International audienceWe show that a version of the Discontinuous Galerkin Method (DGM) can be included in the Gradient Discretisation Method (GDM) framework. We prove that it meets the main mathematical gradient discretisation properties on any kind of polytopal mesh, and that it is identical to the Symmetric Interior Penalty Galerkin (SIPG) method in the case of first order polynomials. A numerical study shows the effect of the numerical parameter included in the scheme

Eymard, Robert

Guichard, Cindy

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HAL Id: hal-01442922https://hal.archives-ouvertes.fr/hal-01442922Submitted on 21 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.DGM, an item of GDMRobert Eymard, Cindy GuichardTo cite this version:Robert Eymard, Cindy Guichard. DGM, an item of GDM. FVCA 2017 - International Conference onFinite Volumes for Complex Applications VIII, 2017, Lille, France. ￿hal-01442922￿DGM, an item of GDMRobert Eymard and Cindy GuichardAbstract We show that a version of the Discontinuous Galerkin Method (DGM) canbe included in the Gradient Discretisation Method (GDM) framework. We provethat it meets the main mathematical gradient discretisation properties on any kind ofpolytopal mesh, and that it is identical to the Symmetric Interior Penalty Galerkin(SIPG) method in the case of first order polynomials. A numerical study shows theeffect of the numerical parameter included in the scheme.Key words: Gradient Discretisation method, Discontinuous Galerkin methodMSC (2010): 65M08, 65N08, 35Q301 IntroductionDiscontinuous Galerkin (DG) methods are being more and more studied. Theypresent the advantage to be suited to elliptic and parabolic problems, while openingthe possibility to closely approximate weakly regular functions on general meshes.Although the convergence of DG methods has been proved on a variety of prob-lems (see [2] and references therein), note that the stabilisation of the classical DGschemes has to be specified in each case. On the other hand, convergence and errorestimate results for a wide variety of numerical methods applied to some elliptic,parabolic, coupled, linear and nonlinear problems are proved on the generic “gradi-Robert EymardUniversité Paris-Est Marne-la-Vallée, Laboratoire d’Analyse et de Mathématiques Appliquées,UPEC, UPEM, UMR8050 CNRS, F-77454, Marne-la-Vallée, Francee-mail: robert.eymard@u-pem.frCindy GuichardANGE project-team, Laboratoire Jacques-Louis Lions (LJLL), UMR 7598 CNRS, Sorbonne Uni-versites, UPMC Univ. Paris 6, and INRIA de Paris, and CEREMA, Francee-mail: guichard@ljll.math.upmc.fr12 Robert Eymard and Cindy Guichardent scheme” issued from the Gradient Discretisation Method framework, assumingthat a very small number of core properties hold true (see [3]).The aim of this paper is to show that, from the DG setting, we can build a Gradi-ent Discretisation which satisfies all these core properties. This is done on generalpolytopal meshes in any space dimension. This work immediately extends the rangeof problems which can be handled by Discontinuous Galerkin methods to all forwhich the Gradient Discretisation is shown to converge (degenerate parabolic prob-lems, two-phase flow problems,. . . ). Note that the gradient scheme resulting fromthe Discontinuous Galerkin Gradient Discretisation (DGGD) may be not identicalto the corresponding stabilised DG scheme proposed in the literature, although weshow in this paper that it is identical to the Symmetric Interior Penalty Galerkin(SIPG) method in the P1(Rd) case for the following elliptic problem:u ∈ H10 (Ω), ∀v ∈ H10 (Ω),∫ΩΛ(x)∇u(x) ·∇v(x)dx =∫Ωf (x)v(x)dx, (1)where:• Ω is an open bounded polytopal connected subset of Rd (d ∈ N?), (2a)• Λ is a measurable function from Ω to the set of d×dsymmetric matrices and there exists λ ,λ > 0 such that,for a.e. x ∈Ω , Λ(x) has eigenvalues in [λ ,λ ], (2b)• f ∈ L2(Ω). (2c)This paper is organised as follows. In Section 2, we give a gradient discretisationversion of Discontinuous Galerkin schemes. We then prove in Section 3 that thisgradient discretisation satisfies the core properties which are sufficient for conver-gence and error estimates results. A short numerical example finally shows the rolethe numerical parameter used in the design of the scheme on its accuracy (Section4).2 Discontinuous Galerkin Gradient Discretisation (DGGD)We consider a polytopal mesh of Ω , in the sense of [3, Definition 7.2], defined bythe triplet T= (M ,F ,P).The set M is a finite family of non empty connected polytopal open disjointsubsets of Ω . For K ∈M , |K|> 0 is the measure of K and hK denotes the diameterof K.The set F contains the “faces” of the mesh – “edges” in 2D. x For all σ ∈F , theset Mσ , which contains the elements of M having σ as face or edge, has exactlyone element if σ ∈Fext (the exterior faces) or two elements if σ ∈Fint (the interiorDGM, an item of GDM 3faces). For all K ∈M , FK ⊂F contains the faces of K, and for any σ ∈FK , wedenote by nK,σ the (constant) unit vector normal to σ outward to K.We denote by P the family of points P = (xK)K∈M , such that for all K ∈M ,K ∈M is strictly star-shaped with respect to xK (see Figure 1). This implies thatthe orthogonal distance dK,σ between xK and σ ∈FK is such that dK,σ > 0. For allK ∈M and σ ∈FK , we denote by DK,σ the cone with vertex xK and basis σ .The size of the polytopal mesh is defined by:hM = sup{hK ,K ∈M }. (3)Finally, for a given polytopal mesh T we define a number that measures theregularity properties of the mesh:ηT = max{hKhL+hLhK, σ ∈Fint , Mσ = {K,L}}∪{hKdK,σ,K ∈M ,σ ∈FK}. (4)dK,σ ′dK,σ ′′nK,σ ′KnK,σ ′′σ ′σ ′′σxKK(β )xKDK,σσyK,σ (x)D(β )K,σxFig. 1 A cell K of a polytopal mesh and notation on DK,σLet us now define the Discontinuous Galerkin Gradient Discretisation (DGGD)for the approximation of (1) in the sense of [5, 3]. For a given value k ∈N?, and for agiven p ∈]1,+∞[, we define the space XD ,0 of all functions v ∈ Lp(Ω) such that, forall K ∈M , v|K ∈ Pk(Rd), the latter denoting the space of polynomial function withdegree less or equal to k (recall that the dimension of Pk(Rd) is (k+d)!k!d! , and thereforethe dimension of XD ,0 is equal to(k+d)!k!d! #M ). We denote by D = (XD ,0,ΠD ,∇D ),where ΠD = Id. Let us now define ∇D . Let β ∈]0,1[ be given. For v ∈ XD ,0, forK ∈M and for any σ ∈FK , we set4 Robert Eymard and Cindy Guichard∇Dv(x) = ∇v|K(x) for a.e. x ∈ D(β )K,σ ,∇Dv(x) = ∇v|K(x)+d [v]K,σ (yK,σ (x))(1−β d)dK,σnK,σ for a.e. x ∈ DK,σ \D(β )K,σ ,(5)where (see Figure 1):•D(β )K,σ := {x ∈ DK,σ ,(x− xK) ·nK,σdK,σ< β} and K(β ) =⋃σ∈FKD(β )K,σ , (6)(note that we have |DK,σ \D(β )K,σ |=1−β dd dK,σ |σ |),• yK,σ (x)∈σ is the intersection between σ and the line joining xK and x; it satisfiesx = xK +(x− xK) ·nK,σdK,σ(yK,σ (x)− xK),• for all K ∈M , we denote by∀y ∈ σ , [v]K,σ (y) = 12 (v|L(y)− v|K(y)) if Mσ = {K,L},[v]K,σ (y) = 0− v|K(y) if Mσ = {K}.(7)Remark 1. It is possible to consider βK,σ instead of a constant β , without changingthe mathematical analysis done in this paper. It is also possible to consider the moregeneral definition for the discrete gradient∇Dv(x) = ∇v|K(x)+ψ((x− xK) ·nK,σdK,σ)[v]K,σ (yK,σ (x))dK,σnK,σ for a.e. x ∈ DK,σ ,where ψ : ]0,1[→ R+ is a bounded measurable function such that ψ(s) = 0 on]0,β [ and ∫ 11−βψ(s)sd−1ds = 1.Then the following mathematical analysis holds as well.Remark 2 (Piecewise constant reconstruction). One can for example replace ΠDby Π̂D such that, for all K ∈M , and a.e. x ∈ K, Π̂Dv(x) = 1|K|∫K v(x)dx, whichprovides a piecewise constant reconstruction, choosing a basis including the valueat the centre of gravity of K.Using the DGGD D = (XD ,0,ΠD ,∇D ), the gradient scheme for the discretisationof (1) is given by: find u ∈ XD ,0 such that∫ΩΛ(x)∇Du(x) ·∇Dv(x)dx =∫Ωf (x)ΠDv(x)dx,∀v ∈ XD ,0. (8)Owing to the properties proved in Section 3, the DGGD scheme then satisfies theconvergence and error estimates properties detailed in [3].DGM, an item of GDM 5Link with the SIPG schemeIn the case k = 1, the gradient of any element of XD ,0 restricted to K ∈M is constantin K. Let us assume that Λ follows the same property. Then the left hand side of (8)can be computed in this particular case:∫ΩΛ(x)∇Du(x) ·∇Dv(x)dx = ∑K∈M(∫KΛK∇u|K ·∇v|Kdx+ ∑σ∈FK(∫σΛK([u]K,σ (y)∇v|K +[v]K,σ (y)∇u|K)·nK,σ dγ(y)+d ΛKnK,σ ·nK,σ(1−β d)dK,σ∫σ[u]K,σ (y)[v]K,σ (y)dγ(y))).We then recover the SIPG scheme as presented in [4] or [2], the penalty coefficientτσ (term σe|e|β0 of [4, eqn. (11)], termηhFof [2, eqn. (4.12)]) being equal, in thepreceding relation, toτσ =d4(1−β d)(ΛKnK,σ ·nK,σdK,σ+ΛLnL,σ ·nL,σdL,σ)if Mσ = {K,L},andτσ =d(1−β d)ΛKnK,σ ·nK,σdK,σif Mσ = {K}.Note that τσ has a minimum value letting β → 0, which can be compared, for ex-ample, to that given by [2, Lemma 4.12]. In our setting, it does not depend on theregularity of the mesh nor on the maximum cardinal of FK (in the DGGD scheme,we don’t handle separately the case d = 1 and the cases d > 1).3 Mathematical properties of the DGGD methodIn this paper, we denote, for ξ ∈Rd by |ξ |= (∑di=1 ξ 2i )1/2 the Euclidean norm of ξ .Lemma 1. Let n ∈ N and β ∈]0,1[ be given. Let T be a polytopal mesh. Then thereholds∀v ∈ Pn(Rd), ∀K ∈M , ∀σ ∈FK ,∫DK,σ|v(x)|pdx ≤ (n+1)p−1β d+pnCp,n∫D(β )K,σ|v(x)|pdx,where Cp,n only depends on p, n and d, and where D(β )K,σ is defined by (6).Proof. This lemma is proved thanks to the change of variable x = xK + s(y− xK),where y ∈ σ and s ∈]0,β [ (we then have dx = dK,σ sd−1dγ(y)ds).6 Robert Eymard and Cindy GuichardLemma 2. There exists A > 0, only depending on β , p, k and d, such that∀v ∈ XD ,0,1A‖v‖DG,p ≤ ‖∇Dv‖Lp(Ω)d ≤ A‖v‖DG,p, (9)where‖v‖pDG,p = ∑K∈M(∫K|∇v|K(x)|pdx+ ∑σ∈FK1dp−1K,σ∫σ|[v]K,σ (y)|pdγ(y)). (10)Remark 3 (DG norm). Note that Definition (10) for the DG norm is slightly differentfrom [1, eqn. (5)] or [2, eqn. (5.1)], with the use of dK,σ instead that of diam(σ), andwith notation (7) for the jump at the faces of the mesh. This allows the applicationof discrete functional analysis results without regularity hypotheses on the polytopalmesh.Proof. We apply the inequality |a+b|p ≤ (1+ cp′)p−1(|a|p + | bc |p) with 1p +1p′ =1, a = ∇v|K(x) +d [v]K,σ (yK,σ (x))(1−β d)dK,σnK,σ and b = −∇v|K(x), for some c > 0 chosenaccounting for Lemma 1, applied to the components of ∇v|K , which are polynomialtoo.From an adaptation of the discrete functional analysis results provided in [1] toour polytopal mesh framework, we conclude on one hand that ‖∇D · ‖Lp(Ω)d is anorm on XD ,0, and on the other hand the two following lemmas.Lemma 3 (coercivity). Let D be a DGGD. We define CD ≥ 0 byCD = maxv∈XD ,0\{0}‖ΠDv‖Lp(Ω)‖∇Dv‖Lp(Ω)d. (11)Then there exists CP only depending on |Ω |, β , p, k and d such that CP ≥CD , whichmeans that any sequence (D)m∈N is coercive in the sense of [3, Definition 2.2].Lemma 4 (compactness). Let (D)m∈N be a sequence of DGGD. Then, for all(vm)m∈N such that, for all m ∈ N, vm ∈ XDm,0 and such that the sequence(‖∇Dmvm‖Lp(Ω))m∈N is bounded, the sequence (vm)m∈N is relatively compact inLp(Ω), which means that any sequence (D)m∈N is compact in the sense of [3, Defi-nition 2.8].Lemma 5 (GD-consistency). Let (D)m∈N be a sequence of DGGD such that hTmtends to 0 as m→∞ while ηTm remains bounded. We define SD :W1,p0 (Ω)→ [0,+∞)by∀ϕ ∈W 1,p0 (Ω), SD (ϕ) = minv∈XD ,0(‖ΠDv−ϕ‖Lp(Ω)+‖∇Dv−∇ϕ‖Lp(Ω)d). (12)Then it holds ∀ϕ ∈C∞c (Ω), limm→∞ SDm(ϕ) = 0,which is a sufficient condition for the GD-consistency of (D)m∈N in the sense of [3,Definition 2.4] thanks to [3, Lemma 2.13].DGM, an item of GDM 7Proof. Let ϕ ∈C∞c (Ω), and let M be an upper bound of D2ϕ := (∂ 2i jϕ)1≤i, j≤d onΩ . We let D = Dm for a given m, and we consider v ∈ XD ,0 defined by∀K ∈M , ∀x ∈ K, v|K(x) = ϕ(xK)+∇ϕ(xK) · (x− xK).Indeed, v|K ∈ Pk(Rd) since k≥ 1. We then perform Taylor expansions at the secondorder of the function ϕ , which allow to conclude the existence of C≥ 0, increasinglydepending on ηT, such that‖∇ϕ−∇Dv‖Lp(Ω)d ≤CMhM (|Ω |)1/p.Lemma 6 (limit conformity). Let (D)m∈N be a sequence of DGGD such that hTmtends to 0 as m→ ∞. Let p′ = pp−1 and define WD : Wp′div(Ω)→ [0,+∞) by∀ϕ ∈W p′div(Ω) ,WD (ϕ ) = maxv∈XD ,0\{0}∣∣∣∣∫Ω(∇Dv(x) ·ϕ (x)+ΠDv(x)divϕ (x))dx∣∣∣∣‖∇Dv‖Lp(Ω)d.(13)Then it holds ∀ϕ ∈C∞(Rd)d , limm→∞WDm(ϕ ) = 0,which is a sufficient condition for the limit conformity of (D)m∈N in the sense of [3,Definition 2.6] thanks to [3, Lemma 2.14] since (D)m∈N is coercive.Proof. The proof relies on the coefficient of [v]K,σ , which ensures that the terms atthe faces of the mesh behave as ChM .4 Numerical resultsThe aim of this section is to assess the influence of the parameter β ∈]0,1[ on theaccuracy of the gradient scheme (8) issued from the DGGD for the discretisationof (1). We consider the 1D case Ω =]0,1[, and the polytopal mesh T defined, forN ∈N? and h = 1N , by M = {](i−1)h, ih[, i = 1, . . . ,N}, F = {{ih}, i = 0, . . . ,N},P = {(i− 12 )h, i = 1, . . . ,N}. We consider one of the test cases studied in [4], thatis Problem (1) with Λ = Id and u(x) = cos(8πx)−1 (hence f (x) = (8π)2 cos(8πx)).Considering first degree polynomials, the set XD ,0 is a vector space with dimension2N. In the following tables (where “order” is the convergence order with respectto the size of the mesh), the columns “FE” correspond to the conforming P1 FiniteElement solution, and we check that the results provided by “[4]” with σn = 4.5,which corresponds to β = 1−1/σn for the interior faces, and β = 1−2/σn for theexterior faces, are close to ours:8 Robert Eymard and Cindy GuichardN\β 0 0.5 0.9 0.99 FE [4]10 0.496 0.241 0.347 0.394 0.399 0.247order 1.438 1.529 1.734 1.843 1.85520 0.183 0.083 0.104 0.110 0.110 0.083order 1.092 1.706 1.909 1.959 1.96440 0.086 0.026 0.028 0.028 0.028 0.024order 1.013 1.894 1.973 1.989 1.99180 0.043 0.007 0.007 0.007 0.007order 0.999 1.967 1.992 1.997 1.998160 0.021 0.002 0.002 0.002 0.002N\β 0 0.5 0.9 0.99 FE [4]10 13.233 11.533 11.360 11.349 11.348 11.777order 0.172 0.781 0.862 0.863 0.86320 11.743 6.714 6.251 6.240 6.240 6.421order 0.010 1.004 0.966 0.965 0.96540 11.666 3.348 3.199 3.197 3.197 3.253order -0.008 1.034 0.992 0.991 0.99180 11.728 1.635 1.609 1.608 1.608order -0.007 1.014 0.998 0.998 0.998160 11.781 0.810 0.805 0.805 0.805L2 error of the solution L2 error of the broken gradientAlthough we did not prove that the linear systems are invertible when β = 0, wenote that in practice a solution is obtained but that the broken gradient does notseem to converge. In this very regular case, the L2 error is the lowest for β = 0.5 butthe convergence seems slightly better for β closer to 1, and it tends to the results ofthe finite element method as β → 1.5 ConclusionThe version of the DG method included in the GDM framework has the advantagesto be samely defined for d = 1 and d > 1, to hold on any polytopal mesh providedthat the grid block are strictly star-shaped, to involve Discrete Functional Analysisresults which do not depend on the regularity of the mesh, and to apply on anyproblem on which the GDM is shown to converge. This version is identical to theSIPG method in the case k = 1. The differences with the SIPG scheme in the casek > 1 remain to be assessed.References1. Di Pietro, D.A., Ern, A.: Discrete functional analysis tools for discontinuous Galerkin meth-ods with application to the incompressible Navier-Stokes equations. Math. Comp. 79(271),1303–1330 (2010). DOI 10.1090/S0025-5718-10-02333-1. URL https://dx-doi-org.fennec.u-pem.fr/10.1090/S0025-5718-10-02333-12. Di Pietro, D.A., Ern, A.: Mathematical aspects of discontinuous Galerkin methods,Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69. Springer,Heidelberg (2012). DOI 10.1007/978-3-642-22980-0. URL https://dx-doi-org.fennec.u-pem.fr/10.1007/978-3-642-22980-03. Droniou, J., Eymard, R., Gallouët, T., Guichard, C., Herbin, R.: The gradient discretisationmethod (2016). URL https://hal.archives-ouvertes.fr/hal-01382358. Working paper or preprint4. Epshteyn, Y., Rivière, B.: Estimation of penalty parameters for symmetric interior penaltyGalerkin methods. J. Comput. Appl. Math. 206(2), 843–872 (2007)5. Eymard, R., Guichard, C., Herbin, R.: Small-stencil 3d schemes for diffusive flows in porousmedia. M2AN 46, 265–290 (2012)

DGM, an item of GDM

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DGM, an item of GDM

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