We propose a modified local discontinuous Galerkin (LDG) method for
second--order elliptic problems that does not require extrinsic penalization to
ensure stability. Stability is instead achieved by showing a discrete
Poincar\'e--Friedrichs inequality for the discrete gradient that employs a
lifting of the jumps with one polynomial degree higher than the scalar
approximation space. Our analysis covers rather general simplicial meshes with
the possibility of hanging nodes.Comment: Accepted for publication in the conference proceedings of Numerical
  Mathematics and Advanced Applications ENUMATH 2015. Typo correcte

John, Lorenz

Neilan, Michael

Smears, Iain

English

arXiv

Lorenz John

Michael Neilan

Iain Smears

Crossref

Stable Discontinuous Galerkin FEM Without Penalty Parameters

International audienceWe propose a modified local discontinuous Galerkin (LDG) method for second–order elliptic problems that does not require extrinsic penalization to ensure stability. Stability is instead achieved by showing a discrete Poincaré–Friedrichs inequality for the discrete gradient that employs a lifting of the jumps with one polynomial degree higher than the scalar approximation space. Our analysis covers rather general simplicial meshes with the possibility of hanging nodes

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HAL Id: hal-01428664https://hal.inria.fr/hal-01428664Submitted on 6 Jan 2017HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Stable Discontinuous Galerkin FEM Without PenaltyParametersLorenz John, Michael Neilan, Iain SmearsTo cite this version:Lorenz John, Michael Neilan, Iain Smears. Stable Discontinuous Galerkin FEM Without PenaltyParameters. Numerical Mathematics and Advanced Applications ENUMATH 2015, Sep 2015, Ankara,Turkey. pp.XIV, 643, ￿10.1007/978-3-319-39929-4_17￿. ￿hal-01428664￿Stable discontinuous Galerkin FEM withoutpenalty parametersLorenz John1, Michael Neilan2, and Iain Smears31 Technische Universität München, München, 80333, Germany, john@ma.tum.de2 University of Pittsburgh, Pittsburgh, PA 15260, United States,neilan@pitt.edu3 INRIA Paris-Rocquencourt, Le Chesnay, 78153, France, iain.smears@inria.frAbstract. We propose a modified local discontinuous Galerkin (LDG) method forsecond–order elliptic problems that does not require extrinsic penalization to ensurestability. Stability is instead achieved by showing a discrete Poincaré–Friedrichsinequality for the discrete gradient that employs a lifting of the jumps with onepolynomial degree higher than the scalar approximation space. Our analysis coversrather general simplicial meshes with the possibility of hanging nodes.1 IntroductionIt is well–known that the local discontinuous Galerkin (LDG) method forsecond–order elliptic problems can be formulated, in part, by replacing thedifferential operators in the variational formulation by their discrete counter-parts [3–5]. For example, on the space of discontinuous piecewise polynomialsof degree at most k, the discrete gradient operator is composed of the element-wise gradient corrected by a lifting of the jumps into the space of piecewisepolynomial vector fields. The original formulation of the LDG method [3] em-ploys liftings of same polynomial degree k as the scalar finite element space,while liftings of order k−1 have also been considered, see the textbook [5] andthe references therein. Part of the motivation for these choices of the order ofthe lifting is the correspondence to the order of the element-wise gradient andreasons of ease of implementation. However, unlike the continuous gradientacting on the space H10 , the discrete gradient operators with liftings of orderk− 1 or k fail to satisfy a discrete Poincaré–Friedrichs inequality. Therefore,the LDG method requires additional penalization with user–defined penaltyparameters to ensure stability.In this note, we construct a modified LDG method with guaranteed sta-bility without the need for extrinsic penalization. This result is obtained bysimply increasing the polynomial degree of the lifting operator to order k+ 1and exploiting properties of the piecewise Raviart–Thomas–Nédélec finite ele-ment space. Our analysis covers the case of meshes with hanging nodes undera mild condition of face regularity which we introduce in this work. We recallthat the order of the lifting in the LDG method does not alter the dimensionor stencil of the resulting stiffness matrix. As a result, the proposed method2 John et al.has a negligible increase of computational cost and inherits the advantagesof the standard LDG method in terms of locality and conservativity.The rest of the paper is organized as follows. In Sec. 2 we give the no-tation used throughout the manuscript and state some preliminary results.We define the lifted gradient operator with increased polynomial degree inSec. 3 and show that the L2 norm of this operator is equivalent to a discreteH1 norm on piecewise polynomial spaces. We establish by means of a coun-terexample that the increased polynomial degree is necessary to obtain thisstability estimate in Sec. 4. In Sec. 5 we propose and study the modified LDGmethod in the context of the Poisson equation.2 NotationLet Ω ⊂ Rd, d ∈ {2, 3}, be a bounded polytopal domain with Lipschitzboundary ∂Ω. Let {Th}h>0 be a shape- and contact–regular sequence ofsimplicial meshes on Ω, as defined in [5, Definition 1.38]. For each elementK ∈ Th, let hK := diamK, with h = maxK∈Th hK for each mesh Th. We de-fine the faces of the mesh as in [5, Definition 1.16], and we collect all interiorand boundary faces in the sets F ih and Fbh, respectively, and let Fh := F ih∪Fbhdenote the skeleton of Th. In particular, F ∈ F ih if F has positive (d − 1)-dimensional Hausdorff measure and if F = ∂K1 ∩ ∂K2 for two distinct meshelements K1 and K2. For an element K ∈ Th, we denote F(K) the set of facesof K, i.e. E ∈ F(K) if E is the closed convex hull of d vertices of the simplexK. Note that on a mesh with hanging nodes, a mesh face may be a propersubset of an element face, see Fig. 1, hence the notions of mesh faces andelement faces do not need to coincide. In this work, the meshes are allowedto have hanging nodes, provided that they satisfy the following notion of faceregularity.Definition 1. A face F ∈ Fh is called regular with respect to the elementK if F ∈ F(K). We say that the mesh Th is face regular if every face of Fhis a regular face with respect to at least one element of Th.Fig. 1 illustrates the notion of face regularity with two examples. Weremark that any matching mesh is face regular. On a face regular mesh, anyboundary face is necessarily regular with respect to the element to which itbelongs. It appears that meshes of practical interest are most likely to be faceregular, so this restriction is rather mild in practice.For integrable functions φ defined piecewise on either Th or Fh, we usethe convention∫Ωφ dx =∑K∈Th∫Kφ dx,∫Fhφds =∑F∈Fh∫Fφ ds.For the integer k ≥ 1, we define the discontinuous finite element spacesVh,k as the space of real-valued piecewise-polynomials of degree at most k onStable DGFEM without penalty parameters 3F2F3F1KF̄2F̄4F̄3F̄1F̄5Fig. 1. Face regularity of meshes: the mesh on the left has interior facesF ih = {Fi}3i=1, each of which is regular to at least one element in the senseof Definition 1, even though F2 and F3 fail to be regular with respect to theelement K, since F2 and F3 are only proper subsets of the elemental faceF2∪F3. Since all boundary faces are also regular, the mesh on the left is faceregular in the sense of Definition 1, whereas the mesh on the right is not: themesh face F̄3 fails to be regular with respect to any element of the mesh.Th, and Σh,k+1 the space of vector-valued piecewise-polynomials of degree atmost k + 1 on Th. We define the mesh-dependent norm ‖·‖1,h on Vh,k by‖vh‖21,h :=∑K∈Th‖∇vh‖2L2(K) +∑F∈Fh1hF‖JvhK‖2L2(F ) ∀ vh ∈ Vh,k, (1)where hF := diamF for each face F ∈ Fh.We shall also make use of the (local) Raviart–Thomas–Nédélec space [7]defined byRTNk+1(K) := Pk(K)⊕ P̃k(K)x ⊂ Pk+1(K),where Pk(K) is the space of vector-valued polynomials of degree at mostk on K, and P̃k(K) is the space of real-valued homogeneous polynomials ofdegree k on K. We recall that τh ∈ RTNk+1(K) is uniquely determined bythe moments∫Kτh · µh dx and∫E(τh · nE) vh ds for all µh ∈ Pk−1(K) andvh ∈ Pk(E) for each E ∈ F(K), where nE denotes a unit normal vector of E.We also recall that if all facial moments of τh vanish on an elemental face E,then τh · nE vanishes identically on E.For a face F ∈ Fh belonging to an element Kext, we define the jump andaverage operators byJwK|F := w|Kext − w|Kint , {w} |F :=12(w|Kext + w|Kint), if F ∈ F ih,JwK|F := w|Kext , {w} |F := w|Kext , if F ∈ Fbh,where w is a sufficiently regular scalar or vector-valued function, and in thecase where F ∈ F ih, Kint is such that F = ∂Kext ∩ ∂Kint. Here, the labellingis chosen so that nF is outward pointing with respect to Kext and inward4 John et al.pointing with respect to Kint. Let φ ∈ L2(Fh), then the lifting operatorsrh : L2(Fh)→ Σh,k+1 and rh : L2(Fh)→ Vh,k are defined by∫Ωrh(φ) · σh dx =∫Fhφ {σh · nF } ds ∀σh ∈ Σh,k+1, (2a)∫Ωrh(φ) vh dx =∫Fihφ {vh} ds ∀ vh ∈ Vh,k. (2b)For quantities a and b, we write a . b if and only if there is a positive constantC such that a ≤ Cb, where C is independent of the quantities of interest,such as the element sizes, but possibly dependent on the shape-regularityparameters and polynomial degrees.3 Stability of lifted gradientsWe define the lifted gradient Gh : Vh,k → Σh,k+1 byGh(vh) = ∇hvh − rh(JvhK) ∀ vh ∈ Vh,k, (3)where ∇h denotes the element-wise gradient operator. We note that Gh isusually defined with a lifting using polynomial degrees k or k − 1, see forinstance [5]. However, as we shall see, by increasing the polynomial degree ofthe lifting to k + 1, we obtain the following key stability result.Theorem 2. Let {Th}h>0 denote a shape regular, contact regular and faceregular sequence of simplicial meshes on Ω. Let the norm ‖·‖1,h be definedby (1) and let the lifted gradient operator Gh be defined by (3). Then, we have‖uh‖1,h . ‖Gh(uh)‖L2(Ω) . ‖uh‖1,h ∀uh ∈ Vh,k. (4)Proof. The upper bound ‖Gh(uh)‖L2(Ω) . ‖uh‖1,h is standard and we referthe reader to [5, Sec. 4.3] for a proof. To show the lower bound, consider anarbitrary uh ∈ Vh,k. Since Gh(uh) ∈ Σh,k+1, we have‖Gh(uh)‖L2(Ω) = supτh∈Σh,k+1\{0}∫ΩGh(uh) · τh dx‖τh‖L2(Ω),with the supremum being achieved by the choice τh = Gh(uh). Therefore, toshow (4), it is sufficient to construct a τh ∈ Σh,k+1 such that‖uh‖21,h .∫ΩGh(uh) · τh dx, (5)‖τh‖L2(Ω) . ‖uh‖1,h. (6)Stable DGFEM without penalty parameters 5Let τK ∈ RTNk+1(K) be defined by∫KτK · µh dx =∫K∇uh · µh dx ∀µh ∈ Pk−1(K), (7a)∫E(τK · nE) vh ds ={−∫E1hEJuhK vh ds if E ∈ Fh,0 if E /∈ Fh,(7b)where (7b) holds for all vh ∈ Pk(E), for each element face E ∈ F(K). Inparticular, if the element face E ∈ Fh, i.e. E is also a mesh face, then werequire that nE agrees with the choice of unit normal used to define the jumpand average operators. If E /∈ Fh, then τK · nE vanishes identically on E,and the orientation of nE on the left-hand side of (7b) does not matter. Theglobal vector field τh ∈ Σh,k+1 is defined element-wise by τh|K = τK .Since the mesh Th is assumed to be face regular, for every F ∈ Fhthere exists an element K ∈ Th and an elemental face E ∈ F(K) such thatE = F ; then E satisfies the first condition in (7b). Therefore, the facts that{τh · nF } |F and JuhK|F both belong to Pk(F ) together with (7b) imply thatfor each F ∈ Fh, one of only three situations may arise:1. F is a boundary face and hence F ∈ F(K). In this case, we have{τh · nF } |F = −h−1F JuhK|F .2. F is an interior face which is regular with respect to both elements towhich it belongs. In this case, we have {τh · nF } |F = −h−1F JuhK|F .3. F is an interior face which is regular with respect to only one of the ele-ments to which it belongs. In this case, we have {τh · nF } |F = − 12h−1F JuhK|F ,since τh|K′ · nF ≡ 0 for the element K ′ with respect to which F is notregular.Therefore, since τh ∈ Σh,k+1, the definition of the lifting operator in (2a)implies that∫ΩGh(uh) · τh dx =∑K∈Th∫K∇uh · τh dx−∑F∈Fh∫F{τh · nF } JuhK ds≥∑K∈Th‖∇uh‖2L2(K) +12∑F∈Fh1hF‖JuhK‖2L2(F )≥ 12‖uh‖21,h,where the second line follows from (7) and from the fact that ∇uh|K ∈Pk−1(K) for each K ∈ Th. Hence (5) is satisfied, and we now verify (6). Aclassical scaling argument using the Piola transformation [2, p. 59] yields‖τh‖L2(K) . supµh∈Pk−1(K)\{0}∫Kτh · µh dx‖µh‖L2(K)+∑E∈F(K)supvh∈Pk(E)\{0}h1/2E∫E(τh · nE)vh ds‖vh‖L2(E)∀K ∈ Th.6 John et al.K1K2K3K4Fig. 2. Counterexample of Sec. 4: the domain Ω = (−1, 1)2 and the criss-cross mesh Th considered in the example.Therefore, it follows from (7) that, for each K ∈ Th,‖τh‖2L2(K) . ‖∇uh‖2L2(K) +∑F∈F(K)∩FhhF ‖h−1F JuhK‖2L2(F ). (8)Summing (8) over all elements therefore implies (6). ut4 Counterexample to stability for equal-order liftingsTheorem 2 shows the stability of the lifted gradient operatorGh provided thatthe lifting operator rh has polynomial degree k+ 1. In this section, we verifyby means of a counterexample that the stability estimate does not generallyhold for lower-order liftings, including in particular the case of equal-orderliftings, which are commonly used in practice; our example simplifies a similarcounterexample in [1].Example. Let Ω = (−1, 1)2, and consider the finite element space Vh,kdefined on a criss-cross mesh with four triangles, as depicted in Fig. 2, usingpiecewise linear polynomials, i.e. k = 1. Let uh ∈ Vh,1 be the piecewise linearfunction defined byuh|K1 = y +23, uh|K2 = x−23,uh|K3 = −y +23, uh|K4 = −x−23.Direct calculations show that {uh}|F ≡ 0 on all interior faces F ∈ F ih, andthat∫Kuh dx = 0 for all elements K ∈ Th. Consequently, if the lifting opera-tor r̃h is defined in (2a) with the polynomial degree k+ 1 replaced by k, andif G̃h(uh) := ∇huh − r̃h(JuhK) denotes the equal-order lifted gradient, thenwe have for all τh ∈ Σh,1,∫ΩG̃h(uh) · τh dx =∑K∈Th∫K∇huh · τh dx−∑F∈Fh∫F{τh · nF } JuhK ds= −∑K∈Th∫Kuh(∇h · τh) dx+∑F∈Fih∫F{uh}Jτh · nF K ds = 0.Stable DGFEM without penalty parameters 7Since G̃h(uh) ∈ Σh,1, we deduce that G̃h(uh) = 0, and thus it is found thatno bound of the form ‖uh‖1,h . ‖G̃h(uh)‖L2(Ω) is possible. ut5 A modified LDG method without penalty parametersAs an application of Theorem 2, consider the discretization of the homoge-neous Dirichlet boundary-value problem of the Poisson equation by a mod-ified LDG method [3,4] as follows. For f ∈ L2(Ω), let u ∈ H10 (Ω) be theunique solution of∫Ω∇u · ∇v dx =∫Ωf v dx ∀ v ∈ H10 (Ω). (9)Let the bilinear form ah : Vh,k × Vh,k → R be defined byah(uh, vh) =∫ΩGh(uh) ·Gh(vh) dx ∀uh, vh ∈ Vh,k, (10)where the lifted gradient operator Gh was defined in (3). The bilinear formah(·, ·) defines a modified LDG method for (9): find uh ∈ Vh,k such thatah(uh, vh) =∫Ωf vh dx ∀vh ∈ Vh,k. (11)It follows from Theorem 2 that ah(·, ·) is uniformly stable with respect tothe norm ‖·‖1,h, and thus (11) is well-posed for each h. Moreoever, the dis-crete Poincaré inequality [5] implies that ‖uh‖1,h . ‖f‖L2(Ω) for all h, sothat the numerical solutions uh are uniformly bounded with respect to themesh-dependent norms ‖·‖1,h. The a priori error analysis for the numericalmethod defined by (11) may be developed following the frameworks of [3,5,6],although for reasons of space we do not present the arguments here.An interesting feature of the modified LDG method (11) is that it doesnot require any additional stabilization, such as added penalty terms of theform∫FhσFhFJuhKJvhK ds for some user-defined parameter σF . The absence ofsuch penalty terms enables us to show the following discrete conservationproperty. We define the lifted divergence Dh : Σh,k+1 → Vh,k byDh(σh) = divh σh − rh(Jσh · nF K), σh ∈ Σh,k+1, (12)where divh denotes the element-wise divergence operator, and where rh is thescalar lifting operator defined in (2b). We note that we have the integration-by-parts identity∫Ωσh ·Gh(vh) dx = −∫ΩDh(σh) vh dx ∀ vh ∈ Vh,k, σh ∈ Σh,k+1, (13)8 John et al.which should be compared with the analogous continuous identity betweenthe spaces H10 (Ω) and H(div, Ω). Therefore, the numerical scheme (11) canbe equivalently expressed in the strong form−∫ΩDh(Gh(uh)) vh dx =∫Ωf vh dx, (14)which implies that the numerical solution uh ∈ Vh,k solves−Dh(Gh(uh)) = Πkhf, (15)in the pointwise sense on each element K, where Πkhf denotes the element-wise L2-projection of f into Vh,k. Although we have shown here how thelifted gradient operator Gh of degree k + 1 may be used to achieve a stablediscretization of the Poisson equation, it is by no means restricted to thismodel problem, as the lifted gradients may be used to discretize the second-order terms of more general differential operators.6 ConclusionsIn this article, we studied an intrinsically stable modified LDG method with-out additional parameter dependent penalization. For this, we showed thatincreasing the degree of the lifting operator by one order leads to stability ofthe discrete gradient operator on face regular meshes with hanging nodes.AcknowledgementThe work of the third author was partially supported by the NSF grantDMS–1417980 and the Alfred Sloan Foundation.References1. F. Brezzi, M. Manzini, D. Marini, P. Pietra and A. Russo, Discontin-uous finite elements for diffusion problems, Atti del Convegno in Memoria diF. Brioschi, Milano, Istiuto Lombardo di Scienze e Lettere (1997).2. D. Boffi, F. Brezzi, M. Fortin, Mixed finite element methods and applica-tions, Springer Series in Computational Mathematics vol. 44, Springer (2013).3. P. Castillo, B. Cockburn, I. Perugia and D. Schötzau, An a priori erroranalysis of the local discontinuous Galerkin method for elliptic problems, SIAMJ. Numer. Anal. 38:5 (2000), 1676–1706 (electronic).4. B. Cockburn and C.-W. Shu, The local discontinuous Galerkin methodfor time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35:6(1998), 2440–2463 (electronic).5. D. A. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkinmethods, Mathématiques & Applications vol. 69, Springer (2012).6. T. Gudi, A new error analysis for discontinuous finite element methods forlinear elliptic problems, Math. Comp. 79:272 (2010), 2169–2189.7. J.-C. Nédélec, Mixed finite elements in R3, Numer. Math. 35:3 (1980), 315–341.

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We propose a modified local discontinuous Galerkin (LDG) method for second–order elliptic problems that does not require extrinsic penalization to ensure stability. Stability is instead achieved by showing a discrete Poincaré–Friedrichs inequality for the discrete gradient that employs a lifting of the jumps with one polynomial degree higher than the scalar approximation space. Our analysis covers rather general simplicial meshes with the possibility of hanging nodes

John, L

Neilan, M

Smears, I

UCL Discovery

arXiv:1602.04625v2  [math.NA]  9 Mar 2016Stable discontinuous Galerkin FEM withoutpenalty parametersLorenz John1, Michael Neilan2, and Iain Smears31 Technische Universita¨t Mu¨nchen, Mu¨nchen, 80333, Germany, john@ma.tum.de2 University of Pittsburgh, Pittsburgh, PA 15260, United States,neilan@pitt.edu3 INRIA Paris-Rocquencourt, Le Chesnay, 78153, France, iain.smears@inria.frAbstract. We propose a modified local discontinuous Galerkin (LDG) method forsecond–order elliptic problems that does not require extrinsic penalization to ensurestability. Stability is instead achieved by showing a discrete Poincare´–Friedrichsinequality for the discrete gradient that employs a lifting of the jumps with onepolynomial degree higher than the scalar approximation space. Our analysis coversrather general simplicial meshes with the possibility of hanging nodes.1 IntroductionIt is well–known that the local discontinuous Galerkin (LDG) method forsecond–order elliptic problems can be formulated, in part, by replacing thedifferential operators in the variational formulation by their discrete coun-terparts [3,4,5]. For example, on the space of discontinuous piecewise poly-nomials of degree at most k, the discrete gradient operator is composed ofthe element-wise gradient corrected by a lifting of the jumps into the spaceof piecewise polynomial vector fields. The original formulation of the LDGmethod [3] employs liftings of same polynomial degree k as the scalar finiteelement space, while liftings of order k − 1 have also been considered, seethe textbook [5] and the references therein. Part of the motivation for thesechoices of the order of the lifting is the correspondence to the order of theelement-wise gradient and reasons of ease of implementation. However, unlikethe continuous gradient acting on the space H10 , the discrete gradient opera-tors with liftings of order k−1 or k fail to satisfy a discrete Poincare´–Friedrichsinequality. Therefore, the LDG method requires additional penalization withuser–defined penalty parameters to ensure stability.In this note, we construct a modified LDG method with guaranteed sta-bility without the need for extrinsic penalization. This result is obtained bysimply increasing the polynomial degree of the lifting operator to order k+1and exploiting properties of the piecewise Raviart–Thomas–Ne´de´lec finite ele-ment space. Our analysis covers the case of meshes with hanging nodes undera mild condition of face regularity which we introduce in this work. We recallthat the order of the lifting in the LDG method does not alter the dimensionor stencil of the resulting stiffness matrix. As a result, the proposed method2 John et al.has a negligible increase of computational cost and inherits the advantagesof the standard LDG method in terms of locality and conservativity.The rest of the paper is organized as follows. In Sec. 2 we give the no-tation used throughout the manuscript and state some preliminary results.We define the lifted gradient operator with increased polynomial degree inSec. 3 and show that the L2 norm of this operator is equivalent to a discreteH1 norm on piecewise polynomial spaces. We establish by means of a coun-terexample that the increased polynomial degree is necessary to obtain thisstability estimate in Sec. 4. In Sec. 5 we propose and study the modified LDGmethod in the context of the Poisson equation.2 NotationLet Ω ⊂ Rd, d ∈ {2, 3}, be a bounded polytopal domain with Lipschitzboundary ∂Ω. Let {Th}h>0 be a shape- and contact–regular sequence ofsimplicial meshes on Ω, as defined in [5, Definition 1.38]. For each elementK ∈ Th, let hK := diamK, with h = maxK∈Th hK for each mesh Th. We de-fine the faces of the mesh as in [5, Definition 1.16], and we collect all interiorand boundary faces in the sets F ih and Fbh, respectively, and let Fh := Fih∪Fbhdenote the skeleton of Th. In particular, F ∈ Fih if F has positive (d − 1)-dimensional Hausdorff measure and if F = ∂K1 ∩ ∂K2 for two distinct meshelements K1 andK2. For an element K ∈ Th, we denote F(K) the set of facesof K, i.e. E ∈ F(K) if E is the closed convex hull of d vertices of the simplexK. Note that on a mesh with hanging nodes, a mesh face may be a propersubset of an element face, see Fig. 1, hence the notions of mesh faces andelement faces do not need to coincide. In this work, the meshes are allowedto have hanging nodes, provided that they satisfy the following notion of faceregularity.Definition 1. A face F ∈ Fh is called regular with respect to the elementK if F ∈ F(K). We say that the mesh Th is face regular if every face of Fhis a regular face with respect to at least one element of Th.Fig. 1 illustrates the notion of face regularity with two examples. Weremark that any matching mesh is face regular. On a face regular mesh, anyboundary face is necessarily regular with respect to the element to which itbelongs. It appears that meshes of practical interest are most likely to be faceregular, so this restriction is rather mild in practice.For integrable functions φ defined piecewise on either Th or Fh, we usethe convention∫Ωφdx =∑K∈Th∫Kφdx,∫Fhφds =∑F∈Fh∫Fφds.For the integer k ≥ 1, we define the discontinuous finite element spacesVh,k as the space of real-valued piecewise-polynomials of degree at most k onStable DGFEM without penalty parameters 3F2F3F1KF¯2F¯4F¯3F¯1F¯5Fig. 1. Face regularity of meshes: the mesh on the left has interior facesF ih = {Fi}3i=1, each of which is regular to at least one element in the senseof Definition 1, even though F2 and F3 fail to be regular with respect to theelement K, since F2 and F3 are only proper subsets of the elemental faceF2∪F3. Since all boundary faces are also regular, the mesh on the left is faceregular in the sense of Definition 1, whereas the mesh on the right is not: themesh face F¯3 fails to be regular with respect to any element of the mesh.Th, and Σh,k+1 the space of vector-valued piecewise-polynomials of degree atmost k + 1 on Th. We define the mesh-dependent norm ‖·‖1,h on Vh,k by‖vh‖21,h :=∑K∈Th‖∇vh‖2L2(K) +∑F∈Fh1hF‖JvhK‖2L2(F ) ∀ vh ∈ Vh,k, (1)where hF := diamF for each face F ∈ Fh.We shall also make use of the (local) Raviart–Thomas–Ne´de´lec space [7]defined byRTNk+1(K) := Pk(K)⊕ P˜k(K)x ⊂ Pk+1(K),where Pk(K) is the space of vector-valued polynomials of degree at mostk on K, and P˜k(K) is the space of real-valued homogeneous polynomials ofdegree k on K. We recall that τh ∈ RTNk+1(K) is uniquely determined bythe moments∫Kτh · µh dx and∫E(τh · nE) vh ds for all µh ∈ Pk−1(K) andvh ∈ Pk(E) for each E ∈ F(K), where nE denotes a unit normal vector of E.We also recall that if all facial moments of τh vanish on an elemental face E,then τh · nE vanishes identically on E.For a face F ∈ Fh belonging to an element Kext, we define the jump andaverage operators byJwK|F := w|Kext − w|Kint , {w} |F :=12(w|Kext + w|Kint), if F ∈ F ih,JwK|F := w|Kext , {w} |F := w|Kext , if F ∈ Fbh,where w is a sufficiently regular scalar or vector-valued function, and in thecase where F ∈ F ih, Kint is such that F = ∂Kext ∩ ∂Kint. Here, the labellingis chosen so that nF is outward pointing with respect to Kext and inward4 John et al.pointing with respect to Kint. Let φ ∈ L2(Fh), then the lifting operatorsrh : L2(Fh)→ Σh,k+1 and rh : L2(Fh)→ Vh,k are defined by∫Ωrh(φ) · σh dx =∫Fhφ {σh · nF } ds ∀σh ∈ Σh,k+1, (2a)∫Ωrh(φ) vh dx =∫Fihφ {vh} ds ∀ vh ∈ Vh,k. (2b)For quantities a and b, we write a . b if and only if there is a positive constantC such that a ≤ Cb, where C is independent of the quantities of interest,such as the element sizes, but possibly dependent on the shape-regularityparameters and polynomial degrees.3 Stability of lifted gradientsWe define the lifted gradient Gh : Vh,k → Σh,k+1 byGh(vh) = ∇hvh − rh(JvhK) ∀ vh ∈ Vh,k, (3)where ∇h denotes the element-wise gradient operator. We note that Gh isusually defined with a lifting using polynomial degrees k or k − 1, see forinstance [5]. However, as we shall see, by increasing the polynomial degree ofthe lifting to k + 1, we obtain the following key stability result.Theorem 2. Let {Th}h>0 denote a shape regular, contact regular and faceregular sequence of simplicial meshes on Ω. Let the norm ‖·‖1,h be definedby (1) and let the lifted gradient operator Gh be defined by (3). Then, we have‖uh‖1,h . ‖Gh(uh)‖L2(Ω) . ‖uh‖1,h ∀uh ∈ Vh,k. (4)Proof. The upper bound ‖Gh(uh)‖L2(Ω) . ‖uh‖1,h is standard and we referthe reader to [5, Sec. 4.3] for a proof. To show the lower bound, consider anarbitrary uh ∈ Vh,k. Since Gh(uh) ∈ Σh,k+1, we have‖Gh(uh)‖L2(Ω) = supτh∈Σh,k+1\{0}∫ΩGh(uh) · τh dx‖τh‖L2(Ω),with the supremum being achieved by the choice τh = Gh(uh). Therefore, toshow (4), it is sufficient to construct a τh ∈ Σh,k+1 such that‖uh‖21,h .∫ΩGh(uh) · τh dx, (5)‖τh‖L2(Ω) . ‖uh‖1,h. (6)Stable DGFEM without penalty parameters 5Let τK ∈ RTNk+1(K) be defined by∫KτK · µh dx =∫K∇uh · µh dx ∀µh ∈ Pk−1(K), (7a)∫E(τK · nE) vh ds ={−∫E1hEJuhK vh ds if E ∈ Fh,0 if E /∈ Fh,(7b)where (7b) holds for all vh ∈ Pk(E), for each element face E ∈ F(K). Inparticular, if the element face E ∈ Fh, i.e. E is also a mesh face, then werequire that nE agrees with the choice of unit normal used to define the jumpand average operators. If E /∈ Fh, then τK · nE vanishes identically on E,and the orientation of nE on the left-hand side of (7b) does not matter. Theglobal vector field τh ∈ Σh,k+1 is defined element-wise by τh|K = τK .Since the mesh Th is assumed to be face regular, for every F ∈ Fhthere exists an element K ∈ Th and an elemental face E ∈ F(K) such thatE = F ; then E satisfies the first condition in (7b). Therefore, the facts that{τh · nF } |F and JuhK|F both belong to Pk(F ) together with (7b) imply thatfor each F ∈ Fh, one of only three situations may arise:1. F is a boundary face and hence F ∈ F(K). In this case, we have{τh · nF } |F = −h−1F JuhK|F .2. F is an interior face which is regular with respect to both elements towhich it belongs. In this case, we have {τh · nF } |F = −h−1F JuhK|F .3. F is an interior face which is regular with respect to only one of the ele-ments to which it belongs. In this case, we have {τh · nF } |F = −12h−1F JuhK|F ,since τh|K′ · nF ≡ 0 for the element K′ with respect to which F is notregular.Therefore, since τh ∈ Σh,k+1, the definition of the lifting operator in (2a)implies that∫ΩGh(uh) · τh dx =∑K∈Th∫K∇uh · τh dx−∑F∈Fh∫F{τh · nF } JuhKds≥∑K∈Th‖∇uh‖2L2(K) +12∑F∈Fh1hF‖JuhK‖2L2(F )≥12‖uh‖21,h,where the second line follows from (7) and from the fact that ∇uh|K ∈Pk−1(K) for each K ∈ Th. Hence (5) is satisfied, and we now verify (6). Aclassical scaling argument using the Piola transformation [2, p. 59] yields‖τh‖L2(K) . supµh∈Pk−1(K)\{0}∫Kτh · µh dx‖µh‖L2(K)+∑E∈F(K)supvh∈Pk(E)\{0}h1/2E∫E(τh · nE)vh ds‖vh‖L2(E)∀K ∈ Th.6 John et al.K1K2K3K4Fig. 2. Counterexample of Sec. 4: the domain Ω = (−1, 1)2 and the criss-cross mesh Th considered in the example.Therefore, it follows from (7) that, for each K ∈ Th,‖τh‖2L2(K) . ‖∇uh‖2L2(K) +∑F∈F(K)∩FhhF ‖h−1F JuhK‖2L2(F ). (8)Summing (8) over all elements therefore implies (6). ⊓⊔4 Counterexample to stability for equal-order liftingsTheorem 2 shows the stability of the lifted gradient operatorGh provided thatthe lifting operator rh has polynomial degree k+1. In this section, we verifyby means of a counterexample that the stability estimate does not generallyhold for lower-order liftings, including in particular the case of equal-orderliftings, which are commonly used in practice; our example simplifies a similarcounterexample in [1].Example. Let Ω = (−1, 1)2, and consider the finite element space Vh,kdefined on a criss-cross mesh with four triangles, as depicted in Fig. 2, usingpiecewise linear polynomials, i.e. k = 1. Let uh ∈ Vh,1 be the piecewise linearfunction defined byuh|K1 = y +23, uh|K2 = x−23,uh|K3 = −y +23, uh|K4 = −x−23.Direct calculations show that {uh}|F ≡ 0 on all interior faces F ∈ Fih, andthat∫K uh dx = 0 for all elements K ∈ Th. Consequently, if the lifting opera-tor r˜h is defined in (2a) with the polynomial degree k+1 replaced by k, andif G˜h(uh) := ∇huh − r˜h(JuhK) denotes the equal-order lifted gradient, thenwe have for all τh ∈ Σh,1,∫ΩG˜h(uh) · τh dx =∑K∈Th∫K∇huh · τh dx−∑F∈Fh∫F{τh · nF } JuhKds= −∑K∈Th∫Kuh(∇h · τh) dx+∑F∈Fih∫F{uh}Jτh · nF Kds = 0.Stable DGFEM without penalty parameters 7Since G˜h(uh) ∈ Σh,1, we deduce that G˜h(uh) = 0, and thus it is found thatno bound of the form ‖uh‖1,h . ‖G˜h(uh)‖L2(Ω) is possible. ⊓⊔5 A modified LDG method without penalty parametersAs an application of Theorem 2, consider the discretization of the homoge-neous Dirichlet boundary-value problem of the Poisson equation by a mod-ified LDG method [3,4] as follows. For f ∈ L2(Ω), let u ∈ H10 (Ω) be theunique solution of∫Ω∇u · ∇v dx =∫Ωf v dx ∀ v ∈ H10 (Ω). (9)Let the bilinear form ah : Vh,k × Vh,k → R be defined byah(uh, vh) =∫ΩGh(uh) ·Gh(vh) dx ∀uh, vh ∈ Vh,k, (10)where the lifted gradient operator Gh was defined in (3). The bilinear formah(·, ·) defines a modified LDG method for (9): find uh ∈ Vh,k such thatah(uh, vh) =∫Ωf vh dx ∀vh ∈ Vh,k. (11)It follows from Theorem 2 that ah(·, ·) is uniformly stable with respect tothe norm ‖·‖1,h, and thus (11) is well-posed for each h. Moreoever, the dis-crete Poincare´ inequality [5] implies that ‖uh‖1,h . ‖f‖L2(Ω) for all h, sothat the numerical solutions uh are uniformly bounded with respect to themesh-dependent norms ‖·‖1,h. The a priori error analysis for the numericalmethod defined by (11) may be developed following the frameworks of [3,5,6],although for reasons of space we do not present the arguments here.An interesting feature of the modified LDG method (11) is that it doesnot require any additional stabilization, such as added penalty terms of theform∫FhσFhFJuhKJvhKds for some user-defined parameter σF . The absence ofsuch penalty terms enables us to show the following discrete conservationproperty. We define the lifted divergence Dh : Σh,k+1 → Vh,k byDh(σh) = divh σh − rh(Jσh · nF K), σh ∈ Σh,k+1, (12)where divh denotes the element-wise divergence operator, and where rh is thescalar lifting operator defined in (2b). We note that we have the integration-by-parts identity∫Ωσh ·Gh(vh) dx = −∫ΩDh(σh) vh dx ∀ vh ∈ Vh,k, σh ∈ Σh,k+1, (13)8 John et al.which should be compared with the analogous continuous identity betweenthe spaces H10 (Ω) and H(div, Ω). Therefore, the numerical scheme (11) canbe equivalently expressed in the strong form−∫ΩDh(Gh(uh)) vh dx =∫Ωf vh dx, (14)which implies that the numerical solution uh ∈ Vh,k solves−Dh(Gh(uh)) = Πkhf, (15)in the pointwise sense on each element K, where Πkhf denotes the element-wise L2-projection of f into Vh,k. Although we have shown here how thelifted gradient operator Gh of degree k + 1 may be used to achieve a stablediscretization of the Poisson equation, it is by no means restricted to thismodel problem, as the lifted gradients may be used to discretize the second-order terms of more general differential operators.6 ConclusionsIn this article, we studied an intrinsically stable modified LDG method with-out additional parameter dependent penalization. For this, we showed thatincreasing the degree of the lifting operator by one order leads to stability ofthe discrete gradient operator on face regular meshes with hanging nodes.AcknowledgementThe work of the third author was partially supported by the NSF grantDMS–1417980 and the Alfred Sloan Foundation.References1. F. Brezzi, M. Manzini, D. Marini, P. Pietra and A. Russo, Discontin-uous finite elements for diffusion problems, Atti del Convegno in Memoria diF. Brioschi, Milano, Istiuto Lombardo di Scienze e Lettere (1997).2. D. Boffi, F. Brezzi, M. Fortin, Mixed finite element methods and applica-tions, Springer Series in Computational Mathematics vol. 44, Springer (2013).3. P. Castillo, B. Cockburn, I. Perugia and D. Scho¨tzau, An a priori erroranalysis of the local discontinuous Galerkin method for elliptic problems, SIAMJ. Numer. Anal. 38:5 (2000), 1676–1706 (electronic).4. B. Cockburn and C.-W. Shu, The local discontinuous Galerkin methodfor time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35:6(1998), 2440–2463 (electronic).5. D. A. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkinmethods, Mathe´matiques & Applications vol. 69, Springer (2012).6. T. Gudi, A new error analysis for discontinuous finite element methods forlinear elliptic problems, Math. Comp. 79:272 (2010), 2169–2189.7. J.-C. Ne´de´lec, Mixed finite elements in R3, Numer. Math. 35:3 (1980), 315–341.

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Stable discontinuous Galerkin FEM without penalty parameters

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