe underlined notation v T “ p v T , p v F q F P F T q for a generic element v T P U kT . If U “ T P T h or U “ F P F h , we define the L -projector π ,lU : L p U q Ñ P l p U q such that, for any v P L p U q , π ,lU v is the unique element of P l p U q satisfying @ w P P l p U q : ż U p π ,lU v ´ v q w “ . (24)When applied to vector-valued function, it is understood that π ,lU acts component-wise. The localinterpolation operator I kT : W , p T q Ñ U kT is then given by @ v P W , p T q : I kT v : “ p π ,kT v, p π ,kF v q F P F T q . (25)Local DOFs are collected in the following global space obtained by patching interface values: U kh : “ ˜ ą T P T h P k p T q ¸ ˆ ˜ ą F P F h P k p F q ¸ . A generic element of U kh is denoted by v h “ pp v T q T P T h , p v F q F P F h q and, for all T P T h , v T “p v T , p v F q F P F T q is its restriction to T . We also introduce the notation v h for the broken polynomialfunction in P k p T h q : “ (cid:32) v P L p Ω q : v | T P P k p T q @ T P T h ( obtained from element-based DOFs bysetting v h | T “ v T for all T P T h . The global interpolation operator I kh : W , p Ω q Ñ U kh is such that @ v P W , p Ω q : I kh v : “ pp π ,kT v q T P T h , p π ,kF v q F P F h q . (26) For U “ T P T h or U “ F P F h , we denote henceforth by p¨ , ¨q U the L - or p L q d -inner producton U . The HHO method hinges on the local discrete gradient operator G kT : U kT Ñ P k p T q d suchthat, for all v T “ p v T , p v F q F P F T q P U kT , G kT v T solves the following problem: For all φ P P k p T q d , p G kT v T , φ q T : “ ´p v T , div φ q T ` ÿ F P F T p v F , φ ¨ n T F q F . (27)Existence and uniqueness of G kT v T immediately follow from the Riesz representation theorem in P k p T q d for the standard L p T q d -inner product. The right-hand side of (27) mimicks an integrationby parts formula where the role of the scalar function inside volumetric and boundary integrals isplayed by element-based and face-based DOFs, respectively. This recipe for the gradient recon-struction is justified by the commuting property in the following proposition. Proposition 10 (Commuting property) . For all v P W , p T q , it holds that G kT I kT v “ π ,kT p ∇ v q . (28)9 roof. Plugging the definition (25) of I kT into (27), it is inferred for all φ P P k p T q d that p G kT I kT v, φ q T “ ´p π ,kT v, div φ q T ` ÿ F P F T p π ,kF v, φ ¨ n T F q F “ ´p v, div φ q T ` ÿ F P F T p v, φ ¨ n T F q F , where, to cancel the projectors in the second line, we have used (24) together with the fact thatdiv φ P P k ´ p T q Ă P k p T q and that φ | F ¨ n T F P P k p F q for all F P F T . Integrating by parts theright-hand side, we conclude that ż T p G kT I kT v ´ ∇ v q¨ φ “ @ φ P P k p T q d . Comparing with (24) the conclusion follows.For further use, we note the following formula inferred from (27) integrating by parts the firstterm in the right-hand side: For all v T P U kT and all φ P P k p T q d , p G kT v T , φ q T “ p ∇ v T , φ q T ` ÿ F P F T p v F ´ v T , φ ¨ n T F q F . (29)We also define the local potential reconstruction operator p k ` T : U kT Ñ P k ` p T q such that, for all v T P U kT , ż T p ∇ p k ` T v T ´ G kT v T q¨ ∇ w “ w P P k ` p T q , ż T p p k ` T v T ´ v T q “ . (30)As already noticed in Ref. [21] (cf., in particular, Eq. (17) therein), we have the following relationwhich establishes a link between the potential reconstruction p k ` T composed with the interpolationoperator I kT defined by (25) and the elliptic projector π ,k ` T defined by (2): p k ` T ˝ I kT “ π ,k ` T . (31)The local gradient and potential reconstructions give rise to the global gradient operator G kh : U kh Ñ P k p T h q d and potential reconstruction p k ` h : U kh Ñ P k ` p T h q such that, for all v h P U kh , p G kh v h q | T “ G kT v T and p p k ` h v h q | T “ p k ` T v T for all T P T h . (32) For all T P T h , we define the local function A T : U kT ˆ U kT Ñ R such thatA T p u T , v T q : “ ż T a p x , G kT u T p x qq ¨ G kT v T p x q d x ` s T p u T , v T q , (33a)with s T : U kT ˆ U kT Ñ R the stabilisation term such that s T p u T , v T q : “ ÿ F P F T h ´ pF ż F ˇˇ δ kT F u T ˇˇ p ´ δ kT F u T δ kT F v T . (33b)In (33b), the scaling factor h ´ pF ensures the dimensional homogeneity of the terms composing A T ,and the face-based residual operator δ kT F : U kT Ñ P k p F q is defined such that, for all v T P U kT , δ kT F v T : “ π ,kF p v F ´ p p k ` T v T q | F q ´ p π ,kT p v T ´ p k ` T v T qq | F . (33c)10 global function A h : U kh ˆ U kh Ñ R is assembled element-wise from local contributions settingA h p u h , v h q : “ ÿ T P T h A T p u T , v T q . (33d)Boundary conditions are strongly enforced by considering the following subspace of U kh : U kh, : “ ! v h P U kh | v F ” @ F P F b h ) . (33e)The HHO approximation of problem (22) reads:Find u h P U kh, such that, for all v h P U kh, , A h p u h , v h q “ ż Ω f v h . (33f)For a discussion on the existence and uniqueness of a solution to (33) we refer the reader toTheorem 4.5 and Remark 4.7 in Ref. [16]. We state in this section an error estimate in terms of the following discrete W ,p -seminorm on U kh : } v h } ,p,h : “ ˜ ÿ T P T h } v T } p ,p,T ¸ p , (34)where, for all T P T h , } v T } ,p,T : “ ´ } ∇ p k ` T v T } pL p p T q d ` s T p v T , v T q ¯ p . Proposition 11 (Norm }¨} ,p,h ) . The map }¨} ,p,h defines a norm on U kh, .Proof. The semi-norm property is trivial, so it suffices to prove that, for all v h P U kh, , } v h } ,p,h “ ùñ v h “ . Let v h P U kh, be such that } v h } ,p,h “
0. The semi-norm equivalence proved inLemma 5.2 of Ref. [16] (see also (40) below) shows that ÿ T P T h } ∇ v T } pL p p T q d ` ÿ T P T h ÿ F P F T h ´ pF } v F ´ p v T q | F } pL p p F q “ . Hence, all p v T q T P T h are constant polynomials and, for any F P F T , v F ” p v T q | F . Starting fromboundary mesh elements T P T h for which there exists F P F T X F b h , using the fact that v F ” F P F b h , and proceeding from neighbour to neighbour towards the interior of the domain,we infer that v T ” T P T h and v F ” F P F h .The regularity assumptions on the exact solution are expressed in terms of the broken W s,p -spacesdefined by W s,p p T h q : “ t v P L p p Ω q : @ T P T h , v P W s,p p T qu , which we endow with the norm } v } W s,p p T h q : “ ˜ ÿ T P T h } v } pW s,p p T q ¸ p . Notice that, if v P W s,p p T h q for h “ h and h “ h with h , h P H , then } v } W s,p p T h q “} v } W s,p p T h q . Our main result is summarised in the following theorem, whose proof makes use ofthe approximation results for the elliptic projector stated in Theorems 1 and 2; cf. Remark 17 forfurther insight into their role. 11 heorem 12 (Error estimate) . Let the assumptions in (20) hold, and let u solve (22) . Let apolynomial degree k ě , a mesh T h , and a set of faces F h be fixed, and let u h solve (33) . Assumethe additional regularity u P W k ` ,p p T h q and a p¨ , ∇ u q P W k ` ,p p T h q d (with p “ pp ´ ), and definethe quantity E h p u q as follows: • If p ě , E h p u q : “ h k ` | u | W k ` ,p p T h q ` h k ` p ´ ´ | u | p ´ W k ` ,p p T h q ` | a p¨ , ∇ u q| p ´ W k ` ,p p T h q d ¯ ; (35a) • If p ă , E h p u q : “ h p k ` qp p ´ q | u | p ´ W k ` ,p p T h q ` h k ` | a p¨ , ∇ u q| W k ` ,p p T h q d . (35b) Then, there exists a real number C ą depending only on Ω , k , the mesh regularity parameter (cid:37) defined in Assumption 9, the coefficients p , β a , λ a , γ a , ζ a defined in (20) , and an upper bound of } f } L p p Ω q such that } I kh u ´ u h } ,p,h ď C E h p u q . (36) Proof.
See Section 3.4.1.Some remarks are of order.
Remark
13 (Orders of convergence) . From (36), it is inferred that the approximation error in thediscrete W ,p -norm scales as the dominant terms in E h , namely h k ` p ´ if p ě , (37) h p k ` qp p ´ q if 1 ă p ă
2. (38)Let us discuss how these orders compare with some known results for P approximations of the p -Laplacian, starting from conforming approximations. In Theorem 5.3.5 of Ref. [12], an order h p ´ is established in the case p ě
2, which is identical to (37) with k “
0. The case 1 ă p ă h ´ p is proved. This order is better than (38) for k “
0, but the proof relies on the fact that the P finite element method is a conforming method(see Eq. (6.5) in this reference). These latter rates are improved to order h in Ref. [5], but undera C ,α regularity assumption on u . The case p ě h estimate is obtained in W ,q for some q ă | f | (and, thus, on | ∇ u | ). All these analyses strongly use the conformity of the P element.Let us now consider nonconforming approximations, to which the HHO method proposed in thiswork belongs. The rates of convergence established in Ref. [2] for the DDFV method are identicalto (37)–(38) for k “
0. Similar considerations apply to the Crouzeix–Raviart approximationconsidered in Ref. [38], which is strongly linked to our HHO methods for k “ ă p ď
2. This suggests, in turn, that the order of convergence depends on the polynomialdegree k , on the index p , and on the conformity properties of the method. We emphasize that,despite the reduced order of convergence with respect to the best estimate for conforming P schemes and 1 ă p ă
2, the HHO method proposed here has the key advantage of supportinggeneral meshes, as well as arbitrary orders of approximation.
Remark
14 (Role of the various terms) . There is a nice parallel between the various error termsin (35) and the error estimate obtained for the gradient discretisation method in Ref. [27]. In theframework of the gradient discretisation method [29, 32], the accuracy of a scheme is essentially12ssessed through two quantities: a measure W D of the default of conformity of the scheme, and ameasure S D of the consistency of the scheme. In (35), the terms involving | a p¨ , ∇ u q| W k ` ,p p T h q d estimate the contribution to the error of the default of conformity of the method, and the termsinvolving | u | W k ` ,p p T h q come from the consistency error of the method.From the convergence result in Theorem 12, we can infer an error estimate on the potentialreconstruction p k ` h u h and on its jumps measured through the stabilisation function s T . Corollary 15 (Convergence of the potential reconstruction) . Under the notations and assump-tions in Theorem 12, and denoting by ∇ h the broken gradient on T h , we have ˜ } ∇ h p u ´ p k ` h u h q} pL p p Ω q d ` ÿ T P T h s T p u T , u T q ¸ p ď C ` E h p u q ` h k ` | u | W k ` ,p p T h q ˘ , (39) where C has the same dependencies as in Theorem 12.Proof. See Section 3.4.2.
Remark
16 (Variations) . Following Remark 4.4 in Ref. [16], variations of the HHO scheme (33)are obtained replacing the space U kT defined by (23) by U l,kT : “ P l p T q ˆ ˜ ą F P F h P k p F q ¸ , for k ě l P t k ´ , k, k ` u . For the sake of simplicity, we consider the case l “ k ´ k ě k “ l “ k ´ I kT naturally has to be replaced with I l,kT v : “ p π ,lT v, p π ,kF v q F P F T q . The definitions (27) of G kT and (30) of p k ` T remain formally thesame (only the domain of the operators changes), and a close inspection shows that both keyproperties (28) and (31) remain valid for all the proposed choices for l (replacing, of course, I kT with I l,kT in both cases). In the expression (33b) of the penalization bilinear form s T , we replace theface-based residual δ kT F defined by (33c) with a new operator δ l,kT F : U l,kT Ñ P k p F q such that, forall v T P U l,kT , δ l,kT F v T : “ π ,kF r v F ´ p p k ` T v T q | F ´ p π ,lT p v T ´ p k ` T v T qq | F s . Up to minor modifications,the proof of Theorem 12 remains valid, and therefore so is the case for the error estimates (36)and (39).
In this section, we write A À B for A ď M B with M having the same dependencies as C inTheorem 12. The notation A « B means A À B and B À A . The proof is split into several steps. In
Step 1 we obtain an initial estimate involving, on the left-hand side, a and s T , and, on the right-hand side, a sum of four terms. In Step 2 we prove that theleft-hand side of this estimate provides an upper bound of the approximation error } I kh u ´ u h } ,p,h .Then, in Steps 3–5 , we estimate each of the four terms in the right-hand side of the originalestimate. Combined with the result of
Step 2 , these estimates prove (36).Throughout the proof, to alleviate the notation, we write O p X q for a quantity that satisfies | O p X q| À X , and we abridge I kh u into p u h . 13e will need the following equivalence of local seminorms, established in Lemma 5.2 of Ref. [16]:For all v T P U kT , } v T } ,p,T « ˆ } ∇ v T } pL p p T q d ` ÿ F P F T h ´ pF } v F ´ v T } pL p p F q ˙ p « ˆ } G kT v T } pL p p T q d ` s T p v T , v T q ˙ p . (40) Step 1.
Initial estimate.
Let v h be a generic element of U kh, , and denote by v T P U kT its restrictionto a generic mesh element T P T h . In this step, we estimate the error made when using p u h , insteadof u h , in the scheme, namely E h p v h q : “ ÿ T P T h ż T ” a p x , G kT p u T q ´ a p x , G kT u T q ı ¨ G kT v T ` ÿ T P T h p s T p p u T , v T q ´ s T p u T , v T qq . (41)Let T P T h be fixed. Setting T ,T : “ } a p¨ , G kT p u T q ´ a p¨ , ∇ u q} L p p T q d , (42)by the H¨older inequality we infer ż T a p x , G kT p u T p x qq¨ G kT v T p x q d x “ ż T a p x , ∇ u p x qq¨ G kT v T p x q d x ` O p T ,T q} G kT v T } L p p T q d . To benefit from the definition (27) of G kT v T , we approximate a p¨ , ∇ u q by its L -orthogonal proje-tion on the polynomial space P k p T q d . We therefore introduce T ,T : “ } a p¨ , ∇ u q ´ π ,kT a p¨ , ∇ u q} L p p T q d , (43)and we have ż T a p x , G kT p u T p x qq¨ G kT v T p x q d x “ ż T π ,kT a p x , ∇ u p x qq¨ G kT v T p x q d x ` O p T ,T ` T ,T q} G kT v T } L p p T q d . (44)Using (29) with φ “ π ,kT a p¨ , ∇ u q , the first term in the right-hand side rewrites ż T π ,kT a p x , ∇ u p x qq¨ G kT v T p x q d x “ p π ,kT a p¨ , ∇ u q , ∇ v T q T ` ÿ F P F T p π ,kT a p¨ , ∇ u q¨ n T F , v F ´ v T q F . We now want to eliminate the projectors π ,kT , in order to utilise the fact that u is a solution to (5).In the first term, the projector π ,kT can be cancelled simply by observing that ∇ v T P P k ´ p T q d Ă P k p T q d , whereas for the second term we introduce an error that we want to control by T ,T : “ ˜ ÿ F P F T h F } a p¨ , ∇ u q ´ π ,kT a p¨ , ∇ u q} p L p p F q ¸ p (45)14this quantity is well defined since a p¨ , ∇ u q P W ,p p T q d by assumption). We therefore have, usingthe H¨older inequality, ż T π ,kT a p x , ∇ u p x qq¨ G kT v T p x q d x “ p a p¨ , ∇ u q , ∇ v T q T ` ÿ F P F T p a p¨ , ∇ u q¨ n T F , v F ´ v T q F ` O p T ,T q ˜ ÿ F P F T h ´ pF } v F ´ v T } pL p p F q ¸ p . We plug this expression into (44) and use the equivalence of seminorms (40) to obtain ż T a p x , G kT p u T p x qq¨ G kT v T p x q d x “ p a p¨ , ∇ u q , ∇ v T q T ` ÿ F P F T p a p¨ , ∇ u q¨ n T F , v F ´ v T q F ` O p T ,T ` T ,T ` T ,T q} v T } ,p,T . Integrating by parts the first term in the right-hand side and writing ´ div p a p¨ , ∇ u qq “ f in T ,we arrive at ż T a p x , G kT p u T p x qq¨ G kT v T p x q d x “p f, v T q T ` ÿ F P F T p a p¨ , ∇ u q¨ n T F , v F q F ` O p T ,T ` T ,T ` T ,T q} v T } ,p,T . We then sum over T P T h , use a p¨ , ∇ u q¨ n T F “ ´ a p¨ , ∇ u q¨ n T F on every interface F P F i h suchthat F P F T X F T for distinct mesh elements T , T P T h (this is because ´ div p a p¨ , ∇ u qq P L p p Ω q )together with v F “ F P F b h to infer ÿ T P T h ÿ F P F T p a p¨ , ∇ u q¨ n T F , v F q F “ , invoke the scheme (33), and use the H¨older inequality on the O terms to write ÿ T P T h ż T ” a p x , G kT p u T p x qq ´ a p x , G kT u T p x qq ı ¨ G kT v T p x q d x ´ ÿ T P T h s T p u T , v T q“ O p T ` T ` T q} v h } ,p,h where, for i P t , , u , we have set T i : “ ˜ ÿ T P T h T p i,T ¸ p . (46)Finally, introducing the last error term T : “ sup v h P U kh , v h ‰ h ř T P T h s T p p u T , v T q} v h } ,p,h , (47)we have E h p v h q “ O p T ` T ` T ` T q} v h } ,p,h . (48)15 tep 2. Lower bound for E h p p u h ´ u h q . Let, for the sake of conciseness, e h : “ p u h ´ u h . The goal of this step is to find a lower bound for E h p e h q in terms of the error measure } e h } ,p,h . To this end, we let v h “ e h in the definition (41)of E h and distinguish two cases.Case p ě
2: Using for all T P T h the bound (72) below with ξ “ G kT p u T and η “ G kT u T for the firstterm in the right-hand side of (41), the definition (33b) of s T and, for all F P F T , the bound (74)below with t “ δ kT F p u T and r “ δ kT F u T for the second, and concluding by the norm equivalence(40), we have E h p e h q Á ÿ T P T h ˜ } G kT e T } pL p p T q d ` ÿ F P F T h ´ pF } δ kT F e T } pL p p F q ¸ Á } e h } p ,p,h . (49)Case p ă
2: Let an element T P T h be fixed. Applying (71) below to ξ “ G kT p u T and η “ G kT u T ,integrating over T and using the H¨older inequality with exponents p and ´ p , we get } G kT e T } pL p p T q d À ˆż T r a p x , G kT p u T p x qq ´ a p x , G kT u T p x qqs¨ G kT e T p x q d x ˙ p ˆ ´ } G kT p u T } pL p p T q d ` } G kT u T } pL p p T q d ¯ ´ p . Summing over T P T h and using the discrete H¨older inequality, we obtain } G kh e h } pL p p Ω q d À E h p e h q p ˆ ´ } G kh p u h } pL p p Ω q d ` } G kh u h } pL p p Ω q d ¯ ´ p . (50)A similar reasoning starting from (73) with t “ h ´ pp F δ kT F p u T and r “ h ´ pp F δ kT F u T , integrating over F , summing over F P F T and using the H¨older inequality gives s T p e T , e T q À p s T p p u T , e T q ´ s T p u T , e T qq p p s T p p u T , p u T q ` s T p u T , u T qq ´ p . Summing over T P T h and using the discrete H¨older inequality, we get ÿ T P T h s T p e T , e T q À E h p e h q p ˆ ˜ ÿ T P T h s T p p u T , p u T q ` ÿ T P T h s T p u T , u T q ¸ ´ p . (51)Combining (50) and (51), and using the seminorm equivalence (40) leads to } e h } p ,p,h À E h p e h q p ˆ ´ } p u h } p ,p,h ` } u h } p ,p,h ¯ ´ p . From the W ,p -boundedness of I kT and the a priori bound on } u h } ,p,h proved in Propositions 7.1and 6.1 of Ref. [16], respectively, we infer that } p u h } ,p,h À } u } W ,p p Ω q À } u h } ,p,h À } f } p p ´ q L p p Ω q À , (52)so that } e h } ,p,h À E h p e h q . (53)In conclusion, combining the initial estimate (48) with v h “ e h with the bounds (49) (if p ě p ă p ě } e h } ,p,h À O ˆ T p ´ ` T p ´ ` T p ´ ` T p ´ ˙ , If p ă } e h } ,p,h À O p T ` T ` T ` T q . (54)16 tep 3. Estimate of T . Recall that, by (46) and (42), T “ ˜ ÿ T P T h } a p¨ , G kT p u T q ´ a p¨ , ∇ u q} p L p p T q d ¸ p . Notice also that, by (28), G kT p u T “ G kT I kT u “ π ,kT p ∇ u q . Thus, using the approximation propertiesof π ,kT summarised in Lemma 18 below (with v “ B i u for i “ , . . . , d ), we infer } G kT p u T ´ ∇ u } L p p T q d À h k ` T | u | W k ` ,p p T q . (55)Case p ě
2: Assume first p ą
2. Recalling (20e), and using the generalised H¨older inequality withexponents p p , p, r q such that p “ p ` r (that is r “ pp ´ ) together with (55) yields, for all T P T h , } a p¨ , G kT p u T q ´ a p¨ , ∇ u q} L p p T q d À } G kT p u T ´ ∇ u } L p p T q d ´ } G kT p u T } p ´ L p p T q d ` } ∇ u } p ´ L p p T q d ¯ À h k ` T | u | W k ` ,p p T q ´ } G kT p u T } p ´ L p p T q d ` } ∇ u } p ´ L p p T q d ¯ . This relation is obviously also valid if p “
2. We then sum over T P T h and use, as before, thegeneralised H¨older inequality, the estimate } G kh p u h } L p p Ω q d À } p u h } ,h,p À } u } W ,p p Ω q (see (40) andProposition 7.1 in Ref. [16]), and (52) to infer T À h k ` | u | W k ` ,p p T h q ´ } G kh p u h } p ´ L p p Ω q d ` } u } p ´ W ,p p Ω q ¯ À h k ` | u | W k ` ,p p T h q . Case p ă
2: By (68) below, } a p¨ , G kT p u T q ´ a p¨ , ∇ u q} L p p T q d À } G kT p u T ´ ∇ u } p ´ L p p T q d . Use then (55)and sum over T P T h to obtain T À h p k ` qp p ´ q | u | p ´ W k ` ,p p T h q .In conclusion, we obtain the following estimates on T :If p ě T À h k ` | u | W k ` ,p p T h q , If p ă T À h p k ` qp p ´ q | u | p ´ W k ` ,p p T h q . (56) Step 4.
Estimate of T ` T . Owing to (46) together with the definitions (43) and (45) of T ,T and T ,T , we have T p ` T p “ ÿ T P T h ˜ } a p¨ , ∇ u q ´ π ,kT p a p¨ , ∇ u qq} p L p p T q d ` ÿ F P F T h F } a p¨ , ∇ u q ´ π ,kT p a p¨ , ∇ u qq} p L p p F q d ¸ . Using the approximation properties (62) and (63) of π ,kT with v replaced by the components of a p¨ , ∇ u q , p instead of p , and m “ s “ k `
1, we get T p ` T p À h p k ` q p | a p¨ , ∇ u q| p W k ` ,p p T h q d . Taking the power 1 { p of this inequality and using p a ` b q p ď a p ` b p leads to T ` T À h k ` | a p¨ , ∇ u q| W k ` ,p p T h q d . (57)17 tep 5. Estimate of T . Recall that T is defined by (47). Using the H¨older inequality, we have for all T P T h , s T p p u T , v T q À s T p p u T , p u T q p s T p v T , v T q p . Hence, using again the H¨older inequality, since ř T P T h s T p v T , v T q ď } v h } p ,p,h by (40), T À ˜ ÿ T P T h s T p p u T , p u T q ¸ p . (58)We proceed in a similar way as in Lemma 4 of Ref. [21] to estimate s T p p u T , p u T q . Let F P F T . We usethe definition (33c) of the face-based residual operator δ kT F together with the triangle inequality,the relation π ,kF π ,kT “ π ,kT , the L p p F q -boundedness (64) of π ,kF , the equality p k ` T p u T “ p k ` T I kT u “ π ,k ` T u (cf. (31)), the trace inequality (19), and the L p p T q - and W ,p p T q -boundedness (64) of π ,kT to write } δ kT F p u T } L p p F q ď } π ,kF p u ´ p k ` T p u T q} L p p F q ` } π ,kT p u ´ p k ` T p u T q} L p p F q À } u ´ π ,k ` T u } L p p F q ` h ´ p T } π ,kT p u ´ π ,k ` T u q} L p p T q ` h ´ p T | π ,kT p u ´ π ,k ` T u q| W ,p p T q À } u ´ π ,k ` T u } L p p F q ` h ´ p T } u ´ π ,k ` T u } L p p T q ` h ´ p T | u ´ π ,k ` T u | W ,p p T q . (59)The optimal W s,p -estimates on the elliptic projector (3) and (4) therefore give, for all F P F T , } δ kT F p u T } L p p F q À h k ` ´ p T | u | W k ` ,p p T q . Raise this inequality to the power p , multiply by h ´ pF , use h ´ pF h p k ` q p ´ T À h ´ p `p k ` q p ´ F “ h p k ` q pF À h p k ` q p , and sum over F P F T to obtain s T p p u T , p u T q À h p k ` q p | u | pW k ` ,p p T q . (60)Substituted into (58), this gives T À h p k ` qp p ´ q | u | p ´ W k ` ,p p T h q . (61) Conclusion.
Use (56), (57), and (61) in (54).
Remark
17 (Role of Theorems 1 and 2) . Theorems 1 and 2 are used through Eqs. (3) and (4) in
Step 5 of the above proof to derive a bound on the stabilisation term s T , when its arguments arethe interpolate of the exact solution.The following optimal approximation properties for the L -orthogonal projector were used in Step4 of the above with U “ T P T h . Lemma 18 ( W s,p -approximation for π ,lU ) . Let U be as in Theorem 2. Let s P t , . . . , l ` u and p P r , `8s . Then, there exists C depending only on d , (cid:37) , l , s and p such that, for all v P W s,p p U q , @ m P t , . . . , s u : | v ´ π ,lU v | W m,p p U q ď Ch s ´ mU | v | W s,p p U q (62) and, if s ě , @ m P t , . . . , s ´ u : h p U | v ´ π ,lU v | W m,p p F U q ď Ch s ´ mU | v | W s,p p U q , (63) with F U , W m,p p F U q and corresponding seminorm as in Theorem 2. roof. This result is a combination of Lemmas 3.4 and 3.6 from Ref. [16]. We give here analternative proof based on the abstract results of Section 2.1. By Lemma 4 with P “ P l p U q , wehave the following boundedness property for π ,lU : For all v P L p U q , } π ,lU v } L p p U q ď C } v } L p p U q with real number C ą d , (cid:37) , and l . The estimate (62) is then an immediateconsequence of Lemma 3 with q “ ,lU “ π ,lU . To prove (63), proceed as in Theorem 2using (62) in place of (3). Corollary 19 ( W s,p -boundedness of π ,lU ) . With the same notation as in Lemma 18, it holds, forall v P W s,p p U q , | π ,lU v | W s,p p U q ď C | v | W s,p p U q . (64) Proof.
Use the triangle inequality to write | π ,lU v | W s,p p U q ď | π ,lU v ´ v | W s,p p U q ` | v | W s,p p U q andconclude using (62) with m “ s for the first term. Let an element T P T h be fixed and set, as in the proof of Theorem 12, p u T : “ I kT u . Recalling thedefinition (33b) of s T , and using the inequality p a ` b q p ď p ´ a p ` p ´ b p , (65)it is inferred s T p u T , u T q “ ÿ F P F T h ´ pF ż F | δ kT F u T | p “ ÿ F P F T h ´ pF ż F | δ kT F p u T ` δ kT F p u T ´ p u T q| p À s T p p u T , p u T q ` s T p u T ´ p u T , u T ´ p u T q . (66)On the other hand, inserting p k ` T p u T ´ π ,k ` T u “ } ∇ p u ´ p k ` T u T q} pL p p T q d À } ∇ p u ´ π ,k ` T u q} pL p p T q d ` } ∇ p k ` T p p u T ´ u T q} pL p p T q d . (67)Summing (66) and (67), and recalling the definition (34) of }¨} ,p,T , we obtain } ∇ p u ´ p k ` T u T q} pL p p T q d ` s T p u T , u T qÀ } ∇ p u ´ π ,k ` T u q} pL p p T q d ` s T p p u T , p u T q ` } p u T ´ u T } p ,p,T . The result follows by summing this estimate over T P T h and invoking Theorem 1 for the firstterm in the right-hand side, (60) for the second, and (36) for the third. For the sake of completeness, we present here some new numerical examples that demonstrate theorders of convergence achieved by the HHO method in practice. The test were run using the hho software platform . Agence pour la Protection des Programmes deposit number IDDN.FR.001.220005.000.S.P.2016.000.10800
We first complete the test cases proposed in Ref. [16] by considering the exact solution of Section4.4 therein for an exponent p strictly smaller than 2. More precisely, we solve on the unit squaredomain Ω “ p , q the p -Laplace Dirichlet problem with p “ corresponding to the exact solution u p x q “ exp p x ` πx q , with suitable right-hand side f inferred from the expression of u . With this choice, the gradientof u is nonzero, which prevents dealing with singularities.We consider the matching triangular, Cartesian, locally refined, and (predominantly) hexagonalmesh families depicted in Figure 1 and polynomial degrees ranging from 0 to 3. The three formermesh families have been used in the FVCA5 benchmark [36], whereas the latter is taken fromRef. [23]. The local refinement in the third mesh family has no specific meaning for the problemconsidered here: its purpose is to demonstrate the seamless treatment of nonconforming interfaces.We report in Figure 2 the error } I kh u ´ u h } ,p,h versus the meshsize h . The observed orders ofconvergence seem to suggest that our estimate (36) is sharp. For k “
0, superconvergence isobserved on the Cartesian mesh family and, to a lesser extent, on the locally refined mesh family.This kind of superconvergence phenomena have already been observed in the past for the Poissonproblem corresponding to p “ The test case of the previous section had already been solved in Ref. [16] for different vales of p greater or equal than 2. We therefore consider here a different manufactured solution. We solve onthe unit square domain Ω “ p , q the homogeneous p -Laplace Dirichlet problem correspondingto the exact solution u p x q “ sin p πx q sin p πx q , with p P t , , u and source term inferred from u (cf. (21) for the expression of a in this case).We consider the same mesh families and polynomial orders as in the previous section.We report in Figure 3 the error } I kh u ´ u h } ,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 Ref. [21] (aknown superconvergence phenomenon is observed on the Cartesian mesh for k “ p “ , k “ p “
3, the observed orders of convergence in the last refinementsteps are inferior to the predicted value for smooth solutions, which can likely be ascribed to the20 ´ . ´ ´ . ´ ´ ´ ´ ´ (a) Triangular ´ . ´ ´ . ´ ´ ´ ´ (b) Cartesian ´ . ´ ´ . ´ ´ ´ ´ (c) Loc. ref. ´ . ´ . ´ ´ . ´ . ´ . ´ . ´ ´ ´ ´ (d) Hexagonal Figure 2: } I kh u ´ u h } ,p,h versus h for the test of Section 3.5.1 and the mesh families of Figure1 with p “ . The slopes represent the orders of convergence expected from Theorem 12, i.e. p k ` q for k P t , . . . , u (resp. blue dots, red squares, brown dots, black stars).21 ´ . ´ ´ . ´ ´ ´ ´ (a) Triangular, p “ ´ . ´ ´ . ´ ´ ´ ´ ´ (b) Triangular, p “ ´ . ´ ´ . ´ ´ ´ ´ ´ (c) Triangular, p “ ´ . ´ ´ . ´ ´ ´ ´ ´ ´ ´ ´ (d) Cartesian, p “ ´ . ´ ´ . ´ ´ ´ ´ (e) Cartesian, p “ ´ . ´ ´ . ´ ´ ´ (f) Cartesian, p “ ´ . ´ ´ . ´ ´ ´ ´ ´ ´ ´ ´ (g) Loc. ref., p “ ´ . ´ ´ . ´ ´ ´ ´ (h) Loc. ref., p “ ´ . ´ ´ . ´ ´ ´ (i) Loc. ref., p “ ´ . ´ . ´ ´ . ´ . ´ . ´ . ´ ´ ´ ´ ´ ´ ´ (j) Hexagonal, p “ ´ . ´ . ´ ´ . ´ . ´ . ´ . ´ ´ ´ ´ (k) Hexagonal, p “ ´ . ´ . ´ ´ . ´ . ´ . ´ . ´ ´ ´ (l) Hexagonal, p “ Figure 3: } I kh u ´ u h } ,p,h versus h for the test of Section 3.5.2 and the mesh families of Figure 1. Theslopes represent the orders of convergence expected from Theorem 12, i.e. k ` p ´ for k P t , . . . , u (resp. blue dots, red squares, brown dots, black stars) and p P t , , u .22 ´ . ´ ´ . ´ ´ ´ ´ ´ ´ (a) Triangular ´ . ´ ´ . ´ ´ ´ ´ ´ (b) Cartesian ´ . ´ ´ . ´ ´ ´ ´ ´ (c) Loc. ref. ´ . ´ . ´ ´ . ´ . ´ . ´ . ´ ´ ´ ´ ´ (d) Hexagonal Figure 4: } I kh u ´ u h } ,p,h versus h for the test of Section 3.5.2 and the mesh families of Figure1 with p “ . The slopes represent the orders of convergence expected from Theorem 12, i.e. p k ` q for k P t , . . . , u (resp. blue dots, red squares, brown dots, black stars). In this case, theorder of convergence is limited by the regularity of a p¨ , ∇ u q .23iolation of the regularity assumption on a p¨ , ∇ u q (cf. Theorem 12), due to the lack of smoothnessof a for that p .For the sake of completeness, we consider also in this case the exponent p “ . Unlike in theprevious section, the regularity required by the error estimates in Theorem 12 does not seem tohold for the exact solution. This can be clearly seen in Figure 4, where the observed convergencerate seems optimal for k P t , u , while it stagnates for k P t , u . For this value of p , the derivativesof ξ Ñ | ξ | p ´ ξ are singular at ξ “ , which prevents a p¨ , ∇ u q from having the W k ` ,p regularityrequired in (35b). A Inequalities involving the Leray–Lions operator
This section collects inequalities involving the Leray–Lions operator adapted from Ref. [27].
Lemma 20.
Assume (20c) , (20e) , and p ď . Then, for a.e. x P Ω and all p ξ , η q P R d ˆ R d , | a p x , ξ q ´ a p x , η q| ď p γ a ` p ´ β a ` β a q| ξ ´ η | p ´ . (68) Proof.
Let r ą
0. If | ξ | ě r and | η | ě r then, using (20e) and p ´ ď
0, we have | a p x , ξ q ´ a p x , η q| ď γ a | ξ ´ η |p| ξ | p ´ ` | η | p ´ q ď γ a r p ´ | ξ ´ η | . (69)Otherwise, assume for example that | η | ă r . Then | ξ | ď | ξ ´ η | ` r and thus, owing to (20c), | a p x , ξ q ´ a p x , η q| ď | a p x , ξ q ´ a p x , q| ` | a p x , q ´ a p x , η q|ď β a p| ξ | p ´ ` | η | p ´ qď β a p| ξ ´ η | ` r q p ´ ` β a r p ´ . (70)Combining (69) and (70) shows that, in either case, | a p x , ξ q ´ a p x , η q| ď γ a r p ´ | ξ ´ η | ` β a p| ξ ´ η | ` r q p ´ ` β a r p ´ . Taking r “ | ξ ´ η | concludes the proof of (68). Lemma 21.
Under Assumption (20f) we have, for a.e. x P Ω and all p ξ , η q P R d ˆ R d , • If p ă , | ξ ´ η | p ď ζ ´ p a p p ´ q ´ p ´ r a p x , ξ q ´ a p x , η qs ¨ r ξ ´ η s ¯ p ´ | ξ | p ` | η | p ¯ ´ p ; (71) • If p ě , | ξ ´ η | p ď ζ ´ a r a p x , ξ q ´ a p x , η qs ¨ r ξ ´ η s . (72) Proof.
Estimate (71) is obtained by raising (20f) to the power p { p| ξ | ` | η |q p ď p ´ p| ξ | p ` | η | p q . To prove (72), we simply write | ξ ´ η | p ď | ξ ´ η | p| ξ | ` | η |q p ´ . Remark . The (real-valued) mapping a : t ÞÑ | t | p ´ t corresponds to the p -Laplace operator indimension 1, and it therefore satisfies (20f). Hence, by Lemma 21,If p ă | t ´ r | p ď C `“ | t | p ´ t ´ | r | p ´ r ‰ r t ´ r s ˘ p p| t | p ` | r | p q ´ pp , (73)If p ě | t ´ r | p ď C “ | t | p ´ t ´ | r | p ´ r ‰ r t ´ r s , (74)where C depends only on p . 24 cknowledgment This work was partially supported by Agence Nationale de la Recherche project HHOMM (ANR-15-CE40-0005) and by the Australian Research Council’s Discovery Projects funding scheme(project number DP170100605).
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