Analysis of a Multiscale Discontinuous Galerkin Method for Convection Diffusion Problems

Transcription

1 Analysis of a Multiscale Discontinuous Galerkin Method for Convection Diffusion Problems A. Buffa,.J.R. Hughes, G. Sangalli Istituto di Matematica Applicata e ecnologie Informatiche del C.N.R. Via Ferrata 1, 7100 Pavia, Italy he University of exas at Austin Institute for Computational Engineering and Sciences 1 University Station C000 Austin, X , U.S.A. Abstract We study a multiscale discontinuous Galerkin method introduced in [10] that reduces the computational complexity of the discontinuous Galerkin method, seemingly without adversely affecting the quality of results. For a stabilized variant we are able to obtain the same error estimates for the advection-diffusion equation as for the usual discontinuous Galerkin method. We assess the stability of the unstabilized case numerically and find that the inf-sup constant is positive, bounded uniformly away from zero, and very similar to that for the usual discontinuous Galerkin method. 1 Introduction he discontinuous Galerkin method has undergone rapid development in recent years (see, e.g., [6] and [5]). Although it has been shown to possess advantageous properties in a number of circumstances, its practical utility has been limited by the much larger number of degreesof-freedom it requires compared with continuous Galerkin methods [8]. his problem has persisted since the inception of the method and has only been recently addressed with the development of a multiscale discontinuous Galerkin method [10] that has the computational structure of a continuous method. he new method utilizes local, element-wise problems to develop a transformation between the parameterization of the discontinuous space and a related, smaller, continuous space. he transformation enables a direct construction of the global matrix problem in terms of the degrees-of-freedom of the continuous space. In the multiscale interpretation, the continuous field is viewed as the coarse scales and the discontinuous field is viewed as the sum of the coarse and fine scales. he discontinuous 1

2 part of the solution can be determined by element-wise post-processing of the continuous solution. In [10] it was shown numerically that the new method at least retains the quality of the discontinuous Galerkin method, and in some instances improves upon it, while at the same time it has the potential to significantly reduce computational cost. A more general framework encompassing the ideas is presented in [3]. In this paper we initiate the mathematical analysis of the method developed in [10]. In Section we present the boundary-value problem under consideration, namely, advectiondiffusion, and give general definitions necessary for subsequent developments. In Section 3 we introduce a discontinuous Galerkin (DG) method that employs interior penalty stabilization and allows for symmetric, neutral, and skew-symmetric treatment of element interface terms corresponding to the diffusion operator. We also introduce a stabilized variant (SDG) that accounts for control of the streamline derivative on element interiors. he DG method is shown to be coercive with respect to the norm induced by its bilinear form, referred to as the DG-norm, and, likewise, the SDG method is shown to be coercive with respect to the SDG-norm induced by its bilinear form. However, the DG-norm is weak in that, in the advective limit, it only controls jumps on element interfaces. In [7], convergence of the DG method in the DG-norm was proved by utilizing the L -interpolant, circumventing the need for a stronger stability condition. Here we prove that the DG method is inf-sup stable with respect to the SDG-norm and this enables us to prove its convergence in the SDG norm by standard means. In Section 4 we present the multiscale generalizations of DG and SDG, referred to as MDG and SMDG, respectively. We define the local, element-wise problems, which amount to the DG method on individual elements with weakly-enforced boundary conditions specified by the shared degrees-of-freedom of the continuous representation, and we define the interscale transfer spaces which emanate from the solutions of the local problems. We prove the infsup stability of the local problems in term of the SDG-norm, without streamline-derivative stabilization in the local problems. We also establish the approximation properties of the interscale transfer spaces. With these, and the fact that SDG is coercive on the discontinuous space, we are able to prove convergence and establish the same error estimates for SMDG as SDG (and DG). However, the proof for MDG poses an additional obstacle, namely, DG is inf-sup stable with respect to the SDG-norm on the entire discontinuous space but not necessarily inf-sup stable on the interscale transfer subspace. his problem remains open. However, a numerical assessment of the situation is made in Section 5 where the inf-sup constant is calculated for a class of boundary-value problems over a broad range of advection and diffusion parameters, and meshes. For the cases considered, we find that the MDG method is inf-sup stable with respect to the SDG-norm, and the values of the inf-sup constant are very similar to those for the DG method. hese results are consistent with the numerical evaluations performed in [10]. We also assess the stability behavior of the methods in terms of the interior penalty parameter and confirm that MDG behaves in similar fashion to DG. Results for SMDG are analogous to those for MDG and thus are omitted for brevity. Conclusions are drawn in Section 6.

3 Preliminaries.1 Problem description Let Ω be a bounded polygonal domain in R n d. he strong form of the boundary value problem we are interested in is the following: κ φ + a φ = f in Ω φ = g on Γ where κ 0 is the diffusion coefficient, a is the solenoidal velocity vector field defined on Ω and Γ = Ω is the boundary on which Dirichlet conditions are imposed. More general boundary conditions may be considered as well, see [9] and [10]. We assume that the values of the diffusion coefficient κ and the velocity field a ensure wellposedness of (1). Additional assumptions on these coefficients will be set later.. General definitions We introduce the following partition of the boundary: Γ = {x Γ : a(x) n(x) 0} () Γ + = {x Γ : a(x) n(x) > 0} (3) where n is the outward unit normal with respect to Γ. Γ will be referred to as the inflow boundary and Γ + as the outflow boundaries. Let { h } h be a family of partitions of Ω into elements. Each h is assumed to be admissible (i.e., non-overlapping elements, their union reproduces the domain, etc.), and shape regular (i.e., the elements verify a minimum angle condition, uniformly with respect to h). he elements h are either triangles/quadrilaterals in two dimensions or tetrahedra/hexaedra in three dimensions. Let h denote the diameter of and h = max h h. We denote by E h the set of all edges of h (including edges on the boundary Γ) and by Eh o the set of internal edges (excluding edges on the boundary Γ) and, by abuse of notation, we denote by Γ both the boundary Ω and the collection of edges lying on it. We also define a partition of the element boundary : Γ = {x : a(x) n(x) 0} (4) Γ + = {x : a(x) n(x) > 0} (5) Here Γ represent the element inflow/outflow boundary, respectively, so that = Γ = Γ + Γ. In order to derive a discontinuous Galerkin formulation, following [1], jumps and averages for scalar and vector fields have to be defined on the edges in E h. herefore, consider an interior edge e Eh o, and denote by + and the downwind and upwind elements that share it, respectively, and by n + and n their respective outward-pointing unit normals. Given a scalar field ν, possibly discontinuous across e, we set ν ± = ν ± on e and define ν = 1 (ν+ + ν ) [ν ] = ν + n + + ν n. (6) 3 (1)

4 Analogously, for a vector field τ we set τ ± = τ ± on e and define τ = 1 (τ + + τ ) [τ ] = τ + n + + τ n. (7) he previous definitions are specialized on the edges on Γ as: ν = ν, [ν ] = ν n, τ = τ, e Γ. (8) We will extensively make use of the following biased identity (based on [1, formula (3.3)]): τ n ν = ( ) ν ± [τ ] + [ν ] τ h e Eh o e e + (9) ν τ n. e Γ e In what follows, C is a constant, possibly different at each occurrence, which is independent of h and of the coefficients κ and a. Moreover, α β means α Cβ, while α β means α β and β α. We suppose that κ and a are constant on each element h. We make use of the following notation: κ = κ, a = a and a = a. Finally, we assume that for any pair of elements + and sharing an edge, 3 he Discontinuous Galerkin method 3.1 Method description κ + κ. (10) Given a positive index k, the following approximation space is introduced: V h = {v L (Ω) : v P k ( ), h } (11) where P k ( ) is the space of polynomials of degree at most k supported on. A possible DG (discontinuous Galerkin) formulation for (1) is: find φ DG V h such that B DG (φ DG, µ) = L DG (g, f; µ) µ V h, (1) where B DG (ν, µ) = µ (aν κ ν) h ( [µ] (aν κ ν ) + sκ µ [ν ] ) + e Eh o e + sκ µ nν + κ ν nµ e Γ e + µνa n + ε κ [µ] [ν ], h e Γ + e E h e 4 e

5 and L DG (g, f; µ) = µf + ( ) κ ε µg + sκ µ ng Ω e Γ e h e a nµg; e Γ e s is either 1, 0, or 1, (corresponding to symmetric, neutral and skew-symmetric interior penalty methods) and for each e Eh o, we set h = + +, while for e Γ we set e h = e. Remark 3.1 Notice that on each internal edge e Eh o the normal component of the velocity field a is continuous, owing to the assumption div(a) = 0. It will be useful to write the bilinear form B DG (, ) as a sum of two contributions: the diffusive part, and the convective part, B DG (ν, µ) = BD DG (ν, µ) + BC DG (ν, µ), (13) where B DG D (ν, µ) = h B DG C (ν, µ) = h µ κ ν [µ] κ ν + sκ µ [ν ] (14) e Eh o e + sκ µ nν + κ ν nµ + ε κ [µ] [ν ] e Γ e e E e h h [µ] aν + µνa n. (15) e Γ + µ aν + e E o h e e We also define the DG-norm ν DG = ν D + ν C, (16) where ν D = h (κ ν H 1 ( ) + h κ ν H ( ) ) ( ) + ε e Eh h 1 κ [ν ] L (e), ν C = e E h a n 1/ [ν ] L (e). (17) he DG formulation is consistent: let φ be the solution of (1), then it is easy to verify that B DG (φ, µ) = L DG (g, f; µ) µ V h. As far as the stability is concerned, we first recall that the form B DG (, ) is coercive with respect to the DG-norm, as stated in the next proposition. 5

6 Proposition 3. For each value of s, there exists positive ε such that, for all ε > ε, there exists α DG > 0 such that B DG (µ, µ) α DG µ DG, for all µ V h. Moreover, α DG is independent of the mesh-size h, and the coefficients κ and a. Proof: he coercivity of the convection term easily follow by integration by parts: B DG C (µ, µ) 1 µ C. Moreover, analogously to the stability proof provided in [1], there exists ε, such that, under the assumption ε > ε, the coercivity of the diffusive term holds, that is, B DG D (µ, µ) α DG µ D. (18) Actually, when s = 1 (skew-symmetric case) the result holds for any ε > 0. he coercivity as given in Proposition 3. is enough to provide an estimate of the form φ φ DG DG C [ (a ) ] h k+1 + κ h k φ H k+1 ( ), (19) h which can be obtained reasoning as in [7], for example. On the other hand, if the convection dominates and the exact solution φ is smooth, the quantity φ φ DG DG is basically a measure of the jumps of the discrete solution. In this case the estimate (19) gives very little information on the error φ φ DG. In order to improve the control of the error, we can add an SUPG like stabilization to the DG formulation (as it was first done in [11] for linear hyperbolic problems). hen, we set B SDG (ν, µ) = B DG (ν, µ) + τ (L ν)(a µ), (0) h L SDG (g, f; µ) = L DG (g, f; µ) + h τ where L ν = κ ν + a ν on and τ of the error analysis, the required asymptotic behavior of τ in the convectiondominated regime (i.e., when (i.e., when h a κ 1). We simply set κ h a 1) and τ h κ f(a µ), (1) is a stabilization parameter. For the purpose is τ h a in the diffusion-dominated regime { } h τ = τ min, h, () a κ where τ is a positive real number at our disposal. he SDG (stabilized discontinuous Galerkin) formulation reads: find φ SDG V h such that B SDG (φ SDG, µ) = L SDG (g, f; µ) µ V h. (3) 6

7 For the theoretical analysis of the SDG scheme (3), we will need the following SDG-norm and the related ν SDG = ν DG + h τ a ν L ( ), (4) ν SDG = ν SDG + e E 0 h a n 1/ ν L (e) + h τ 1 ν L ( ). (5) It is immediate that the SDG formulation is consistent. Moreover, the problem (3) admits a unique solution under suitable assumptions, as a consequence of the following known result. Proposition 3.3 For each value of s, there exist positive τ and ε such that, for all τ < τ and ε > ε, there exists α SDG > 0 such that B SDG (µ, µ) α SDG µ SDG, µ V h, (6) where α SDG is independent of the mesh-size h, and the coefficients κ and a. Moreover B SDG (ν, µ) ν SDG µ SDG, ν V h + H 1 (Ω), µ V h. (7) Proof: We first note that, due to (), τ κ µ L ( ) τκ h µ H ( ) µ D. (8) h h hanks to Proposition 3., when ε is greater than a suitable ε we have B SDG (µ, µ) α DG µ SDG τ (κ µ)(a µ). h By the Cauchy-Schwarz inequality and (8), (6) is proved by choosing τ sufficiently small. In order to prove (7), we proceed in a standard way as follows. B SDG (ν, µ) = B DG D (ν, µ) + BC DG (ν, µ) + τ (a µ)(l ν) = I + II + III. h We estimate the three terms separately. First, reasoning similarly to [1], I = B DG D (ν, µ) ν D µ D. (9) Second, by the Cauchy-Schwarz inequality: II = a µ ν + [µ] aν + µνa n h e Eh o e e Γ + e µ SDG a n 1/ ν L (e) + τ 1 ν L ( ) e Eh 0 Γ+ h µ SDG ν SDG. hird, again by the Cauchy-Schwarz inequality, and (8): III = τ (a µ)( κ ν + a ν) µ SDG ν SDG. (31) h 7 1/ (30)

8 3. Error estimate We first provide an error estimate for the SDG method (3). Proposition 3.4 Let φ be the solution of (1), and assume φ H k+1 (Ω). Let φ SDG be given by (3). Under the assumption of Proposition 3.3, the following error estimate holds: ( ( φ φ SDG SDG a h k+1 h 1/ ) + κ h k φ H k+1 ( )). (3) Proof: Let φ I V h be the usual nodal interpolant of φ. Using coercivity and continuity, (6) and (7), together with consistency, we get α SDG φ SDG φ I SDG B SDG (φ SDG φ I, φ SDG φ I ) = B SDG (φ φ I, φ SDG φ I ) φ φ I SDG φ SDG φ I SDG. (33) For the usual local estimates on the interpolation error φ φ I we readily obtain ( 1/ ( ) φ φ I SDG κ h k + τ a h k + τ 1 hk+ + τ κ h k φ H k+1 ( )). (34) h When choosing the stabilization parameter τ see that (3) follows. according to (), by direct comparison, we For the pure discontinuous Galerkin method (1), a suitable control on the streamline derivative can be obtained, as was first studied in [1] for the pure convection (scalar hyperbolic) equation. In the following result, we prove an inf-sup condition for the bilinear form B DG (, ) with respect to the SDG-norm. his improves the stability result stated in Proposition 3.. heorem 3.5 here exists ε such that for all ε ε, where β DG is independent of h, κ, a, and the domain. B DG (ν, µ) inf sup β DG > 0; (35) ν V h µ V h ν SDG µ SDG Proof: Given ν V h, we choose µ = ν + γ h τ (a ν) = ν + γµ where γ is a positive parameter at our disposal. Note that µ V h, as the velocity field is piecewise constant on h. We prove the following: µ SDG ν SDG, (36) B(ν, µ) β ν SDG. (37) We start by proving (36). o this end, we need to estimate the different terms of µ SDG. Recall that, from (), τ τ h κ, (38) 8

9 and Using (39), we have τ τ h a. (39) τ a (τ a ν) L ( ) τ 3 a (a ν) L ( ) (τc inv) τ a ν L ( ), (40) where C inv is the constant of the local inverse inequality, giving h τ a µ L ( ) ν SDG. Consider an internal edge e Eh o, and denote by and + the adjacent upwind and downwind elements. We have [µ ] L (e) µ L (e) + µ + L (e) τ (a ν) L (e) + τ + (a ν) + L (e). (41) Using the trace inequality, ξ L (e) ξ L ( ) ξ L ( ), which holds for all ξ H 1 ( ), and then using the inverse inequality, and we also have From (41) and (4) we obtain (a ν) ± L (e) C invh 1 ± a ν L ( ± ). (4) a n 1/ [µ ] L (e) τ 1/ + a ν L ( + ) + τ 1/ a ν L ( ). Similarly, for a boundary edge e Γ, if e then Summarizing, we have proved a n 1/ [µ ] L (e) τ 1/ a ν L ( ). µ C h τ a ν L ( ). (43) By the inverse inequality, as in (40), we have κ µ L ( ) κ a τ ν H ( ) C invκ ν L ( ). (44) On the other hand, recalling (10), (41) (4) implies that, for each e E o h : κ h [µ ] L (e) κ + ν L ( + ) + κ ν L ( ), (45) or, for e Γ, κ h [µ ] L (e) κ ν L ( ). (46) 9

10 his proves that µ D C D ν D (47) where C D is a constant indenpendent of the mesh size and the problem parameters. We turn now to the proof of (37). First of all, we have BC DG (ν, ν) = 1 a n 1/ [ν ] L (e) e E h On the other hand, by integration by parts, BC DG (ν, µ ) = ( ) τ a ν τ (a ν) + [aν ], (48) h and, using (4), h τ (a ν) + [aν ] h τ a n 1/ (a ν) + L ( ) a n 1/ [ν ] L ( ) Using these estimates, we have: ( BC DG (ν, µ) 1 γ λ ) a n 1/ [ν ] L (e) + γ e E h 1 τ a n 1/ (a ν) + L λ ( ) h + λ a n 1/ [ν ] L ( ) h C C inv τ a ν L λ ( ) + λ a n 1/ [ν ] L (e). h e E h ( 1 C C ) inv τ a ν L λ ( ). (49) h For the estimation of the diffusion part BD DG (ν, µ), we use coercivity (18), continuity (e.g., see (9)) of BD DG (, ) and the estimate (47), to obtain: BD DG (ν, µ) = BD DG (ν, ν) + γbd DG (ν, µ ) β 1 ν D γ β µ D ν D (β 1 γc D β ) ν D. (50) Summing equations (49) and (50), and setting β = C D β we obtain: B DG (ν, µ) (β 1 γβ ) ν D ( + 1 γ λ ) ( a n 1/ [ν ] L (e) + γ 1 C C ) inv τ a ν L λ ( ). e E h h { he theorem is then proved by choosing λ = CC inv and γ = min λ 1, β } 1. β From heorem 3.5 we deduce the following error estimate for the DG scheme. 10

11 Corollary 3.6 Let φ be the solution of (1), and assume φ H k+1 (Ω); let φ DG be the solution of (1). We have ( 1/ ( ) φ φ DG SDG a h k+1 + κ h k φ H k+1 ( )). (51) h Proof: Let φ I V h be the nodal interpolant of φ and let ζ = φ DG φ I. Let µ V h be the test function provided by (36) (37). Using consistency and Proposition 3.3, β ζ SDG B DG (ζ, µ) = B DG (φ φ I, µ) We deduce (51) by the triangle inequality. φ φ I SDG µ SDG ( 1/ ( ) a h k+1 + κ h k φ H k+1 ( )) ζ SDG. h 4 he Multiscale Discontinuous Galerkin method In this section, we present a reduction technique, referred to as the MDG (multiscale discontinuous Galerkin) method, which was first introduced in [10]. Furthermore, a stabilized variant of this method, referred as SMDG (stabilized multiscale discontinuous Galerkin), will be introduced subsequently. he main idea is the following: (i) Solve (1) or (3) on a suitable subspace of V h preserving the stability and approximation properties, (ii) Use a multiscale paradigm and local problems to perform the elimination of degrees-of-freedom for both the test and trial spaces. 4.1 Method description We introduce the space V h = V h H 1 (Ω). he local problems read: ν V h, find ν V h such that for all h, b (ν, µ) = l ( ν, f; µ), (5) where we have set: b (ν, µ) = κ ν µ (κ ν nµ sκ µ nν) + ε Γ µ aν + (1 + δ) µνa n, l ( ν, f; µ) = Γ Γ + µ νa n + δ Γ + + sκ µ n ν + Γ 11 µ νa n + ε Γ f µ. κ h µ ν Γ κ h µν (53)

12 Observe that (5) is a DG formulation for the local problem L ν = f on, with ν = ν on the boundary. Comparing the local DG formulation (5) with the global DG formulation (1), notice that the former has an extra term, which depends on a new parameter δ > 0. his new term is needed for implementation purposes (see [10]). We denote by h : V h L (Ω) V h the operator which associates to each ( ν, f) V h L (Ω) the solution ν of the local problems (5) on each element h. he stability of (5), which is stated below (in Proposition 4.4), implies that the problems (5) admit unique solutions on each element h, that is, the operator h is well defined. h represents the interscale transfer operator, and the associated interscale transfer spaces are the (affine) manifold and the (linear) manifold h (V h, f) = { h ( ν, f) ν V h }, h (V h, 0) = { h ( ν, 0) ν V h }. With this notation, the MDG method reads: find φ MDG h (V h, f) such that: B DG (φ MDG, µ) = L DG (g, f; µ) for all µ h (V h, 0). (54) Its stabilized version SMDG reads: find φ SMDG h (V h, f) such that: B SDG (φ SMDG, µ) = L SDG (g, f; µ) for all µ h (V h, 0). (55) Notice that SMDG is an SUPG stabilization of MDG. Remark 4.1 he spaces h (V h, f) and h (V h, 0) can be parameterized by means of the degrees-of-freedom of V h lying on the skeleton Σ = e Eh e. Remark 4. he MDG method can be interpreted as a multiscale technique. Both trial and test discontinuous functions ν V h can be split into a continuous coarse scale ν plus a discontinuous fine scale ν = ν ν. Performing integration by parts in (5), we find that ν satisfies b (ν, µ) = (f L ν)µ, µ V h ( ). (56) Equation (56) suggests a relationship between the MDG approach and the RFB ( Residual- Free Bubble) approach (see, e.g., [4]). Consider, for the sake of simplicity, the case of lowest order approximation k = 1. Actually ν in (56) can be understood as the DG approximation of the exact residual-free bubble ν bubble, which satisfies L ν bubble = f L ν on, with ν bubble = 0 on the boundary. A DG approximation of the exact residual-free bubble has been used also in the DB ( discontinuous bubble) implementation of the RFB formulation (see [13]). he major difference between MDG and DB is that for the latter the space of test functions was V h instead of h (V h, 0). he relation between those two approaches deserves further investigation. 1

13 4. Approximation properties of h (V h, f) he first step in the analysis of problems (54) and (55) is the study of the approximation properties of the interscale transfer affine space h (V h, f). heorem 4.3 (Approximation) Let φ be the solution of (1); then there exists ν h (V h, f) such that ( 1/ ( ) φ ν SDG a h k+1 + κ h k φ H k+1 ( )). (57) h Before proving heorem 4.3, we need some lemmas. On each element h, we introduce the following local norm: ν SDG( ) := κ ν H 1 ( ) + h κ ν H ( ) + τ a ν L ( ) + εh 1 κ ν L ( ) + a n 1/ ν L ( ). (58) In what follows, we set V h ( ) = V h (note that this is nothing other than the space of degree k polynomials on ). he first lemma states that the local problems (5) are stable. Lemma 4.4 (Local Stability) here exists positive ε and δ such that for all ε ε and δ δ, b (ν, µ) inf sup β b > 0, h, (59) ν V h ( ) ν SDG( ) µ SDG( ) µ V h ( ) and the constant β b is independent of, κ and a. Proof: If δ = 0, then (59) is a particular case of (35) where the domain is, (endowed with a one-element mesh) instead of Ω. hen, for δ = 0, given ν V h ( ) there exists µ V h ( ) such that µ SDG( ) ν SDG( ), b (ν, µ) β DG ν (60) SDG( ). If δ 0, given ν V h ( ) and for the same µ V h ( ) of in (60), we have µ SDG( ) ν SDG( ), b (ν, µ) β DG ν SDG( ) + Γ + δµνa n. (61) Moreover Γ + δµνa n δ µ SDG( ) ν SDG( ). hen, for δ δ = β DG /, from (61) we get which gives (59), for β b = β DG /. µ SDG( ) ν SDG( ), b (ν, µ) β DG ν SDG( ), (6) 13

14 he local problems are consistent: let φ be the solution of (1), then b (φ, µ) = l (φ Γ, f, µ) µ V h, h. (63) In the following lemma we state a Poincaré-like estimate for the norm SDG( ). Lemma 4.5 For each element h, and each function ν H 1 ( ) the following estimate holds: τ 1 ν L ( ) ν SDG( ). (64) Proof: Fix an element h. Because of the definition () of τ, (64) is a consequence of the two Poincaré estimates a ν L h ( ) h a ν L a ( ) + a n 1/ ν L ( ), (65) κ h ν L ( ) κ ν H 1 ( ) + h 1 κ1/ ν L ( ). (66) he inequality (66) is a consequence of the standard Poincaré inequality plus a scaling argument. herefore, we concentrate on the less common (65). Let η be the solution of the problem: a η = 1 on, and η Γ = 0. It is easy to verify that η L ( ) h. Given v H 1 ( ), we estimate L a ( ) as follows: ν L ( ) = ν a η = a (ν ) η + a n η ν η L ( )( a ν L ( ) ν L ( ) + a n 1/ ν L ( ) ) h a ( a ν L ( ) ν L ( ) + a n 1/ ν L ( ) ) h a a ν L ( ) + 1 ν L ( ) + h a a n 1/ ν L ( ). Γ + he inequality (65) follows, dividing both sides by h a. Proof of heorem 4.3. Let φ I V h be the nodal interpolant of φ and let ν be the solution of the following local problems: b (ν, µ) = l (φ I Γ, f, µ) µ V h, h. We have ν h (V h, f) and we will show that ν verifies the estimate (57). First, we prove that φ ν SDG h φ ν SDG( ). (67) 14

15 It is immediate that φ ν SDG + e E 0 h Γ+ a n 1/ (φ ν) L (e) h φ ν SDG( ), (68) and, making use of (64), we also have h τ 1 ν L ( ) h φ ν SDG( ). (69) herefore, from (67) and the usual triangle inequality, we get φ ν SDG h φ φ I SDG( ) + h φ I ν SDG( ) = I + II. Let us concentrate on II first. Fix a generic h. Consistency (63) implies: b (φ ν, µ) = l (φ φ I, 0; µ) µ V h ( ). (70) By Lemma 4.4, there exists µ V h ( ) such that µ SDG( ) φ I ν SDG( ) and We have φ I ν SDG( ) b (φ I ν, µ) = b (φ I φ, µ) + b (φ ν, µ) = b (φ I φ, µ) + l (φ φ I, 0; µ). b (φ I φ, µ) ( φ I φ SDG( ) + τ 1 φi φ L ( )) µ SDG( ), l (φ φ I, µ) φ I φ SDG( ) µ SDG( ). hanks to (71) (7), and the Poincaré estimate (64), we obtain (71) (7) φ I ν SDG( ) φ I φ SDG( ) + τ 1 φi φ L ( ) φ I φ SDG( ). Squaring and summing over all the elements, we end up with II I. Finally, observe that, by using the standard estimates for the interpolation error, we easily get ( 1/ ( ) I a h k+1 + κ h k φ H k+1 ( )). h his gives (57). 15

16 4.3 Error estimate An optimal error estimate for the SMDG method readily follows from heorem 4.3 and Proposition 3.3: heorem 4.6 Let φ and φ SMDG be the solutions of (1) and (55) respectively. Under the same assumption of Proposition 3.3, ( ( φ φ SMDG SDG a h k+1 h 1/ ) + κ h k φ H k+1 ( )). (73) Proof: Let ν h (V h, f) be the approximant of φ given by heorem 4.3, and ζ = φ SMDG ν. Linearity ensures that ζ h (V h, 0), that is, it is an admissible test function for (55). Repeating the same steps as in Proposition 3.4, we obtain the estimate: φ φ SMDG SDG φ ν SDG he statement is then proved by using heorem 4.3. Remark 4.7 he problem of providing an optimal error estimate for MDG remains open. he error estimate (19) for DG, proved in [7], makes use of an interpolant which is the L -projection of φ onto V h, which is not generally available in h (V h, f). On the other hand, the stronger error estimate (51) we have proved in Section 3, still for DG, relies on the validity of the inf-sup condition (35). A similar error analysis for the MDG method would need the following inf-sup condition: inf sup ν h (V h,0) µ h (V h,0) B DG (ν, µ) ν SDG µ SDG β MDG > 0; (74) which is not a consequence of (35). One of the objectives of the next section is the numerical evaluation of the inf-sup constant β MDG in (74). 5 Selection of parameters he stability of the numerical schemes we have considered depends on the parameters ε (which specifies the amount of interior penalty stabilization, in all the formulations) and τ (which specifies the amount of streamline stabilization, in SDG and SMDG). In this section, we want to investigate more in detail the relation between the stability of the schemes and the value of the parameters for a specific model problem. Moreover, we investigate numerically the validity of (74) and we demonstrate that (74) holds, at least for the cases covered by our numerical experiments. We consider a square domain Ω = [0, 1], and a uniform partition h of N N square elements. hen, we select bilinear finite element spaces, discontinuous for V h and globally continuous for V h. We restrict ourselves to the simplest case of constant coefficients κ and a. 16

17 Numerical testing of this configuration has been performed in [10]. Here, we want to measure the stability of the schemes by a numerical evaluation of the inf-sup constant B(ν, µ) inf sup, (75) ν V h µ V h ν SDG µ SDG for the DG and SDG formulations (where B(, ) B DG (, ) and B(, ) B SDG (, ), resp.) and B(ν, µ) inf sup, (76) ν SDG µ SDG ν h (V h,0) µ h (V h,0) for the MDG and SMDG formulations (where B(, ) B DG (, ) and B(, ) B SDG (, ), resp.). he evaluation of (75) and (76) can be performed through a generalized eigenvalue computation (see, e.g., [] for details). In the sequel we assume δ = 0. Very similar results are obtained with the choice δ = 0.01, proposed in [10], which has advantages from the implementation standpoint. 5.1 he interior penalty parameter First, we study the effect of ε, the amount of interior penalty stabilization. We focus on the diffusion-dominated regime, where the interior penalty term plays a role, setting κ = 1 and a = he values of (75) (76) are plotted in Figure 1, respectively, for the DG and MDG schemes (similar result are obtained for the stabilized SDG and SMDG schemes), and for a partition of elements (N = 10). he symmetric version (s = 1), the skewsymmetric version (s = 1), as well as the neutral version (s = 0) are considered. We confirm that the skew-symmetric version is stable for all positive ɛ, while the other two formulations are unstable if the interior penalty stabilization is too small. Nevertheless, the symmetric version attains more accurate numerical solutions and is preferred (see [10]). We also observe that the MDG scheme needs less interior penalty stabilization than the DG scheme. his is not surprising: indeed, roughly speaking, in the diffusive regime, h (V h, 0) is composed of functions that are almost continuous, and therefore the interior penalty stabilization is only needed on the boundary of Ω. 5. he SUPG parameter and the inf-sup stability of MDG Second, we analyze the role of the streamline stabilization. We select, from now on, the symmetric version (s = 1) and we take ε = 6 (this gives sufficient interior penalty stabilization to both DG and SDG, as seen in Section 5.1). We know, from heorem 3.5, that there is no need of streamline stabilization in the DG method. his is confirmed in Figure 3, where (75) is plotted for different κ and a = [cos(θ), sin(θ)], on a grid of We have set τ = 1/ in the definition of SDG. he values of (75) are bounded away from zero, uniformly with respect to the operator coefficients. In Figure 5 we focus the attention on the convectiondominated regime, which is now the most interesting case: we set κ = 10 6 and compute (75) for different a = [cos(θ), sin(θ)], on different uniform meshes of N N elements. We confirm that the inf-sup condition holds uniformly with respect to the mesh-size. 17

18 he major result of this section is the evaluation of the stability of the MDG scheme. Actually, the MDG scheme turns out to be stable with respect to the SDG, for the model case here considered: in Figure 4 we plot the inf-sup constant (76) for different κ and a = [cos(θ), sin(θ)], on the uniform grid, while in Figure 6 we plot (76) in the convection-dominated regime (κ = 10 6 ) for different directions of the convective field a and different uniform meshes. Our conclusion is that, at least for this model case, the MDG scheme is inf-sup stable, that is, condition (74) holds with β MDG independent of the problem coefficients and the mesh-size. From this, and reasoning as in heorem 4.6, we can infer the optimal error estimate for the MDG scheme: ( ( φ φ MDG SDG a h k+1 h 1/ ) + κ h k φ H k+1 ( )). (77) Similar plots and results are obtained for the stabilized SDG and SMDG methods, in accordance with Proposition 3.3, and are omitted inf-sup PSfrag replacements s = 1 s = 0 s = ε Figure 1: Inf-sup constant of the DG method vs. ε 6 Conclusions he mathematical analysis of the multiscale discontinuous Galerkin MDG method introduced in [10] was initiated. his method alleviates a long-standing drawback of discontinuous Galerkin methods, namely, the large size of the solution space. It utilizes local, element-wise 18

19 inf-sup PSfrag replacements s = 1 s = 0 s = ε Figure : Inf-sup constant of the MDG method vs. ε inf-sup PSfrag replacements θ κ 10 5 Figure 3: Inf-sup constant of the DG method vs. a = [cos θ, sin θ] and κ. 19

20 inf-sup PSfrag replacements θ κ 10 5 Figure 4: Inf-sup constant of the MDG method vs. a = [cos θ, sin θ] and κ. 0.5 inf-sup PSfrag replacements θ 0 10 N Figure 5: Inf-sup constant of the DG method vs. a = [cos θ, sin θ] and N. 0

21 0.5 inf-sup PSfrag replacements θ 0 10 N Figure 6: Inf-sup constant of the MDG method vs. a = [cos θ, sin θ] and N. problems to generate an interscale transfer operator, enabling the size of the matrix problem to be significantly reduced, apparently without degradation in the quality of results. We studied MDG and a stabilized version, SMDG. We were able to characterize the approximation properties of the interscale transfer spaces. he corresponding global discontinuous Galerkin methods, DG and SDG, are inf-sup stable and coercive, respectfully, with respect to the norm induced by the bilinear form of SDG. Coercivity is inherited by the interscale transfer subspaces, but not necessarily inf-sup stability. Consequently, we were able to obtain the same error estimates for SMDG as for DG and SDG but the situation for MDG remains open. Numerical evaluations of the inf-sup constant for MDG indicated that it was positive, bounded uniformly away from zero, and very similar to that for DG. hese results are consistent with the numerical calculations performed in [10]. Acknowledgment A. Buffa and G. Sangalli thank the Institute for Computational Engineering and Sciences (U-Austin) for kind hospitality and the financial support of the J. insley Oden Faculty Fellowship Research Program. he work of.j.r. Hughes was supported by Sandia Contract No. A with the University of exas and is gratefully acknowledged. Finally, we thank Guglielmo Scovazzi for helpful discussions. 1

22 References [1] Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5): , 001/0. [] Klaus-Jürgen Bathe, Dena Hendriana, Franco Brezzi, and Giancarlo Sangalli. Inf-sup testing of upwind methods. Internat. J. Numer. Methods Engrg., 48(5): , 000. [3] Pavel Bochev, homas J. R. Hughes, and Guglielmo Scovazzi. A multiscale discontinuous Galerkin method. echnical Report 05-17, ICES, U-Austin, 005. [4] Franco Brezzi and Alessandro Russo. Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci., 4(4): , [5] Bernardo Cockburn. Discontinuous Galerkin Methods for Computational Fluid Mechanics. In E. Stein, R. de Borst, and.j.r. Hughes, editors, Encyclopedia of Computational Mechanics, volume 3, pages Wiley, Chichester (England), 004. [6] Bernardo Cockburn, George E. Karniadakis, and Chi-Wang Shu, editors. Discontinuous Galerkin methods: heory, computation and applications, volume 11 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin, 000. [7] Paul Houston, Christoph Schwab, and Endre Süli. Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal., 39(6): , 00. [8] homas J. R. Hughes, Gerald Engel, Luca Mazzei, and Mats G. Larson. A comparison of discontinuous and continuous Galerkin methods based on error estimates, conservation, robustness and efficiency. In Discontinuous Galerkin methods (Newport, RI, 1999), volume 11 of Lect. Notes Comput. Sci. Eng., pages Springer, Berlin, 000. [9] homas J. R. Hughes, Leopoldo P. Franca, and Gregory M. Hulbert. A new finite element formulation for computational fluid dynamics. VIII. he Galerkin/leastsquares method for advective-diffusive equations. Comput. Methods Appl. Mech. Engrg., 73(): , [10] homas J. R. Hughes, Guglielmo Scovazzi, Pavel Bochev, and Annalisa Buffa. A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method. echnical Report 05-16, ICES, U-Austin, 005. Submitted to Comput. Meth. Appl. Mech. Engrg. [11] Claes Johnson, Uno Nävert, and Juhani Pitkäranta. Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg., 45(1-3):85 31, [1] Claes Johnson and Juhani Pitkäranta. An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp., 46(173):1 6, [13] Giancarlo Sangalli. A discontinuous residual-free bubble method for advection-diffusion problems. J. Engrg. Math., 49():149 16, 004.