A quasi-optimal variant of the Hybrid High-Order method for elliptic PDEs with H −1 loads
aa r X i v : . [ m a t h . NA ] A p r A QUASI-OPTIMAL VARIANT OF THE HYBRID HIGH-ORDERMETHOD FOR ELLIPTIC PDES WITH H − LOADS
ALEXANDRE ERN AND PIETRO ZANOTTI
Abstract.
Hybrid High-Order methods for elliptic diffusion problems havebeen originally formulated for loads in the Lebesgue space L (Ω). In this pa-per we devise and analyze a variant thereof, which is defined for any load inthe dual Sobolev space H − (Ω). The main feature of the present variant isthat its H -norm error can be bounded only in terms of the H -norm besterror in a space of broken polynomials. We establish this estimate with thehelp of recent results on the quasi-optimality of nonconforming methods. Weprove also an improved error bound in the L -norm by duality. Compared toprevious works on quasi-optimal nonconforming methods, the main noveltiesare that Hybrid High-Order methods handle pairs of unknowns, and not asingle function, and, more crucially, that these methods employ a reconstruc-tion that is one polynomial degree higher than the discrete unknowns. Theproposed modification affects only the formulation of the discrete right-handside. This is obtained by properly mapping discrete test functions into H (Ω). Introduction
Hybrid High-Order (HHO) methods have been introduced in [13] for diffusionproblems and in [12] for locking-free linear elasticity. These methods employ faceunknowns and cell unknowns and are devised from two local operators, a recon-struction operator and a stabilization operator. HHO methods support generalmeshes (with polyhedral cells and nonmatching interfaces), they are locally con-servative and are robust in various regimes of practical interest, and they offercomputational benefits resulting from the local elimination of cell unknowns bystatic condensation. The realm of applications of HHO methods has been vigor-ously expanded over the last few years; for brevity, we only mention [6, 1, 2] fornonlinear solid mechanics and refer the reader to the bibliography therein. AnOpen-Source library for HHO methods based on generic programming is also avail-able [8]. Finally, we mention that HHO methods are closely related to hybridizablediscontinuous Galerkin methods [10] and to nonconforming virtual element methods[3], as shown in [9].In the present work, we focus on the Poisson model problem which reads asfollows: Given f ∈ H − (Ω) , find u ∈ H (Ω) such that ∀ w ∈ H (Ω) ˆ Ω ∇ u · ∇ w = h f, w i H − (Ω) × H (Ω) . (1.1) Mathematics Subject Classification.
Key words and phrases. hybrid methods, quasi-optimality, rough loads, arbitrary order, gen-eral meshes.
Although the model problem (1.1) is posed for general loads in the dual Sobolevspace H − (Ω), the devising and analysis of HHO methods in [13] requires thatthe load is in the Lebesgue space L (Ω). In particular, H -norm error estimateswith optimal decay rates have been derived in [13] for smooth solutions in H p (Ω)(where p ≥ u ∈ H s (Ω), s > , which is reasonable forthe model problem (1.1) if f ∈ L (Ω). Moreover, improved L -norm error estimatescan also be derived by means of the Aubin–Nitsche duality argument. These resultswere recently extended in [15] to the regularity requirement u ∈ H s (Ω), s > L q (Ω) with q > d d , where d is the spacedimension, that is q > d = 2 and q > if d = 3. Therein, quasi-optimalerror estimates were established in an augmented norm that is stronger than the H -norm.The above discussion shows that a theoretical gap still remains in the analysisof the HHO methods. One option to fill this gap would be to bound the H -normerror only in terms of the H -norm best error in the underlying discrete space. Infact, such quasi-optimal estimate would not require regularity assumptions beyond H (Ω) for the solution and H − (Ω) for the load. Notably, the abstract theory of[23] on the quasi-optimality of nonconforming methods indicates that an estimatein this form can be expected for a variant of the original HHO method of [13].This is the main achievement of the present work. In particular, the modified HHOmethod is defined and stable for arbitrary loads in H − (Ω), as well as properlyconsistent.Quasi-optimality in the energy norm has been previously achieved by variants ofclassical nonconforming methods [24] and discontinuous Galerkin and other interiorpenalty methods [25] for second- and fourth-order elliptic problems. Similarly to[24, 25], our modification of the original HHO method affects only the discretizationof the load. In the novel HHO method, the discrete test functions are transformedthrough an averaging operator to achieve stability and bubble smoothers to en-force consistency. The main novelties concerning the analysis of nonconformingquasi-optimal methods is that we extend the abstract framework of [23] so as tohandle pairs of functions (one defined in the computational domain and one on themesh skeleton) and that we deal with the presence of a reconstruction operatorthat is one polynomial degree higher than the discrete unknowns. In addition toquasi-optimality in the H -norm, we also show that the Aubin–Nitsche duality ar-gument still allows one to derive improved L -norm error estimates for the modifiedHHO method. Finally, owing to the results from [9], we notice that the presentfindings thus provide a way to achieve the same quasi-optimal properties by ap-propriately modifying the discrete load in hybridizable discontinuous Galerkin andnonconforming virtual element methods.This paper is organized as follows. In section 2, we briefly summarize the abstractframework from [23] for quasi-optimal nonconforming methods. In section 3 weoutline the main ideas and results concerning HHO methods on simplicial mesheswith loads in L (Ω). In section 4, we present and analyze a quasi-optimal variantof the HHO method. This section contains the main results of this work. Finally,in section 5, we show how the results of section 4 can be extended to the setting ofpolytopic meshes. QUASI-OPTIMAL HHO METHOD WITH H − LOADS 3 Abstract framework for quasi-optimality
In this section we summarize the framework of [23] in a form that is convenientto guide the design of the method proposed in section 4. Moreover, we recall thenotion of quasi-optimality and a couple of related results.Let V be a Hilbert space with scalar product a . Denote by V ∗ the topologicaldual space of V and consider the elliptic variational problem(2.1) Given ℓ ∈ V ∗ , find u ∈ V such that ∀ w ∈ V a ( u, w ) = h ℓ, w i V ∗ × V which is uniquely solvable, according to the Riesz representation theorem.Let S be a finite-dimensional linear space and assume that a can be extended toa scalar product e a on V + S , inducing the extended energy norm k · k := pe a ( · , · ).Let E : S → V be a linear operator and consider the following approximationmethod for (2.1):(2.2) Given ℓ ∈ V ∗ , find U ∈ S such that ∀ σ ∈ S e a ( U, σ ) = h ℓ, Eσ i V ∗ × V which is uniquely solvable, due to the positive-definiteness of e a on S . We say that S is a nonconforming space and (2.2) is nonconforming method whenever S * V .Since U ∈ S , the approximation error u − U satisfies inf s ∈ S k u − s k ≤ k u − U k ,showing that the best error inf s ∈ S k u − s k is an intrinsic benchmark for (2.2). Hence,we say that the method (2.2) is quasi-optimal for (2.1) in the norm k · k if there isa constant C ≥ k u − U k ≤ C inf s ∈ S k u − s k and C is independent of u and U . In this case, we refer to the best value of C asthe quasi-optimality constant of (2.2) in the norm k · k . Remark E ) . We call E smoother , because its actionoften increases the regularity of the elements of S . An immediate observation isthat the use of a smoother makes the duality h ℓ, Eσ i V ∗ × V in (2.2) well-defined forall ℓ ∈ V ∗ , irrespective of the possible nonconformity of S . Notice also that E is bounded, because S is finite-dimensional. Thus, we infer that (2.2) is a stablemethod, in that(2.4) k U k ≤ k E k L ( S,V ) k ℓ k V ∗ = k E k L ( S,V ) k u k . Moreover, the operator norm of E is the best possible constant in this inequality foran arbitrary load ℓ ∈ V ∗ . The above stability (2.4) is necessary for quasi-optimality,owing to the triangle inequality, and the quasi-optimality constant can be boundedfrom below in terms of the operator norm of E . Remark . The computation of U from (2.2) requires theevaluation of E on each element of the basis { φ , . . . , φ n } of S at hand. Hence,it is highly desirable that the duality h ℓ, Eφ i i V ∗ × V , i = 1 , . . . , n , can be evaluatedwith O(1) operations. To this end, a sufficient condition is that each Eφ i is locallysupported and can be obtained from φ i with O(1) operations.Conforming Galerkin methods for (2.1) fit into this abstract framework with S ⊆ V e a = a E = Idand are quasi-optimal in the energy norm, according to the so-called C´ea’s lemma[7]. Still, quasi-optimality can be achieved also if S is nonconforming, depending A. ERN AND P. ZANOTTI on the interplay of e a and E . In fact, [23, Theorem 4.14] states that the following algebraic consistency (2.5) ∀ s ∈ V ∩ S, σ ∈ S e a ( s, Eσ ) = e a ( s, σ ) . is necessary and sufficient for the existence of a constant C so that (2.3) holds.This is actually equivalent to prescribe that the solution u of (2.1) solves also (2.2),whenever u ∈ V ∩ S .It is worth noticing that (2.5) is, possibly, a mild or trivial condition, as itinvolves only conforming trial functions s ∈ V ∩ S . Thus, it is not a surprise thatadditional informations are actually needed in order to access the size of the quasi-optimality constant. For instance, Remark 2.1 reveals a lower bound in terms ofthe operator norm of the employed smoother. Moreover, one may expect that thequasi-optimality constant depends also on the discrepancy of the left- and right-hand sides of (2.5) for nonconforming trial functions s ∈ S \ V . This claim can beconfirmed with the help of [23, Theorem 4.19].In section 4, we actually build on a generalized version of (2.5), because thesetting considered there does not exactly fit into the framework described here.3. The HHO method on simplicial meshes
In this section we recall the HHO method for (1.1) proposed in [13] and some ofits properties. In order to avoid unnecessary technicalities, we restrict our attentionto matching simplicial meshes, here and in the next section. The extension of ourresults to more general meshes is addressed in section 5.3.1.
Discrete problem.
Let Ω ⊆ R d , d ∈ { , } , be an open and bounded polyg-onal/polyhedral set with Lipschitz-continuous boundary. Let M = ( K ) K ∈M bea matching simplicial mesh of Ω, i.e., all cells of M are d -simplices and, for any K ∈ M with vertices { a , . . . , a d } and for all K ′ ∈ M , the intersection K ∩ K ′ iseither empty or the convex hull of a subset of { a , . . . , a d } .We denote by F i the set of all interfaces of M . Since the mesh is matching,any interface F ∈ F i is such that F = K ∩ K for some K , K ∈ M , and F is afull face of both K and K . Similarly, we collect the boundary faces into F b andobserve that each F ∈ F b satisfies F = K ∩ ∂ Ω for some K ∈ M . Then, the set F := F i ∪ F b consists of all faces, and the skeleton of M is given byΣ := [ F ∈F F. For each K ∈ M , we denote by F K the set of all faces of K , i.e. the faces F ∈ F such that F ⊆ K . We indicate by h K and h F the diameters of K and F ,respectively. Moreover, we write n K for the outer normal unit vector of K .For p ∈ N := N ∪ { } , K ∈ M and F ∈ F , let P p ( K ) and P p ( F ) be the spacesof all polynomials of total degree ≤ p in K and F , respectively. The correspondingbroken spaces on M and Σ are given by P p ( M ) := { s M : Ω → R | ∀ K ∈ M ( s M ) | K ∈ P p ( K ) } , P p (Σ) := { s Σ : Σ → R | ∀ F ∈ F ( s Σ ) | F ∈ P p ( F ) , ∀ F ∈ F b ( s Σ ) | F = 0 } . QUASI-OPTIMAL HHO METHOD WITH H − LOADS 5
We shall make use of the L -orthogonal projections Π M : L (Ω) → P p ( M ) andΠ Σ : L (Σ) → P p (Σ), which are defined such that ˆ K q Π M v M = ˆ K qv M and ˆ F r Π Σ v Σ = ˆ F rv Σ for all K ∈ M , q ∈ P p ( K ), v M ∈ L (Ω) and for all F ∈ F i , r ∈ P p ( F ), v Σ ∈ L (Σ),respectively.The HHO space of degree p is the Cartesian productˆ S p H := P p ( M ) × P p (Σ) , so that the elements of ˆ S p H are pairs ˆ s = ( s M , s Σ ). The first component of ˆ s isintended to approximate the solution u of (1.1) in each simplex of M , whereas thesecond component is intended to approximate the trace of u on each face composingthe skeleton Σ. Notice that the face component of a member of ˆ S p H incorporatesthe boundary condition of (1.1). In what follows, we denote both pairs and spacesof pairs using a hat symbol. Moreover, we drop the superscript p and simply writeˆ S H to alleviate the notation. The subscript ’H’ serves to distinguish the HHO spacefrom its abstract counterpart in section 2.The first constitutive ingredient of the HHO method is a suitable higher-order reconstruction. This is realized through the linear operator R : ˆ S H → P p +1 ( M ),which is uniquely determined by the conditions(3.1a) ∀ q ∈ P p +1 ( K ) ˆ K ∇R ˆ s · ∇ q = − ˆ K s M ∆ q + ˆ ∂K s Σ ∇ q · n K and(3.1b) ˆ K R ˆ s = ˆ K s M for all K ∈ M and ˆ s = ( s M , s Σ ) ∈ ˆ S H . The local problem in (3.1a) is uniquelysolvable up to an additive constant, which is then fixed by (3.1b). The computationof R ˆ s can be performed element-wise because, for each K ∈ M , the restriction( R ˆ s ) | K depends only on ( s M ) | K and ( s Σ ) | ∂K .The second constitutive ingredient of the HHO method is the following stabi-lization bilinear form defined on ˆ S H × ˆ S H :(3.2) θ (ˆ s, ˆ σ ) := X K ∈M X F ∈F K ˆ F Π Σ ( s Σ − ( S ˆ s ) | K )Π Σ ( σ Σ − ( S ˆ σ ) | K ) , with arbitrary ˆ s = (ˆ s M , ˆ s Σ ) and ˆ σ = (ˆ σ M , ˆ σ Σ ) in ˆ S H and with the stabilizationoperator S : ˆ S H → P p +1 ( M ) such that(3.3) S ˆ s := s M + (Id − Π M ) R ˆ s. Since both R and Π M can be computed element-wise, the operator S inherits thisproperty.Denote by ∇ M the broken gradient on M , whose action on an element-wise H -function v is given by ( ∇ M v ) | K := ∇ ( v | K ) for all K ∈ M . The HHO bilinearform on ˆ S H × ˆ S H can be written as follows: b H (ˆ s, ˆ σ ) := ˆ Ω ∇ M R ˆ s · ∇ M R ˆ σ + θ (ˆ s, ˆ σ ) . A. ERN AND P. ZANOTTI
It can be verified that, for all ˆ s ∈ ˆ S H , the semi-norm θ (ˆ s, ˆ s ) penalizes the dis-crepancy between the face component of ˆ s and the trace of the cell componenton the skeleton Σ. This, in turn, enforces positive-definiteness of b H , as stated inLemma 3.1 below.Assume for the moment that the load f of (1.1) is in L (Ω). The HHO methodof [13] for the Poisson problem reads(3.4) Find ˆ U ∈ ˆ S H such that ∀ ˆ σ = ( σ M , σ Σ ) ∈ ˆ S H b H ( ˆ U , ˆ σ ) = ˆ Ω f σ M . Discrete stability and approximation properties.
We now aim at as-sessing the stability of the form b H and the approximation properties of the spaceˆ S H . For all K ∈ M , denote by r K the radius of the largest ball inscribed in K .The shape parameter γ = γ ( M ) of the mesh M is defined as the largest positivereal number such that(3.5) ∀ K ∈ M γ r K ≤ h K . We indicate by C γ and C γ,p two generic functions of the quantities indicated by thesubscripts, nondecreasing in each argument, which do not need to be the same ateach occurrence. Sometimes, we use the abbreviation A . B in place of A ≤ C γ,p B .For instance, if K ∈ M and F ∈ F K , the so-called discrete and continuous traceinequalities read ∀ q ∈ P p +1 ( K ) h − F k q k L ( F ) ≤ C γ,p h − K k q k L ( K ) (3.6) ∀ v ∈ H ( K ) h − F k v k L ( F ) ≤ C γ (cid:0) h − K k v k L ( K ) + k∇ v k L ( K ) (cid:1) (3.7)see, e.g., [11, Lemmata 1.46 and 1.49]. Recall also that, if v ∈ H ( K ) and ´ K v = 0,we have the Poincar´e–Steklov inequality [4](3.8) k v k L ( K ) ≤ π − h K k∇ v k L ( K ) . The following result implies that the HHO bilinear form b H is nondegenerate andensures that the problem (3.4) is uniquely solvable. A proof can be found in [13,Lemma 4]. Lemma 3.1 (Coercivity of b H ) . For all ˆ s = ( s M , s Σ ) ∈ ˆ S H , we have k ˆ s k S H ≤ C γ,p b H (ˆ s, ˆ s ) , where the norm k · k ˆ S H is defined as (3.9) k ˆ s k S H := X K ∈M k∇ s M k L ( K ) + X F ∈F K h − F k s Σ − ( s M ) | K k L ( F ) ! . Next, we examine the approximation properties of the HHO space underlying(3.4). To this end, we consider the interpolant ˆ I : H (Ω) → ˆ S H defined as follows:(3.10) ˆ I v := (Π M v, Π Σ v ) . As the reconstruction R maps the elements of ˆ S H into piecewise polynomials ofdegree ( p + 1), we compare, in particular, the approximation in the mapped space R ( ˆ S H ) with the one in the broken space P p +1 ( M ), with respect to the H -norm QUASI-OPTIMAL HHO METHOD WITH H − LOADS 7 and the L -norm. For this purpose, we make use of the broken elliptic projection E : H (Ω) → P p +1 ( M ), which is obtained by imposing(3.11a) ∀ q ∈ P p +1 ( K ) ˆ K ∇E v · ∇ q = ˆ K ∇ v · ∇ q and(3.11b) ˆ K E v = ˆ K v for all K ∈ M and v ∈ H (Ω).Recall the definitions of R and ˆ I from (3.1) and (3.10), respectively, and let v ∈ H (Ω) be given. We have ´ K ∇R ˆ I v · ∇ q = − ´ K Π M v ∆ q + ´ ∂K Π Σ v ∇ q · n K = ´ K ∇ v · ∇ q for all K ∈ M and q ∈ P p +1 ( K ). Furthermore, ´ K R ˆ I v = ´ K Π M v = ´ K v . Then, comparing with (3.3) and (3.11), we derive the identities(3.12) R ◦ ˆ I = E and S ◦ ˆ I = E + Π M (Id − E )which can be used to assess the approximation properties of ˆ S H . Lemma 3.2 (Interpolation errors) . For all v ∈ H (Ω) , we have (3.13a) k ∇ M ( v − R ˆ I v ) k L (Ω) + θ (ˆ I v, ˆ I v ) . X K ∈M inf q ∈ P p +1 ( K ) k∇ ( v − q ) k L ( K ) , (3.13b) k v − R ˆ I v k L (Ω) ≤ X K ∈M (cid:18) h K π (cid:19) inf q ∈ P p +1 ( K ) k∇ ( v − q ) k L ( K ) . Proof.
The proof follows from [13] and is briefly sketched for completeness. Let v ∈ H (Ω) be given. The first summand in the left-hand side of (3.13a) can berewritten using the first part of (3.12) and the element-wise H -orthogonality of E ,which imply that(3.14) k∇ ( v − R ˆ I v ) k L ( K ) = k∇ ( v − E v ) k L ( K ) = inf q ∈ P p +1 ( K ) k∇ ( v − q ) k L ( K ) for all K ∈ M . Concerning the other summand, the second part of (3.12) reveals S ˆ I v = E v + Π M ( v − E v ). Inserting this identity into (3.2), we infer that(3.15) θ (ˆ I v, ˆ I v ) = X K ∈M X F ∈F K h − F k Π Σ ( v − ( E v ) | K ) + Π M ( v − E v ) | K k L ( F ) . Consider any K ∈ M and F ∈ F K . We exploit the boundedness of Π Σ in the L ( F )-norm, the trace inequality (3.7), the identity (3.11b), and the Poincar´e–Steklov inequality (3.8) to infer that h − F k Π Σ ( v − ( E v ) | K ) k L ( F ) ≤ C γ k∇ ( v − E v ) k L ( K ) . Next, we invoke the discrete trace inequality (3.6), the boundedness of Π M in the L ( K )-norm, the identity (3.11b), and the Poincar´e–Steklov inequality (3.8) toobtain h − F k Π M ( v − E v ) | K k L ( F ) ≤ C γ,p k∇ ( v − E v ) k L ( K ) . A. ERN AND P. ZANOTTI
Hence, we derive the claimed bound (3.13a) inserting these estimates into (3.15)and using again (3.14). Finally, the identity (3.11b), the first identity in (3.12), andthe Poincar´e–Steklov inequality (3.8) yield k v − R ˆ I v k L (Ω) ≤ X K ∈M (cid:18) h K π (cid:19) k∇ ( v − E v ) k L ( K ) . This proves (3.13b), in combination with (3.14). (cid:3)
The first estimate in Lemma 3.2 has the remarkable property that the left- andthe right-hand sides are equivalent, according to the inclusion R ˆ I v ∈ P p +1 ( M ).The other estimate does not enjoy the same property, because the right-hand siderequires higher regularity of v than the left-hand side. Note also that, in bothestimates, the right-hand side involves only local best errors on the simplices of M , in the spirit of [22]. This entails that the best approximation in ˆ S H needs onlypiecewise (and not global) regularity of v to achieve convergence with a certaindecay rate.Both estimates in Lemma 3.2 are possible benchmarks for any approximationmethod based on the HHO space. Indeed, if u ∈ H (Ω) solves (1.1) and ˆ U ∈ ˆ S H is the approximation resulting from a given HHO method, one may ask whether ˆ U fulfills the same error bounds as ˆ I u , possibly up to more pessimistic constants. Thiswould guarantee that the method under examination reproduces the approximationproperties of the underlying space.Unfortunately, the H - and the L -norm errors of the HHO method (3.4) cannotbe bounded like the corresponding interpolation errors in Lemma 3.2. In fact, al-though it is certainly possible to relax the assumption that the load f is in L (Ω),as done in [15], the duality ´ Ω f σ M in the right-hand side of (3.4) cannot be contin-uously extended to general loads f ∈ H − (Ω) and arbitrary discrete test functionsˆ σ = ( σ M , σ Σ ) ∈ ˆ S H , because we possibly have σ M / ∈ H (Ω). As a consequence, anyerror bound of (3.4) must involve additional regularity beyond f ∈ H − (Ω) and u ∈ H (Ω). Motivated by this observation, we aim at designing a variant of (3.4)with improved approximation properties.4. A quasi-optimal variant of the HHO method
In this section we exploit the abstract framework of section 2 to design a newHHO method, which is quasi-optimal for (1.1) in the semi-norm involved in theleft-hand side of (3.13a). According to Remark 2.1, this requires, in particular,the use of a smoother in the discretization of the load. Hence, we first pointout a condition on the smoother that is sufficient for quasi-optimality. Then, weconstruct a smoother fulfilling such a condition and derive broken H - and L -normerror estimates.4.1. The HHO method with smoothing.
Let M be the simplicial mesh intro-duced in section 3.1 and recall that the space ˆ S H consists of pairs ˆ s = ( s M , s Σ ),where the first component s M ∈ P p ( M ) is an element-wise polynomial on M ,whereas the second component s Σ ∈ P p (Σ) is a face-wise polynomial on Σ. Sincethe abstract framework of section 2 involves the sum of continuous and discretespaces, it is formally convenient to identify any element v ∈ H (Ω) with the pair QUASI-OPTIMAL HHO METHOD WITH H − LOADS 9 ˆ v := ( v, v | Σ ), where v | Σ denotes the trace of v on Σ. In fact, the Poisson problem(1.1) fits into the abstract elliptic problem (2.1) provided we rewrite it as follows:(4.1) Given ℓ H ∈ ˆ V ∗ , find ˆ u ∈ ˆ V s.t. ∀ ˆ w ∈ ˆ V a H (ˆ u, ˆ w ) = h ℓ H , ˆ w i ˆ V ∗ × ˆ V , with the space(4.2a) ˆ V := { ˆ v = ( v M , v Σ ) | v M ∈ H (Ω) , v Σ = ( v M ) | Σ } and the forms(4.2b) a H (ˆ v, ˆ w ) := ˆ Ω ∇ v M · ∇ w M (4.2c) h ℓ H , ˆ w i ˆ V ∗ × ˆ V := h f, w M i H − (Ω) × H (Ω) where ˆ v = ( v M , v Σ ) and ˆ w = ( w M , w Σ ) are in ˆ V and f ∈ H − (Ω) is the loadin (1.1). This way of looking at the model problem (2.1) is instrumental to thederivation of Proposition 4.5, although it may appear a bit artificial at first glance.The intersection of ˆ V and the HHO space can be characterized as follows:ˆ V ∩ ˆ S H = { ˆ v = ( v M , v Σ ) ∈ ˆ V | v M ∈ P p ( M ) } . In particular, any element ˆ v = ( v M , v Σ ) ∈ ˆ V ∩ ˆ S H satisfies(4.3) ˆ I v M = ˆ v, E v M = v M , R ˆ v = v M , S ˆ v = v M , θ (ˆ v, · ) = 0 . Proceeding as in section 2, we look for a symmetric bilinear form e a H on ˆ V + ˆ S H such that e a H | ˆ V = a H and e a H | ˆ S H = b H . In other words, we require that e a H is acommon extension of a H and b H . It is readily seen that we must have(4.4) e a H (ˆ v + ˆ s, ˆ w + ˆ σ ) := ˆ Ω ∇ M ( v M + R ˆ s ) · ∇ M ( w M + R ˆ σ ) + θ (ˆ s, ˆ σ ) , for all ˆ v, ˆ w ∈ ˆ V and ˆ s, ˆ σ ∈ ˆ S H . To check that e a H is indeed well-defined, assume thatˆ v + ˆ s = ˆ v ′ + ˆ s ′ for some ˆ v ′ ∈ ˆ V and ˆ s ′ ∈ ˆ S H . Then, we have ˆ v − ˆ v ′ = ˆ s ′ − ˆ s ∈ ˆ V ∩ ˆ S H ,so that (4.3) implies v M − v ′M = R (ˆ v − ˆ v ′ ) = R (ˆ s ′ − ˆ s ) and θ (ˆ s − ˆ s ′ , · ) = 0.Rearranging terms, we infer that e a H (ˆ v + ˆ s, · ) = e a H (ˆ v ′ + ˆ s ′ , · ). This observation andthe symmetry of e a H confirm our claim.Let E H : ˆ S H → H (Ω) be a linear operator. Motivated by Remark 2.1, weconsider the following variant of the HHO method (3.4):Given f ∈ H − (Ω) , find ˆ U ∈ ˆ S H s.t. ∀ ˆ σ ∈ ˆ S H b H ( ˆ U , ˆ σ ) = h f, E H ˆ σ i H − (Ω) × H (Ω) . (4.5)Note, in particular, that here the right-hand side is defined for all f ∈ H − (Ω).The new HHO method (4.5) fits into the abstract discrete problem (2.2) with(4.6) S = ˆ S H e a = e a H E ˆ σ = ˆ E H ˆ σ := ( E H ˆ σ, ( E H ˆ σ ) | Σ )so that ˆ E H : ˆ S H → ˆ V . Since ˆ S H * ˆ V , this is a nonconforming method. Quasi-optimality.
The extended energy semi-norm induced by the extendedbilinear form e a H is | ˆ v + ˆ s | e a H := pe a H (ˆ v + ˆ s, ˆ v + ˆ s )with ˆ v ∈ ˆ V and ˆ s ∈ ˆ S H . This is the unique common extension of the energy norminduced by a H and the discrete norm induced by b H . We now aim at determiningthe properties of E H that are relevant for the quasi-optimality of (4.5) in the semi-norm |·| e a H . For this purpose, an important preliminary observation is that thesetting proposed above does not fit into the abstract framework of section 2. Infact, the extended bilinear form e a H is only positive semi-definite on the sum ˆ V + ˆ S H ,although its restrictions to ˆ V and ˆ S H are indeed positive definite. The followingresult makes our claim more precise. Lemma 4.1 (Kernel of |·| e a H ) . We have | ˆ v − ˆ s | e a H = 0 for ˆ v = ( v M , v Σ ) ∈ ˆ V and ˆ s ∈ ˆ S H if and only if v M ∈ P p +1 ( M ) and ˆ s = ˆ I v M .Proof. Assume first that v M ∈ P p +1 ( M ) and ˆ s = ˆ I v M . Owing to (3.12), we have R ˆ s = v M = S ˆ s . The first identity implies that k ∇ M ( v M − R ˆ s ) k L (Ω) = 0. Thesecond one and the fact that v M ∈ H (Ω) reveal that θ (ˆ s, ˆ s ) = 0. We concludethat | ˆ v − ˆ s | e a H = 0.Conversely, assume that ˆ v ∈ ˆ V and ˆ s ∈ ˆ S H are such that | ˆ v − ˆ s | e a H = 0. Thisimplies, in particular, that ∇ M ( v M − R ˆ s ) = 0. Therefore, we have v M ∈ P p +1 ( M ).Hence, arguing as above, we infer the identity | ˆ v − ˆ I v M | e a H = 0, and the triangleinequality yields | ˆ s − ˆ I v M | e a H = 0. Since |·| e a H coincides with the norm inducedby b H on ˆ S H , we conclude that ˆ s = ˆ I v M , owing to the coercivity of b H stated inLemma 3.1. (cid:3) Remark e a H ) . Let ˆ v ∈ ˆ V and ˆ s ∈ ˆ S H be such that | ˆ v − ˆ s | e a H = 0.The ’only if’ part of Lemma 4.1 entails that we have two possibilities. If thecell component v M of ˆ v is in P p ( M ), then we have ˆ v = ˆ I v M = ˆ s . If, instead, v M ∈ P p +1 ( M ) \ P p ( M ), then we have ˆ v = ˆ s , because ˆ v is not in ˆ S H . On the onehand, this confirms that e a H is not positive definite on ˆ V + ˆ S H . On the other hand,we see that the difference ˆ v − ˆ s is a nonzero element in the kernel of |·| e a H if andonly if ˆ v and ˆ s are different pairs but v M coincides with the reconstruction of ˆ s .This originates from the fact that ˆ S H is mapped by R into a different space, whichis ’one degree higher’.One possibility to deal with the degeneracy of e a H would be to take the quotientof ˆ V + ˆ S H over the kernel of |·| e a H . Another, actually equivalent, option is to replacethe intersection ˆ V ∩ ˆ S H in the consistency condition (2.5) with the space of all pairsin ˆ V whose distance to ˆ S H vanishes in the semi-norm |·| e a H , i.e.(4.7) ˆ Z := { ˆ z ∈ ˆ V | inf ˆ s ∈ ˆ S H | ˆ z − ˆ s | e a H = 0 } = { ˆ z ∈ ˆ V | | ˆ z − ˆ I z M | e a H = 0 } . Notice that the second equality follows from Lemma 4.1.Quasi-optimality in the semi-norm |·| e a H prescribes that the error of (4.5) vanisheswhenever the corresponding solution of (4.1) belongs to ˆ Z . This is a more restrictiveconsistency condition than (2.5), because ˆ V ∩ ˆ S H is a strict subspace of ˆ Z . QUASI-OPTIMAL HHO METHOD WITH H − LOADS 11
Lemma 4.3 (Consistency conditions) . Assume that ˆ u ∈ ˆ V solves the problem (4.1) and denote by ˆ U ∈ ˆ S H the solution of (4.5) . The following conditions are equivalent: ˆ u ∈ ˆ Z = ⇒ | ˆ u − ˆ U | e a H = 0(4.8a) ˆ u ∈ ˆ Z = ⇒ ˆ U = ˆ I u M (4.8b) ˆ u ∈ ˆ Z = ⇒ (cid:16) ∀ ˆ σ ∈ ˆ S H , e a H (ˆ u, ˆ σ − ˆ E H ˆ σ ) = 0 (cid:17) (4.8c) and are necessary for quasi-optimality in the semi-norm |·| e a H .Proof. Let ˆ u ∈ ˆ Z . The second identity in (4.7) entails that e a H (ˆ u − ˆ I u M , · ) = 0.Comparing also (4.1) with (4.5) and recalling that e a H extends b H , we see that b H (ˆ I u M , ˆ σ ) = e a H (ˆ I u M , ˆ σ ) = e a H (ˆ u, ˆ σ ) ,b H ( ˆ U , ˆ σ ) = h f, E H ˆ σ i H − (Ω) × H (Ω) = e a H (ˆ u, E H ˆ σ ) . These identities reveal that the following is an equivalent reformulation of (4.8c):ˆ u ∈ ˆ Z = ⇒ (cid:16) ∀ ˆ σ ∈ ˆ S H , b H (ˆ I u M − ˆ U , ˆ σ ) = 0 (cid:17) . Thus, we infer that (4.8b) ⇐⇒ (4.8c) owing to the nondegeneracy of b H , whereasthe equivalence (4.8a) ⇐⇒ (4.8b) is a consequence of Lemma 4.1. Finally, the factthat (4.8a) is necessary for quasi-optimality in the semi-norm |·| e a H readily followsfrom (2.3) and the definition of ˆ Z . (cid:3) Recall from (4.6) that the smoother ˆ E H : ˆ S H → ˆ V is obtained by means of thelinear operator E H : ˆ S H → H (Ω) which is used in the right-hand side of (4.5).Owing to the definition of e a H , condition (4.8c) can be further rewritten as follows: ∀ ˆ u ∈ ˆ Z, ˆ σ ∈ ˆ S H ˆ Ω ∇ u M · ∇ M R ˆ σ = ˆ Ω ∇ u M · ∇ E H ˆ σ. Similar conditions can be found in [24, Section 3.3] and [25, Section 3.2] and are en-forced there by means of moment-preserving smoothers, i.e., smoothers preservingcertain moments on the simplices and on the interfaces of M . The integration byparts formula and the definition of the reconstruction allow us to apply the sametechnique also in this context.In what follows, we adopt the convention P − = { } . Lemma 4.4 (Consistency via moment-preserving smoothers) . Let ˆ σ = ( σ M , σ Σ ) be any pair in ˆ S H and assume that the operator E H : ˆ S H → H (Ω) is such that (4.9) ˆ K q ( E H ˆ σ ) = ˆ K qσ M and ˆ F r ( E H ˆ σ ) = ˆ F rσ Σ for all K ∈ M , q ∈ P p − ( K ) and for all F ∈ F i , r ∈ P p ( F ) . Let ˆ E H be defined asin (4.6) . Then, we have (4.10) e a H (ˆ s, ˆ σ − ˆ E H ˆ σ ) = θ (ˆ s, ˆ σ ) , for all ˆ s ∈ ˆ S H . Moreover, (4.8) holds true. Proof.
Let ˆ σ = ( σ M , σ Σ ) ∈ ˆ S H be given. The definitions of R and e a H in (3.1) and(4.4), respectively, yield e a H (ˆ s, ˆ σ ) − θ (ˆ s, ˆ σ ) = X K ∈M − ˆ K (∆ R ˆ s ) σ M + X F ∈F K ˆ F ( ∇R ˆ s · n K ) σ Σ ! for all ˆ s ∈ ˆ S H . Indeed, the fact that R ˆ s ∈ P p +1 ( M ) ensures that R ˆ s is an admissibletest function in (3.1a). Moreover, since R ˆ s is element-wise smooth, we can exploitonce more the definition of e a H and integrate by parts element-wise. We obtain e a H (ˆ s, ˆ E H ˆ σ ) = X K ∈M − ˆ K (∆ R ˆ s ) E H ˆ σ + X F ∈F K ˆ F ( ∇R ˆ s · n K ) E H ˆ σ ! , for all ˆ s ∈ ˆ S H . Comparing this identity with the previous one and invoking assump-tion (4.9), we infer that (4.10) holds true.Next, let ˆ u = ( u M , u Σ ) ∈ ˆ Z . The combination of (3.12) with Lemma 4.1 revealthat R ˆ I u M = u M as well as θ (ˆ I u M , · ) = 0. Setting ˆ s = ˆ I u M in (4.10), we inferthat e a H (ˆ u, ˆ E H ˆ σ ) = e a H (ˆ I u M , ˆ E H ˆ σ ) = e a H (ˆ I u M , ˆ σ ) = e a H (ˆ u, ˆ σ ) , for all ˆ σ ∈ ˆ S H , showing that (4.8) holds true. (cid:3) The importance of the identity (4.10) in Lemma 4.4 goes beyond the fact thatit is instrumental to check the validity of the consistency condition (4.8). Roughlyspeaking, it can be exploited also to bound the consistency error of (4.5) in theso-called second Strang lemma [5]. This is the key ingredient not only to prove thequasi-optimality of (4.5) in the semi-norm |·| e a H , but also to bound the correspondingquasi-optimality constant. Proposition 4.5 (Quasi-optimality) . Assume that ˆ u ∈ ˆ V solves the problem (4.1) and denote by ˆ U ∈ ˆ S H the solution of (4.5) . If the operator E H satisfies (4.9) , thenwe have (4.11) | ˆ u − ˆ U | e a H ≤ q C H inf ˆ s ∈ ˆ S H | ˆ u − ˆ s | e a H , where C H is the smallest constant such that (4.12) ∀ ˆ σ ∈ ˆ S H k ∇ M ( R ˆ σ − E H ˆ σ ) k L (Ω) ≤ C H | ˆ σ | e a H . Proof.
We adapt the approach devised in [25, section 3] to our context. Denote byˆ P : ˆ V → ˆ S H the e a H -orthogonal projection onto ˆ S H , i.e.,(4.13) ∀ ˆ σ ∈ ˆ S H e a H ( ˆ P ˆ v, ˆ σ ) = e a H (ˆ v, ˆ σ )for all ˆ v ∈ ˆ V . Notice that this problem is uniquely solvable (because e a H restrictedto ˆ S H is positive-definite) and that ˆ P ˆ v is the best approximation of ˆ v in ˆ S H withrespect to the semi-norm |·| e a H . The e a H -orthogonality of ˆ P implies that(4.14) | ˆ u − ˆ U | e a H = | ˆ u − ˆ P ˆ u | e a H + | ˆ U − ˆ P ˆ u | e a H . Since e a H is a scalar product on ˆ S H , we have(4.15) | ˆ U − ˆ P ˆ u | e a H = sup ˆ σ ∈ ˆ S H e a H ( ˆ U − ˆ P ˆ u, ˆ σ ) | ˆ σ | e a H . QUASI-OPTIMAL HHO METHOD WITH H − LOADS 13
Let ˆ σ ∈ ˆ S H be arbitrary and recall that the restriction of e a H to ˆ S H coincides with b H . A comparison of problems (2.1) and (4.5) reveals that e a H ( ˆ U − ˆ P ˆ u, ˆ σ ) = e a H (ˆ u, E H ˆ σ ) − e a H ( ˆ P ˆ u, ˆ σ ) = e a H (ˆ u − ˆ P ˆ u, E H ˆ σ ) − θ ( ˆ P ˆ u, ˆ σ ) , where the second identity follows from Lemma 4.4. Rearranging terms in (4.13)and recalling the expression of e a H in (4.4), we infer that θ ( ˆ P ˆ u, ˆ σ ) = ˆ Ω ∇ M ( u M − R ˆ P ˆ u ) · ∇ M R ˆ σ where u M is the cell component of ˆ u . If we insert this identity into the previousone, we infer that e a H ( ˆ U − ˆ P ˆ u, ˆ σ ) = ˆ Ω ∇ M ( u M − R ˆ P ˆ u ) · ∇ M ( E H ˆ σ − R ˆ σ ) . Comparing with (4.15) and recalling the definition of C H in (4.12), we finally obtainthat | ˆ U − ˆ P ˆ u | e a H ≤ C H | ˆ u − ˆ P ˆ u | e a H . We conclude by inserting this inequality into (4.14). (cid:3)
Moment-preserving smoothers.
Motivated by Proposition 4.5, we nowaim at constructing a concrete smoother which fulfills (4.9) and such that theconstant C H in (4.12) is ≤ C γ,p . To make sure that our construction is of practicalinterest, we also require that the smoother is computationally feasible in the senseof Remark 2.2. As before, we denote by d ∈ { , } the space dimension and use theconvention P − = { } . Our construction is inspired by the one in [24, Section 3.3].For all K ∈ M , we denote by Φ K ∈ H (Ω) the element bubble determined by theconditions ( i ) Φ K ≡ \ K , ( ii ) (Φ K ) | K ∈ P d +1 ( K ) and ( iii ) Φ K ( m K ) = 1 at thebarycenter m K of K . We introduce a local linear operator B K : L (Ω) → P p − ( K )by setting(4.16) ∀ q ∈ P p − ( K ) ˆ K q ( B K v M )Φ K = ˆ K qv M , for all v M ∈ L (Ω). Then, the global operator B M : L (Ω) → H (Ω) is definedsuch that(4.17) B M v M := X K ∈M ( B K v M )Φ K . Since ( B M v M ) | K = ( B K v M )Φ K , the operator B M preserves all the moments of v M of degree ≤ p − M , as a consequence of (4.16).Next, let F ∈ F i be an interface and let K , K ∈ M be such that F = K ∩ K .Setting ω F := K ∪ K , we denote by Φ F ∈ H (Ω) the face bubble determinedby the conditions ( i ) Φ F ≡ \ ω F , ( ii ) (Φ F ) | K j ∈ P d ( K j ) for j = 1 , iii ) Φ F ( m F ) = 1 at the barycenter m F of F . We introduce a local linear operator B F : L (Σ) → P p ( F ) setting(4.18) ∀ r ∈ P p ( F ) ˆ F r ( B F v Σ )Φ F = ˆ F rv Σ for all v Σ ∈ L (Σ). For p = 0, it is straightforward to extend B F v Σ from P ( F ) to H (Ω) ∩ P ( M ). For p ≥
1, let L p ( M ) collect the Lagrange nodes of degree p of M . For each z ∈ L p ( M ), let Φ z be the Lagrange basis function of H (Ω) ∩ P p ( M )associated with the evaluation at z , that is Φ z ( z ′ ) = δ zz ′ for all z ′ ∈ L p ( M ). Since M is a matching simplicial mesh, the set { (Φ z ) | F | z ∈ L p ( M ) ∩ F } is theLagrange basis of P p ( F ). Therefore, we have B F v Σ = P z ∈L p ( M ) ∩ F ( B F v Σ )( z )Φ z in F . Motivated by this identity, we define the global operator B Σ : L (Σ) → H (Ω)such that(4.19) B Σ v Σ := P F ∈F i ( B F v Σ )Φ F , p = 0 , P F ∈F i P z ∈L p ( M ) ∩ F ( B F v Σ )( z )Φ z Φ F , p ≥ . Since ( B Σ v Σ ) | F = ( B F v Σ )Φ F for all F ∈ F i and all p ≥
0, the identity (4.18)implies that the operator B Σ preserves all the moments of v Σ of degree ≤ p on eachinterface of M .A proper combination of B M and B Σ provides an operator B which preserves allmoments prescribed in (4.9). Proposition 4.6 (Bubble smoother) . The operator B : L (Ω) × L (Σ) → H (Ω) defined for all ˆ v = ( v M , v Σ ) ∈ L (Ω) × L (Σ) such that (4.20) B ˆ v := B Σ v Σ + B M ( v M − B Σ v Σ ) fulfills (4.9) and satisfies, for all K ∈ M , the following estimate: (4.21) k∇B ˆ v k L ( K ) ≤ C γ,p h − K k v M k L ( K ) + X F ∈F K h − F k v Σ k L ( F ) ! . Proof.
Let ˆ v = ( v M , v Σ ) ∈ L (Ω) × L (Σ). Owing to the definition of B M and(4.16), we have(4.22) ˆ K q ( B ˆ v ) = ˆ K q ( B Σ v Σ ) + ˆ K q ( B K ( v M − B Σ v Σ ))Φ K = ˆ K qv M , for all K ∈ M and q ∈ P p − ( K ). Moreover, since B M ( v M − B Σ v Σ ) vanishes on theskeleton of M , the definition of B Σ and (4.18) reveal that(4.23) ˆ F r ( B ˆ v ) = ˆ F r ( B Σ v Σ ) = ˆ F r ( B F v Σ )Φ F = ˆ F rv Σ , for all F ∈ F i and r ∈ P p ( F ). The above identities confirm that B fulfills (4.9).To verify the claimed H -norm estimate (4.21), fix K ∈ M and ˆ v = ( v M , v Σ ) ∈ L (Ω) × L (Σ). The definition of B M and Φ K ≤ kB M v M k L ( K ) ≤ ˆ K ( B K v M ) v M ≤ kB K v M k L ( K ) k v M k L ( K ) . Hence, we obtain kB M v M k L ( K ) . k v M k L ( K ) by a standard argument with bubblefunctions, see [26]. Next, for p ≥
1, the boundedness of the extension employed in(4.19) and a scaling argument imply that kB Σ v Σ k L ( K ) . X F ∈F K X z ∈L p ( M ) ∩ F |B F v Σ ( z )Φ F ( z ) | . X F ∈F K h F k ( B F v Σ )Φ F k L ( F ) . Apart of the intermediate step, the same estimate holds also for p = 0. Then, forall F ∈ F K , we argue as before, noticing Φ F ≤
1, to infer that k ( B F v Σ )Φ F k L ( F ) ≤ ˆ F ( B F v Σ ) Φ F = ˆ F ( B F v Σ ) v Σ ≤ kB F v Σ k L ( F ) k v Σ k L ( F ) . QUASI-OPTIMAL HHO METHOD WITH H − LOADS 15
This entails kB Σ v Σ k L ( K ) . P F ∈F K h / F k v Σ k L ( F ) , by a standard argument withbubble functions, see [26]. We conclude combining this bound and the previousone with the definition of B in (4.20) and with the inverse estimate k∇B ˆ v k L ( K ) . h − K kB ˆ v k L ( K ) . (cid:3) The bubble smoother B maps into a space of bubble functions, thus generatingspurious oscillations. This simple observation and inequality (4.21) suggest that the H -norm of B ˆ σ cannot be uniformly bounded by the |·| e a H -norm of ˆ σ , irrespectiveof the size of M , for arbitrary ˆ σ ∈ ˆ S H . This claim can be verified arguing as in [24,Remark 3.5]. Therefore, the bubble smoother B should not be used into the HHOmethod (4.5), although it preserves all the moments prescribed in (4.9). In fact, asmentioned in Remark 2.1, the quasi-optimality constant of a quasi-optimal methodis bounded from below in terms of the operator norm of the employed smoother.The inequality (4.21) indicates that we may define a stabilized version of B if wereplace ˆ v with ˆ v − ˆ A ˆ v in (4.20), provided ˆ A ˆ v ∈ ˆ V is locally (at least) a first-orderapproximation of ˆ v . The operator ˆ A can be defined, for instance, through someaveraging technique, in the vein of [20, 18, 14].To make things precise, denote by L ip +1 ( M ) the interior Lagrange nodes of degree p + 1 of M (i.e. the Lagrange nodes not lying on ∂ Ω). For each node z ∈ L ip +1 ( M ),let Φ z be the Lagrange basis function of H (Ω) ∩ P p +1 ( M ) associated with theevaluation at z . We define A : ˆ S H → H (Ω) such that(4.24) A ˆ σ := X z ∈L ip +1 ( M ) ω z X K ∈ ω z ( R ˆ σ ) | K ( z ) ! Φ z , for all ˆ σ = ( σ M , σ Σ ) ∈ ˆ S H , where ω z collects the simplices of M to which z belongsand ω z denotes the cardinality of ω z . The next proposition confirms that we canuse this operator to stabilize the bubble smoother B . We discuss possible variantsof A in Remark 4.8 below. Notice that A should not be directly used in (4.5),because it may not preserve the moments prescribed in (4.9). Proposition 4.7 (Stabilized bubble smoother) . Let B and A be defined as in (4.20) and (4.24) , respectively, and let ˆ A : ˆ S H → ˆ V be defined such that ˆ A ˆ σ :=( A ˆ σ, ( A ˆ σ ) | Σ ) for all ˆ σ = ( σ M , σ Σ ) ∈ ˆ S H . Then, the operator E H : ˆ S H → H (Ω) such that (4.25) E H ˆ σ := A ˆ σ + B (ˆ σ − ˆ A ˆ σ ) fulfills (4.9) and is such that (4.26) k ∇ M ( R ˆ σ − E H ˆ σ ) k L (Ω) ≤ C γ,p X K ∈M X F ∈F K h − F k σ Σ − ( σ M ) | K k L ( F ) . Proof.
According to (4.22), we have ˆ K qE H ˆ σ = ˆ K q ( A ˆ σ − B ˆ A ˆ σ ) + ˆ K q B ˆ σ = ˆ K qσ M , for all K ∈ M , q ∈ P p − ( K ) and ˆ σ ∈ ˆ S H . The fact that E H preserves all themoments of degree ≤ p on the interfaces of M can be verified similarly, with thehelp of (4.23). This confirms that E H fulfills (4.9). Concerning the claimed stability, we first derive a local version of (4.26). To thisend, let K ∈ M and ˆ σ ∈ ˆ S H be given. The triangle inequality readily implies that k∇ ( R ˆ σ − E H ˆ σ ) k L ( K ) ≤ k∇ ( R ˆ σ − A ˆ σ ) k L ( K ) + k∇B (ˆ σ − ˆ A ˆ σ ) k L ( K ) . We estimate the second summand in the right-hand side with the help of Proposi-tion 4.6, the discrete trace inequality (3.6), identity (3.1b) and the Poincar´e-Steklovinequality (3.8): k∇B (ˆ σ − ˆ A ˆ σ ) k L ( K ) . h − K k σ M − A ˆ σ k L ( K ) + X F ∈F K h − F k σ Σ − A ˆ σ k L ( F ) . h − K kR ˆ σ − A ˆ σ k L ( K ) + k∇ ( σ M − R ˆ σ ) k L ( K ) + X F ∈F K h − F k σ Σ − ( σ M ) | K k L ( F ) . We insert this bound into the previous one. An inverse estimate yields k∇ ( R ˆ σ − E H ˆ σ ) k L ( K ) . h − K kR ˆ σ − A ˆ σ k L ( K ) + k∇ ( σ M − R ˆ σ ) k L ( K ) + X F ∈F K h − F k σ Σ − ( σ M ) | K k L ( F ) . (4.27)We estimate the first summand in the right-hand side by means of [14, Lemma 4.3].Invoking also (3.1b), (3.6) and (3.8), we derive h − K kR ˆ σ − A ˆ σ k L ( K ) . X F ∩ K = ∅ h − F k J R ˆ σ K k L ( F ) . X K ′ ∩ K = ∅ k∇ ( σ M − R ˆ σ ) k L ( K ′ ) + X F ′ ∈F K ′ h − F ′ k σ Σ − ( σ M ) | K ′ k L ( F ′ ) where F and K ′ vary in F and M , respectively, and J · K is the jump operator.Moreover, for all K ′ ∈ M , the identity (3.1a) and (3.6) reveal k∇ ( σ M − R ˆ σ ) k L ( K ′ ) . X F ′ ∈F K ′ h − F ′ k σ Σ − ( σ M ) | K ′ k L ( F ′ ) . We insert this bound and the previous one into (4.27). Squaring and summing overall K ∈ M , we infer that k ∇ M ( R ˆ σ − E H ˆ σ ) k L (Ω) . X K ∈M X K ′ ∩ K = ∅ X F ′ ∈F K ′ h − F ′ k σ Σ − ( σ M ) | K ′ k L ( F ′ ) , where K ′ varies in M . We conclude recalling that the number of simplices touchinga given simplex is ≤ C γ . (cid:3) Remark A ) . Instead of taking the average of R ˆ σ at each node z ∈ L ip +1 ( M ), it is possible to fix K z ∈ M with z ∈ K z and set(4.28) A ′ ˆ σ := X z ∈L ip +1 ( M ) ( R ˆ σ ) | K z Φ z in the vein of the Scott–Zhang interpolation [21]. This modification preserves themain properties of A , whereas the operations needed compute A ′ are significantlyreduced, see [25, Lemma 3.3]. One may also replace the reconstruction R ˆ σ by thecell component σ M of ˆ σ , both in (4.24) and (4.28). Hence, for p ≥
1, the sum canbe restricted to the interior Lagrange nodes of degree p (and not p + 1). With this QUASI-OPTIMAL HHO METHOD WITH H − LOADS 17 variant of A and A ′ , the statement of Proposition 4.7 remains unchanged. Yet,the proof of Lemma 4.11 below and the subsequent derivation of an L -norm errorestimate appear to be problematic for p = 0. Remark E H ) . Let E H be as in Proposition 4.7. A computationallyconvenient basis of ˆ S H consists of functions ˆ σ , . . . , ˆ σ N that are supported eitherin one simplex or on one interface of M . The local estimates established in theproof of Proposition 4.7 reveal that the support of E H ˆ σ i , i = 1 , . . . , N , is a subsetof S { K ∈ M | K ∩ supp(ˆ σ i ) = ∅} . Hence, the number of simplices in the supportof E H ˆ σ i is ≤ C γ . Moreover, the construction of E H ˆ σ i from ˆ σ i requires at mostO(1) operations. Therefore, we can evaluate the duality h f, E H ˆ σ i i H − (Ω) × H (Ω) with O(1) operations and the cost for solving (4.5) is at most a constant factortimes the cost for solving (3.4).4.4. Error estimates.
We now consider the HHO method (4.5) with the smoother E H proposed in (4.25) and derive broken H - and L -norm error estimates. Theformer readily follows from the abstract quasi-optimality stated in Proposition 4.5,combined with the approximation properties of the HHO space and Proposition 4.7. Theorem 4.10 (Broken H -norm error estimate) . Let u ∈ H (Ω) solve (1.1) anddenote by ˆ U ∈ ˆ S H the solution of (4.5) with E H as in Proposition 4.7. Then, thefollowing holds true: (4.29) k ∇ M ( u − R ˆ U ) k L (Ω) + θ ( ˆ U , ˆ U ) ≤ C γ,p X K ∈M inf q ∈ P p +1 ( K ) k∇ ( u − q ) k L ( K ) . Furthermore, if u ∈ H m (Ω) with m ∈ { , . . . , p + 2 } , we have (4.30) k ∇ M ( u − R ˆ U ) k L (Ω) + θ ( ˆ U , ˆ U ) ≤ C γ,p X K ∈M h m − K | u | H m ( K ) . Proof.
The combination of Propositions 4.5 and 4.7 ensures that the HHO method(4.5) with E H as in (4.25) is quasi-optimal in the semi-norm |·| e a H . Recalling thedefinition of the semi-norm |·| e a H , the quasi-optimal estimate (4.11) takes the form k ∇ M ( u − R ˆ U ) k L (Ω) + θ ( ˆ U , ˆ U ) ≤ (1 + C H ) inf ˆ s ∈ ˆ S H (cid:16) k ∇ M ( u − R ˆ s ) k L (Ω) + θ (ˆ s, ˆ s ) (cid:17) . Lemma 3.1 and Proposition 4.5 provide also an upper bound on C H . In fact, forall ˆ σ = ( σ M , σ Σ ) ∈ ˆ S H , we have k ∇ M ( R ˆ σ − E H ˆ σ ) k L (Ω) . k ˆ σ k ˆ S H . | ˆ σ | e a H , showing that C H ≤ C γ,p . Thus, we infer that k ∇ M ( u − R ˆ U ) k L (Ω) + θ ( ˆ U , ˆ U ) . inf ˆ s ∈ ˆ S H (cid:16) k ∇ M ( u − R ˆ s ) k L (Ω) + θ (ˆ s, ˆ s ) (cid:17) . We can now derive the first claimed estimate by taking ˆ s = ˆ I u and using inequality(3.13a) in Lemma 3.2. The second estimate easily follows from the first one usingstandard polynomial approximation properties in Sobolev spaces. (cid:3) According to Theorem 4.10, the HHO method (4.5) with the smoother E H pro-posed in (4.25) reproduces the approximation properties of the interpolant ˆ I (see(3.10)) in the semi-norm |·| e a H . In fact, similarly to the first estimate of Lemma 3.2,the right-hand side of (4.29) bounds the left-hand side also from below. Note also that only the minimal regularity u ∈ H (Ω) is involved there and that (4.30)exploits only element-wise regularity of u .Next, we recall from [13, Theorem 10] that an L -norm error estimate of theHHO method (3.4) can be derived via the Aubin–Nitsche duality technique. Weaim at establishing a counterpart of such a result in the present setting. This wouldconfirm, in particular, that the use of a smoother does not generally rule out thepossibility of establishing L -norm error estimates by duality.As before, we denote by u ∈ H (Ω) and ˆ U = ( U M , U Σ ) ∈ ˆ S H the solutions ofproblems (1.1) and (4.5), respectively, with E H as in (4.25). Proceeding as in [13],we let ψ ∈ H (Ω) be the weak solution of(4.31) − ∆ ψ = U M − Π M u in Ω and ψ = 0 on ∂ Ω . By elliptic regularity [17], there are α ∈ ( ,
1] and a constant c ≥ ψ ∈ H α (Ω) with(4.32) k ψ k H α (Ω) ≤ c k U M − Π M u k L (Ω) . As a preliminary step, we derive a supercloseness estimate on the L -norm of U M − Π M u . Unlike [13], we do not need to address the lowest-order case p = 0separately. Lemma 4.11 (Supercloseness L -estimate) . Let u ∈ H (Ω) solve (1.1) and denoteby ˆ U ∈ ˆ S H the solution of (4.5) with E H as in (4.25) . Let α ∈ ( , be such that (4.32) is satisfied. Then, the following holds true with h := max K ∈M h K : (4.33) k U M − Π M u k L (Ω) ≤ C γ,p h α X K ∈M inf q ∈ P p +1 ( K ) k∇ ( u − q ) k L ( K ) . Proof.
We test (4.31) with U M − Π M u and integrate by parts element-wise, ex-ploiting the regularity of ψ . We obtain k U M − Π M u k L (Ω) = − X K,F ˆ F ( U M − Π M u ) ∇ ψ · n K + ˆ Ω ∇ M ( U M − Π M u ) · ∇ ψ = − X K,F ˆ F (( U M − U Σ ) − (Π M u − Π Σ u )) ∇ ψ · n K + ˆ Ω ∇ M ( U M − Π M u ) · ∇ ψ where K and F vary in M and F K , respectively. The second identity follows fromthe observation that each interface of M appears twice in the sum, with oppositeorientations, combined with the fact that both u and ˆ U vanish on the boundaryfaces. Note also that, to alleviate the notation, we write U M and Π M u , insteadof ( U M ) | K and (Π M u ) | K , in the face integrals. Owing to the definition of thereconstruction R in (3.1), we have ˆ Ω ∇ M R ( ˆ U − ˆ I u ) · ∇ M E ψ = − X K,F ˆ F (( U M − U Σ ) − (Π M u − Π Σ u )) ∇E ψ · n K ++ ˆ Ω ∇ M ( U M − Π M u ) · ∇ M E ψ because E ψ ∈ P p +1 ( M ) is an admissible test function in (3.1a). Inserting thisidentity into the previous one and exploiting the H -orthogonality of the broken QUASI-OPTIMAL HHO METHOD WITH H − LOADS 19 elliptic projection, we infer that k U M − Π M u k L (Ω) = − X K,F ˆ F (( U M − U Σ ) − (Π M u − Π Σ u )) ∇ ( ψ − E ψ ) · n K ++ ˆ Ω ∇ M R ( ˆ U − ˆ I u ) · ∇ M E ψ. In order to rewrite the second summand in the right-hand side, we recall the identity
R ◦ ˆ I = E from (3.12). Then, we exploit problems (1.1) and (4.5) and observe thatthe combination of Lemma 4.4 with Proposition 4.7 guarantees the validity of (4.10)which we exploit here as follows: e a H (ˆ I u, ˆ E H ˆ I ψ ) = e a H (ˆ I u, ˆ I ψ ) − θ (ˆ I u, ˆ I ψ ) . Thus, we have ´ Ω ∇ M E u · ∇ E H ˆ I ψ = ´ Ω ∇ M R ˆ I u · ∇ M E ψ , whence we infer that(4.34) ˆ Ω ∇ M R ( ˆ U − ˆ I u ) · ∇ M E ψ = − θ ( ˆ U , ˆ I ψ ) + ˆ Ω ∇ M ( u − E u ) · ∇ E H ˆ I ψ. Therefore, exploiting again the H -orthogonality of E , we obtain(4.35) k U M − Π M u k L (Ω) = T + T + T with T := − X K,F ˆ F (( U M − U Σ ) − (Π M u − Π Σ u )) ∇ ( ψ − E ψ ) · n K T := − θ ( ˆ U , ˆ I ψ ) T := ˆ Ω ∇ M ( u − E u ) · ∇ M ( E H ˆ I ψ − E ψ ) . It remains to bound the three summands T , T and T . The definition of theinterpolant ˆ I and the coercivity stated in Lemma 3.1 entail T . | ˆ U − ˆ I u | e a H X K,F h F k∇ ( ψ − ( E ψ ) | K ) k L ( F ) where K and F vary in M and F K , respectively. Owing to the approximationproperties of the broken elliptic projection, we obtain h F k∇ ( ψ − ( E ψ ) | K ) k L ( F ) . h αK | ψ | H α ( K ) , for all K ∈ M and F ∈ F K . Combining this bound and the previous one with thefirst part of Lemma 3.2 and the H -norm error estimate (4.29), we obtain T . h α | ψ | H α (Ω) X K ∈M inf q ∈ P p +1 ( K ) k∇ ( u − q ) k L ( K ) . Invoking again Lemma 3.2 and (4.29) yields also T . X K ∈M inf q ∈ P p +1 ( K ) k∇ ( ψ − q ) k L ( K ) ! X K ∈M inf q ∈ P p +1 ( K ) k∇ ( u − q ) k L ( K ) ! . h α | ψ | H α (Ω) X K ∈M inf q ∈ P p +1 ( K ) k∇ ( u − q ) k L ( K ) , where the second estimate follows from standard polynomial approximation prop-erties in Sobolev spaces. In order to bound the third summand T in (4.35), we proceed similarly to the proof of (4.26) in Proposition 4.7. Owing to the ap-proximation properties of the broken elliptic projection, we only need to bound k∇ ( E ψ − E H ˆ I ψ ) k L ( K ) . For all K ∈ M , the triangle inequality yields(4.36) k∇ ( E ψ − E H ˆ I ψ ) k L ( K ) ≤ k∇ ( E ψ − A ˆ I ψ ) k L ( K ) + k∇B (ˆ I ψ − ˆ A ˆ I ψ ) k L ( K ) . The definitions of ˆ I and B readily imply that B ˆ I ψ = B ˆ ψ , with ˆ ψ = ( ψ, ( ψ ) | Σ ).This observation, Proposition 4.6 and the multiplicative trace inequality (3.7) yield k∇B (ˆ I ψ − ˆ A ˆ I ψ ) k L ( K ) . h − K k ψ − A ˆ I ψ k L ( K ) + k∇ ( ψ − A ˆ I ψ ) k L ( K ) . Next, we combine [14, Lemma 4.3] with the identity (3.12) and the multiplicativetrace inequality (3.7). We obtain that h − K kE ψ − A ˆ I ψ k L ( K ) + k∇ ( E ψ − A ˆ I ψ ) k L ( K ) . X F ∩ K = ∅ h − F k J E ψ K k L ( F ) . X K ′ ∩ K = ∅ (cid:0) h − K ′ k ψ − E ψ k L ( K ′ ) + k∇ ( ψ − E ψ ) k L ( K ′ ) (cid:1) where F and K ′ vary in F and M , respectively, and J · K is the jump operator. Weinsert this inequality and the previous one into (4.36). Owing to the approximationproperties of the broken elliptic projection, we infer that k∇ ( E ψ − E H ˆ I ψ ) k L ( K ) . X K ′ ∩ K = ∅ h αK ′ | ψ | H α ( K ′ ) . Squaring and summing over all K ∈ M , we finally derive that(4.37) T . h α | ψ | H α (Ω) X K ∈M inf q ∈ P p +1 ( K ) k∇ ( u − q ) k L ( K ) . in view of (3.14) and recalling that the maximum number of simplices touching agiven simplex is ≤ C γ . Collecting the bounds on T , T and T and invoking theelliptic regularity property (4.32) concludes the proof. (cid:3) Theorem 4.12 ( L -norm error estimate) . Let u ∈ H (Ω) solve (1.1) and denoteby ˆ U ∈ ˆ S H the solution of (4.5) with E H as in (4.25) . Let α ∈ ( , be such that (4.32) is satisfied. Then, the following holds true: (4.38) k u − R ˆ U k L (Ω) ≤ C γ,p h α X K ∈M inf q ∈ P p +1 ( K ) k∇ ( u − q ) k L ( K ) where h := max K ∈M h K . Furthermore, if u ∈ H m (Ω) with m ∈ { , . . . , p + 2 } , wehave (4.39) k u − R ˆ U k L (Ω) ≤ C γ,p h α X K ∈M h m − K | u | H m ( K ) . Proof.
We have k u − R ˆ U k L (Ω) ≤ k u − E u k L (Ω) + kR ˆ U − E u k L (Ω) . Concerningthe first summand, the identity (3.12) and the bound (3.13b) imply that(4.40) k u − E u k L (Ω) ≤ X K ∈M (cid:18) h K π (cid:19) inf q ∈ P p +1 ( K ) k∇ ( u − q ) k L ( K ) . QUASI-OPTIMAL HHO METHOD WITH H − LOADS 21
Concerning the other summand, the identity (3.12) implies R ˆ U − E u = R ( ˆ U − ˆ I u ).We fix any K ∈ M and denote by ffl K the integral mean value on K . The identity(3.1b) and the Poincar´e–Steklov inequality (3.8) yield kR ( ˆ U − ˆ I u ) k L ( K ) ≤ kR ( ˆ U − ˆ I u ) − ffl K R ( ˆ U − ˆ I u ) k L ( K ) + k ffl K R ( ˆ U − ˆ I u ) k L ( K ) ≤ π − h K k∇R ( ˆ U − ˆ I u ) k L ( K ) + k U M − Π M u k L ( K ) . Summing over all simplices of M and using the first part of Lemma 3.2 and the H -norm error estimate (4.29), we obtain(4.41) k u − R ˆ U k L (Ω) . X K ∈M h K π inf q ∈ P p +1 ( K ) k∇ ( u − q ) k L ( K ) + k U M − Π M u k L (Ω) . Thus, we derive (4.38) by inserting the bound (4.33) into (4.41). Finally, the esti-mate (4.39) follows from (4.38) and standard polynomial approximation propertiesin Sobolev spaces. (cid:3)
Similarly to Theorem 4.10, estimate (4.38) holds under the minimal regularity u ∈ H (Ω), and (4.39) exploits only the element-wise regularity of u . Still, bothestimates are more pessimistic than (3.13b) in Lemma 3.2, even for α = 1, if M isa graded mesh. This is a general drawback of the estimates derived via the Aubin–Nitsche duality argument. Perhaps, a better result could be obtained with the helpof the technique recently devised in [16, 19].5. Polytopic meshes
Since the HHO methods in [12, 13] are not only defined for matching simplicialmeshes of Ω, but more generally on polyhedral meshes possibly comprising hangingnodes, it is worth asking if we can relax the assumptions on M in the previoussections. To this end, a first inspection reveals that the abstract results of section 4.2on the quasi-optimality of (4.5) build only on the notion of interface and on thenondegeneracy of b H . Of course, both ingredients are in any case needed in thedefinition of the space ˆ S H and for the solution of problem (3.4). Thus, in principle,it appears possible to design HHO methods, that are quasi-optimal in the semi-norm |·| e a H , within a larger class of polytopic meshes.Proceeding as in [12, 13], we now consider meshes M = ( K ) K ∈M of Ω such that • Ω = S K ∈M K and the cardinality of M is finite, • each cell K ∈ M is an open polygon/polyhedron, • for all cells K , K ∈ M with K = K , we have K ∩ K = ∅ .We say that F ⊂ Ω is a face of M if it is a subset, with nonempty relative interior,of some ( d − H F and if one of the following conditionsholds true: either there are two distinct cells K , K ∈ M so that F = K ∩ K ∩ H F or there is one cell K ∈ M so that F = K ∩ ∂ Ω ∩ H F . We collect in the set F i allthe interfaces, i.e. the faces of M fulfilling the first condition.To preserve the validity of the results in section 3.2, we further assume that M is an admissible mesh in the sense of [11, Section 1.4]. More precisely, we requirethat there is a matching simplicial submesh T = ( T ) T ∈T of M , such that • for each simplex T ∈ T , there is a cell K ∈ M such that T ⊆ K and h K . h T . The inequalities stated in (3.6), (3.7) and (3.8) as well as the ones in Lemmata 3.1and 3.2 still hold true under this assumption, possibly up to more pessimistic con-stants, depending on the shape regularity of M and T . We refer to [11, section 1.4]and [13] for a more detailed discussion on this point.The real bottleneck in the extension of our previous results is the construction ofa smoother E H , generalizing the one in Proposition 4.7. For this purpose, one optionis to still write E H as the combination of a bubble smoother, which accommodatesthe conservation of the moments prescribed by Proposition 4.5, and an averagingoperator, that serves to keep under control the constant C H in (4.12).For the sake of completeness, we sketch a possible construction for arbitrary p ≥
0. For all K ∈ M , we can find a simplex T K ∈ T such that T K ⊆ K .Denote by Φ T K ∈ H (Ω) the cell bubble determined by ( i ) Φ T K ≡ \ T K , ( ii )(Φ T K ) | T K ∈ P d +1 ( T K ) and ( iii ) Φ T K ( m T K ) = 1 at the barycenter m T K of T K . Since( q , q ) ´ K q q Φ T K is a scalar product on P p − ( K ), we define the operators B K and B M as in (4.16) and (4.17), respectively, with Φ T K in place of Φ K .For all F ∈ F i , we can find an interface T F of T and T , T ∈ T so that T F ⊆ F and T F = T ∩ T . Set ω T F := T ∪ T and denote by Φ T F ∈ H (Ω) the face bubble obtained prescribing ( i ) Φ T F ≡ \ ω T F , ( ii ) (Φ T F ) | T j ∈ P d ( T j ) for j = 1 , iii ) Φ T F ( m T F ) = 1 at the barycenter m T F of T F . We define the operator B F as in(4.18), with Φ T F in place of Φ F . Then, for all v Σ ∈ L (Σ), we set B Σ v Σ := P F ∈F i ( B F v Σ )Φ T F , p = 0 , P F ∈F i P z ∈L p ( T ) ∩ T F ( B F v Σ )( z )Φ z Φ T F , p ≥ , where L p ( T ) denotes the Lagrange nodes of degree p of T and Φ z is the Lagrangebasis function of H (Ω) ∩ P p ( T ) associated with the evaluation at z .With B M and B Σ as indicated, the bubble smoother B : L (Ω) × L (Σ) → H (Ω)is simply given by (4.20) and fulfills (4.9) and (4.21).Finally, denote by L ip +1 ( T ) the interior Lagrange nodes of degree p + 1 of T . Forall ˆ σ = ( σ M , σ Σ ) ∈ ˆ S H , we consider the averaging(5.1) A ˆ σ := X z ∈L ip +1 ( T ) ω z X T ∈ ω z ( R ˆ σ ) | T ( z ) ! Φ z , where T varies in T and ω z collects the simplices of T to which z belongs.With A and B as indicated, the smoother E H : ˆ S H → H (Ω), defined as inProposition 4.7, fulfills (4.9) and (4.26). The derivation of H - and L -norm errorestimates of the HHO method (4.5) with this smoother proceeds along the samelines as in section 4.4. Remark . The use of the simplicial submesh T in thedefinition of the bubble smoother B is not really necessary. Indeed, one only needsbubble functions attached to the cells and to the interfaces of M and boundedextension operators from each interface to Ω. In contrast, our construction of theaveraging A substantially builds on the submesh. Of course, this can be seen as amain disadvantage, as it restricts applicability of the proposed method to the classof admissible meshes. Still, it must be said that this is just one possible constructionand that the use of alternative averaging operators could be further explored. QUASI-OPTIMAL HHO METHOD WITH H − LOADS 23
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Universit´e Paris-Est, CERAMICS (ENPC), 77455 Marne-la-Vall´ee cedex 2, Franceand INRIA Paris, 75589 Paris, France
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