A Multiscale DG Method for Convection Diffusion Problems

Transcription

1 A Multiscale DG Method for Convection Diffusion Problems Annalisa Buffa Istituto di Matematica Applicata e ecnologie Informatiche - Pavia National Research Council Italy Joint project with h. J.R. Hughes, G. Sangalli, G. Scovazzi, P. Bochev. A. Buffa (IMAI-CNR Italy) MDG method 1 / 27

2 1 Convection diffusion problem Discontinuous Galerkin technique Control on the stream-line derivative - uniform inf-sup condition 2 Variational Multiscale DG method (VMS-DG) Definition of the method Local/Global problems and their solvability Control on the stream-line derivative - error analysis Numerics Relation with RFB approach 3 Conclusions A. Buffa (IMAI-CNR Italy) MDG method 2 / 27

3 Notation and Definitions Lφ = κ φ + a φ = f in Ω φ = g on Γ κ 0 diffusion coefficent, a is the solenoidal velocity vector field; g H 1/2 (Γ) Dirichlet boundary condition We set: Γ = {x Γ : a(x) n(x) 0} Γ + = {x Γ : a(x) n(x) > 0} 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 boundary; Convection dominated regime : a >> κ A. Buffa (IMAI-CNR Italy) MDG method 3 / 27

4 Discontinuous Galerkin discretization: generalities h is a mesh of Ω, E h the set of edges, E 0 h the set of internal edges. Finite element space : V h = {v L 2 (Ω) : v P k (), h } Jumps and averages : e E 0 h +, downwind and upwind triangles ν = 1 2 (ν+ + ν ) [[ν]] = ν + n + + ν n τ = 1 2 (τ + + τ ) [[τ]] = τ + n + + τ n Note that, e.g., ν + (ν ) are the trace values taken from the downwind (upwind) triangle, respectively. A. Buffa (IMAI-CNR Italy) MDG method 4 / 27

5 Construction of an up-wind DG scheme If φ is the solution, then, integrating by parts: f µ = Lφµ = κ φ µ a µφ + (aφ κ φ) n µ Γ We use following biased identity (take τ = aφ κ φ) τ n µ = ( ) µ + [[τ]] + [[µ]] τ + µ τ n. Γ h e Eh o e e Γ which, together with upwind on Γ, brings to (for all µ V h ) f µ = Lφµ = µ a µφ) + Ω Ω h (κ φ [[µ]](aφ κ φ) e Eh 0 e + µκ φ + a nµφ + a nµg Γ Γ + Γ A. Buffa (IMAI-CNR Italy) MDG method 5 / 27

6 Construction of an up-wind DG scheme If φ is the solution, then, integrating by parts: f µ = Lφµ = κ φ µ a µφ + (aφ κ φ) n µ Γ We use following biased identity (take τ = aφ κ φ) τ n µ = ( ) µ + [[τ]] + [[µ]] τ + µ τ n. Γ h e Eh o e e Γ which, together with upwind on Γ, brings to (for all µ V h ) f µ = Lφµ = µ a µφ) + Ω Ω h (κ φ [[µ]](aφ κ φ) e Eh 0 e + µκ φ + a nµφ + a nµg Γ Γ + Γ A. Buffa (IMAI-CNR Italy) MDG method 5 / 27

7 Construction of an up-wind DG scheme If φ is the solution, then, integrating by parts: f µ = Lφµ = κ φ µ a µφ + (aφ κ φ) n µ Γ We use following biased identity (take τ = aφ κ φ) τ n µ = ( ) µ + [[τ]] + [[µ]] τ + µ τ n. Γ h e Eh o e e Γ which, together with upwind on Γ, brings to (for all µ V h ) f µ = Lφµ = µ a µφ) + Ω Ω h (κ φ [[µ]](aφ κ φ) e Eh 0 e + µκ φ + a nµφ + a nµg Γ Γ + Γ A. Buffa (IMAI-CNR Italy) MDG method 5 / 27

8 Up-wind Discontinuous Galerkin method B DG (φ DG, µ) = B DG D (φdg, µ) + B DG C (φdg, µ) = L DG (g, f ; µ) Bilinear Forms BD DG (ν, µ) = BC DG (ν, µ) = Right hand side L DG (g, f ; µ) = µf + Ω e Γ (DG) µ κ ν [[µ]] κ ν + s κ µ [[ν]] h e Eh o e + s κ µ nν κ ν nµ + ε κ [[µ]] [[ν]] e Γ e e E e h h µ aν + [[µ]] aν + µνa n h e Eh o e e Γ + e ( ) κ ε µg + s κ µ ng a nµg. e h e e Γ e back A. Buffa (IMAI-CNR Italy) MDG method 6 / 27

9 Up-wind Discontinuous Galerkin method B DG (φ DG, µ) = B DG D (φdg, µ) + B DG C (φdg, µ) = L DG (g, f ; µ) Bilinear Forms BD DG (ν, µ) = BC DG (ν, µ) = Right hand side L DG (g, f ; µ) = µf + Ω e Γ (DG) µ κ ν [[µ]] κ ν + s κ µ [[ν]] h e Eh o e + s κ µ nν κ ν nµ + ε κ [[µ]] [[ν]] e Γ e e E e h h µ aν + [[µ]] aν + µνa n h e Eh o e e Γ + e ( ) κ ε µg + s κ µ ng a nµg. e h e e Γ e back A. Buffa (IMAI-CNR Italy) MDG method 6 / 27

10 Up-wind Discontinuous Galerkin method B DG (φ DG, µ) = B DG D (φdg, µ) + B DG C (φdg, µ) = L DG (g, f ; µ) Bilinear Forms BD DG (ν, µ) = BC DG (ν, µ) = Right hand side L DG (g, f ; µ) = µf + Ω e Γ (DG) µ κ ν [[µ]] κ ν + s κ µ [[ν]] h e Eh o e + s κ µ nν κ ν nµ + ε κ [[µ]] [[ν]] e Γ e e E e h h µ aν + [[µ]] aν + µνa n h e Eh o e e Γ + e ( ) κ ε µg + s κ µ ng a nµg. e h e e Γ e back A. Buffa (IMAI-CNR Italy) MDG method 6 / 27

11 Up-wind Discontinuous Galerkin method B DG (φ DG, µ) = B DG D (φdg, µ) + B DG C (φdg, µ) = L DG (g, f ; µ) Bilinear Forms BD DG (ν, µ) = BC DG (ν, µ) = Right hand side L DG (g, f ; µ) = µf + Ω e Γ (DG) µ κ ν [[µ]] κ ν + s κ µ [[ν]] h e Eh o e + s κ µ nν κ ν nµ + ε κ [[µ]] [[ν]] e Γ e e E e h h µ aν + [[µ]] aν + µνa n h e Eh o e e Γ + e ( ) κ ε µg + s κ µ ng a nµg. e h e e Γ e back A. Buffa (IMAI-CNR Italy) MDG method 6 / 27

12 Up-wind Discontinuous Galerkin method B DG (φ DG, µ) = B DG D (φdg, µ) + B DG C (φdg, µ) = L DG (g, f ; µ) Bilinear Forms BD DG (ν, µ) = BC DG (ν, µ) = Right hand side L DG (g, f ; µ) = µf + Ω e Γ (DG) µ κ ν [[µ]] κ ν + s κ µ [[ν]] h e Eh o e + s κ µ nν κ ν nµ + ε κ [[µ]] [[ν]] e Γ e e E e h h µ aν + [[µ]] aν + µνa n h e Eh o e e Γ + e ( ) κ ε µg + s κ µ ng a nµg. e h e e Γ e back A. Buffa (IMAI-CNR Italy) MDG method 6 / 27

13 Norms and streamline derivative ν 2 DG = ν 2 D + ν 2 C, where we set: (κ = κ ) ν 2 D = ) ( ) (κ ν 2 H 1 () + h2 κ ν 2 H 2 () + ε e Eh h 1 κ [[ν]] 2 L 2 (e), h ν 2 C = e E h a n 1/2 [[ν]] 2 L 2 (e). Coercivity For ε > ε: B DG C (µ,µ) + B DG D (µ,µ) 1 2 µ 2 C + β 1 µ 2 D Coercivity was used to provide error estimates in the DG norm in Houston et al, 2001, but this norm is very weak in the advection dominated regime. A. Buffa (IMAI-CNR Italy) MDG method 7 / 27

14 Norms and streamline derivative ν 2 DG = ν 2 D + ν 2 C, where we set: (κ = κ ) ν 2 D = ) ( ) (κ ν 2 H 1 () + h2 κ ν 2 H 2 () + ε e Eh h 1 κ [[ν]] 2 L 2 (e), h ν 2 C = e E h a n 1/2 [[ν]] 2 L 2 (e). Coercivity For ε > ε: B DG C (µ,µ) + B DG D (µ,µ) 1 2 µ 2 C + β 1 µ 2 D Coercivity was used to provide error estimates in the DG norm in Houston et al, 2001, but this norm is very weak in the advection dominated regime. A. Buffa (IMAI-CNR Italy) MDG method 7 / 27

15 Control on the stream-line derivative Johnson-Pitkäranta, 86, B. Hughes, Sangalli 05 { } h Control on the streamline derivative : τ = τ min, h2 a κ ν 2 SDG = ν 2 DG + h τ a ν 2 L 2 () We can prove a uniform inf-sup condition : B DG (ν,µ) inf sup β DG > 0; ν V h µ V h ν SDG µ SDG where β DG is independent of h, κ, a, and the domain. Proof : Given ν V h, choose µ = ν + γ h τ (a ν)... A. Buffa (IMAI-CNR Italy) MDG method 8 / 27

16 Control on the stream-line derivative Johnson-Pitkäranta, 86, B. Hughes, Sangalli 05 Remark : DG methods do not need stabilization (e.g., SUPG): the control of the stream-line derivative is built in it. Error estimate : Optimal error estimates in SDG norm can be provided. Indeed: φ φ DG SDG h ( a h 2k+1 ) + κ h 2k φ 2 H k+1 () 1/2. (1) Main drawback of DG methods : he number of d.o.f.! A. Buffa (IMAI-CNR Italy) MDG method 9 / 27

17 Control on the stream-line derivative Johnson-Pitkäranta, 86, B. Hughes, Sangalli 05 Remark : DG methods do not need stabilization (e.g., SUPG): the control of the stream-line derivative is built in it. Error estimate : Optimal error estimates in SDG norm can be provided. Indeed: φ φ DG SDG h ( a h 2k+1 ) + κ h 2k φ 2 H k+1 () 1/2. (1) Main drawback of DG methods : he number of d.o.f.! A. Buffa (IMAI-CNR Italy) MDG method 9 / 27

18 Control on the stream-line derivative Johnson-Pitkäranta, 86, B. Hughes, Sangalli 05 Remark : DG methods do not need stabilization (e.g., SUPG): the control of the stream-line derivative is built in it. Error estimate : Optimal error estimates in SDG norm can be provided. Indeed: φ φ DG SDG h ( a h 2k+1 ) + κ h 2k φ 2 H k+1 () 1/2. (1) Main drawback of DG methods : he number of d.o.f.! A. Buffa (IMAI-CNR Italy) MDG method 9 / 27

19 Multiscale Paradigm: a VMS-DG Method Hughes, Scovazzi, Pavel, B B., Hughes, Sangalli 06 Reduce the number of d.o.f. solve the DG problem on a strict subset of V h, which is built by solving local problems in a multiscale fashion. We introduce the space V h = V h C 0 (Ω) and define a interscale transfer operator h : V h L 2 (Ω) V h : ( ν, f ) V h L 2 (Ω), solve with DG { L ν = f h ν = ν Γ where L is the restriction of L on the triangle. A. Buffa (IMAI-CNR Italy) MDG method 10 / 27

20 Multiscale Paradigm: a VMS-DG Method Hughes, Scovazzi, Pavel, B B., Hughes, Sangalli 06 Reduce the number of d.o.f. solve the DG problem on a strict subset of V h, which is built by solving local problems in a multiscale fashion. We introduce the space V h = V h C 0 (Ω) and define a interscale transfer operator h : V h L 2 (Ω) V h : ( ν, f ) V h L 2 (Ω), solve with DG { L ν = f h ν = ν Γ where L is the restriction of L on the triangle. A. Buffa (IMAI-CNR Italy) MDG method 10 / 27

21 Multiscale Paradigm: a VMS-DG Method Hughes, Scovazzi, Pavel, B B., Hughes, Sangalli 06 Reduce the number of d.o.f. solve the DG problem on a strict subset of V h, which is built by solving local problems in a multiscale fashion. We introduce the space V h = V h C 0 (Ω) and define a interscale transfer operator h : V h L 2 (Ω) V h : ( ν, f ) V h L 2 (Ω), solve with DG { L ν = f h ν = ν Γ where L is the restriction of L on the triangle. Remark he affine space h (V h, f ) represents a subset of V h, parametrized through the d.o.f. of V h : it is strictly contained in V h. A. Buffa (IMAI-CNR Italy) MDG method 10 / 27

22 Multiscale Paradigm: a VMS-DG Method Hughes, Scovazzi, Pavel, B B., Hughes, Sangalli 06 Reduce the number of d.o.f. solve the DG problem on a strict subset of V h, which is built by solving local problems in a multiscale fashion. We introduce the space V h = V h C 0 (Ω) and define a interscale transfer operator h : V h L 2 (Ω) V h : ( ν, f ) V h L 2 (Ω), solve with DG { L ν = f h ν = ν Γ where L is the restriction of L on the triangle. Multiscale reduction We solve the DG problem on the set h (V h, f ). A. Buffa (IMAI-CNR Italy) MDG method 10 / 27

23 Variational Multiscale method - definition Local Problems Given ν V h, ν = h ( ν, f ) is the solution of b (ν, µ) = l ( ν, f ; µ) µ V h, h (LP) where b is the DG bilinear form associated with the operator L. Global Problem 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). (MDG) Questions : 1. Solvability for the local and global problems; 2. Stability of the local and global problems; 3. Approximation properties of the interscale transfer. (affine) space h (V h, f ). A. Buffa (IMAI-CNR Italy) MDG method 11 / 27

24 Variational Multiscale method - definition Local Problems Given ν V h, ν = h ( ν, f ) is the solution of b (ν, µ) = l ( ν, f ; µ) µ V h, h (LP) where b is the DG bilinear form associated with the operator L. Global Problem 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). (MDG) Questions : 1. Solvability for the local and global problems; 2. Stability of the local and global problems; 3. Approximation properties of the interscale transfer. (affine) space h (V h, f ). A. Buffa (IMAI-CNR Italy) MDG method 11 / 27

25 Variational Multiscale method - algebraic structure he algebraic structure is the following: h ( ν, f ) u ν + f f i.e., the final system is of the following type: ub u φ = uf ub f f. A. Buffa (IMAI-CNR Italy) MDG method 12 / 27

26 Variational Multiscale method - algebraic structure he algebraic structure is the following: h ( ν, f ) u ν + f f i.e., the final system is of the following type: ub u φ = u f ub f f. *u B u ν = *u u * B f * f f Figure: size of the final system: N V h N V h Remark : he transfer matrices u (N Vh N V h ) a f (N Vh N Vh ) are build inexpensively element by element. A. Buffa (IMAI-CNR Italy) MDG method 12 / 27

27 Variational Multiscale method - comments Computed quantity If φ MDG is the MDG solution, i.e., B DG (φ MDG, µ) = L DG (g, f ; µ) for all µ h (V h, 0). (MDG) then φ MDG = h ( φ MDG, f ), for some φ MDG V h. Indeed, we compute φ MDG, and φ MDG can be built by post processing (applying the matrix ). We have two discrete quantities, and we can evaluate both errors: φ φ MDG and φ φ MDG A. Buffa (IMAI-CNR Italy) MDG method 13 / 27

28 Variational Multiscale method - comments Computed quantity If φ MDG is the MDG solution, i.e., B DG (φ MDG, µ) = L DG (g, f ; µ) for all µ h (V h, 0). (MDG) then φ MDG = h ( φ MDG, f ), for some φ MDG V h. Indeed, we compute φ MDG, and φ MDG can be built by post processing (applying the matrix ). We have two discrete quantities, and we can evaluate both errors: φ φ MDG and φ φ MDG A. Buffa (IMAI-CNR Italy) MDG method 13 / 27

29 Local problems ν V h, we solve with an up-wind DG scheme: L ν = f on ν = ν on Γ. hus, ν = h ( ν, f ) if ν V h verifies for all h : b (ν, µ) = l ( ν, f ; µ) µ V h (), with b (ν, µ) = κ ν µ (κ ν nµ s κ µ nν) + ε Γ µ aν + (1 + δ)µνa n, l ( ν, f ; µ) = ε Γ Γ Γ + κ µ ν + s κ µ n ν + h Γ µ νa n + δ µ νa n Γ + f µ Γ κ h µν A. Buffa (IMAI-CNR Italy) MDG method 14 / 27

30 Local problems ν V h, we solve with an up-wind DG scheme: L ν = f on ν = ν on Γ. hus, ν = h ( ν, f ) if ν V h verifies for all h : b (ν, µ) = l ( ν, f ; µ) µ V h (), with b (ν, µ) = κ ν µ (κ ν nµ s κ µ nν) + ε Γ µ aν + (1 + δ)µνa n, l ( ν, f ; µ) = ε Γ Γ Γ + κ µ ν + s κ µ n ν + h Γ µ νa n + δ µ νa n Γ + f µ Γ κ h µν A. Buffa (IMAI-CNR Italy) MDG method 14 / 27

31 Local problems ν V h, we solve with an up-wind DG scheme: L ν = f on ν = ν on Γ. hus, ν = h ( ν, f ) if ν V h verifies for all h : b (ν, µ) = l ( ν, f ; µ) µ V h (), with b (ν, µ) = κ ν µ (κ ν nµ s κ µ nν) + ε Γ µ aν + (1 + δ)µνa n, l ( ν, f ; µ) = ε Γ Γ Γ + κ µ ν + s κ µ n ν + h Γ µ νa n + δ µ νa n Γ + f µ Γ κ h µν A. Buffa (IMAI-CNR Italy) MDG method 14 / 27

32 Solvability and Stability for the local problems he local problems read: ν V h, find ν V h : b (ν, µ) = l ( ν, f ; µ) µ V h, h We construct local norms (with control on the stream-line derivative) ν 2 SDG() := κ ν 2 H 1 () + h2 κ ν 2 H 2 () + εh 1 κ ν 2 L 2 (Γ ) + τ a ν 2 L 2 () + a n 1/2 ν 2 L 2 (Γ ). Stability and Solvability here exists positive ε and δ such that for all ε ε and δ δ, inf sup ν V h () µ V h () b (ν, µ) ν SDG() µ SDG() β b > 0, h, A. Buffa (IMAI-CNR Italy) MDG method 15 / 27

33 Solvability and Stability for the local problems he local problems read: ν V h, find ν V h : b (ν, µ) = l ( ν, f ; µ) µ V h, h We construct local norms (with control on the stream-line derivative) ν 2 SDG() := κ ν 2 H 1 () + h2 κ ν 2 H 2 () + εh 1 κ ν 2 L 2 (Γ ) + τ a ν 2 L 2 () + a n 1/2 ν 2 L 2 (Γ ). Stability and Solvability here exists positive ε and δ such that for all ε ε and δ δ, inf sup ν V h () µ V h () b (ν, µ) ν SDG() µ SDG() β b > 0, h, A. Buffa (IMAI-CNR Italy) MDG method 15 / 27

34 Solvability and Stability for the local problems he local problems read: ν V h, find ν V h : b (ν, µ) = l ( ν, f ; µ) µ V h, h We construct local norms (with control on the stream-line derivative) ν 2 SDG() := κ ν 2 H 1 () + h2 κ ν 2 H 2 () + εh 1 κ ν 2 L 2 (Γ ) + τ a ν 2 L 2 () + a n 1/2 ν 2 L 2 (Γ ). Stability and Solvability here exists positive ε and δ such that for all ε ε and δ δ, inf sup ν V h () µ V h () b (ν, µ) ν SDG() µ SDG() β b > 0, h, A. Buffa (IMAI-CNR Italy) MDG method 15 / 27

35 Solvability and Stability for the local problems he local problems read: ν V h, find ν V h : b (ν, µ) = l ( ν, f ; µ) µ V h, h We construct local norms (with control on the stream-line derivative) ν 2 SDG() := κ ν 2 H 1 () + h2 κ ν 2 H 2 () + εh 1 κ ν 2 L 2 (Γ ) + τ a ν 2 L 2 () + a n 1/2 ν 2 L 2 (Γ ). Stability and Solvability here exists positive ε and δ such that for all ε ε and δ δ, inf sup ν V h () µ V h () b (ν, µ) ν SDG() µ SDG() β b > 0, h, A. Buffa (IMAI-CNR Italy) MDG method 15 / 27

36 Approximation properties for the transfer spaces Let φ be the solution of the continuous problem: Lφ = f Ω, φ = g Ω then there exists ν h (V h, f ) such that φ ν 2 SDG h ( a h 2k+1 ) + κ h 2k φ 2 H k+1 () Proof : Select ν = h (φ I, f ), φ I being the interpolant of φ, use the stability of local problems and an ad-hoc Poincaré inequality: τ 1 ν 2 L 2 () ν 2 SDG() { } h τ = τ min, h2. a κ A. Buffa (IMAI-CNR Italy) MDG method 16 / 27

37 Approximation properties for the transfer spaces Let φ be the solution of the continuous problem: Lφ = f Ω, φ = g Ω then there exists ν h (V h, f ) such that φ ν 2 SDG h ( a h 2k+1 ) + κ h 2k φ 2 H k+1 () Proof : Select ν = h (φ I, f ), φ I being the interpolant of φ, use the stability of local problems and an ad-hoc Poincaré inequality: τ 1 ν 2 L 2 () ν 2 SDG() { } h τ = τ min, h2. a κ A. Buffa (IMAI-CNR Italy) MDG method 16 / 27

38 Solvability and Stability for the global problem 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). (MDG) Coercivity in weak norm: B DG (φ MDG, φ MDG ) α φ MDG 2 DG + 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; If we had the inf-sup condition (or if we add a SUPG stabilization), then φ φ MDG 2 SDG h (a h 2k+1 + κ h 2k ) φ 2 H k+1 () A. Buffa (IMAI-CNR Italy) MDG method 17 / 27

39 0.5 inf-sup θ N Figure: Inf-sup constant of the MDG method vs. a = [cos θ,sin θ] and N. A. Buffa (IMAI-CNR Italy) MDG method 18 / 27

40 0.5 inf-sup θ κ 10 5 Figure: Inf-sup constant of the MDG method vs. a = [cos θ,sin θ] and κ on a mesh A. Buffa (IMAI-CNR Italy) MDG method 19 / 27

41 1-D numerical tests Pe L =1; Pe h = Skew (s=+1) Neutral (s=0) Symmetric (s= 1) Pe L =24; Pe h = Pe L =640; Pe h = Figure: blue=mdg (cont), L-blue= MDG (disc), magenta = donor DG back A. Buffa (IMAI-CNR Italy) MDG method 20 / 27

42 1-D numerical tests, convergence (neutral) 10 2 Pe= Pe= Pe= L 2 error Pe= Pe= Pe=700 L 2 error Pe h Pe h Pe h Figure: blue=mdg (cont), L-blue= MDG (disc), magenta = donor DG A. Buffa (IMAI-CNR Italy) MDG method 21 / 27

43 1-D numerical tests, convergence (skew-symmetric) L 2 error 10 2 Pe= Pe= Pe= Pe= Pe= Pe=700 L 2 error Pe h Pe h Pe h Figure: blue=mdg (cont), L-blue= MDG (disc), magenta = donor DG A. Buffa (IMAI-CNR Italy) MDG method 22 / 27

44 2-D numerical tests, internal layer y/l 0 0 x/l y/l 0 0 x/l y/l 0 0 x/l (a) donor DG (b) MDG (cont) (c) MDG (tot) A. Buffa (IMAI-CNR Italy) MDG method 23 / 27

45 2-D numerical tests, section x/l y/l Figure: blue=mdg (cont), L-blue= MDG (disc), magenta = donor DG A. Buffa (IMAI-CNR Italy) MDG method 24 / 27

46 Multiscale interpretation and comparison with RFB Multiscale local problem Given ν = h ( ν, f ), with ν V h, then ν = ν + ν, ν = fast scale When k = 1 (to fix ideas), the fast scale ν solves the local problems: b (ν, µ) = (f L ν)µ, µ V h () hus, ν is a DG discretization for the Residual Free Bubble : L ν bubble = f L ν on, ν bubble Γ = 0. Multiscale global problem Compute φ V h B DG ( φ+φ ( φ,f ), µ+µ ( φ,0)) = L DG (g, f ; µ+µ ( φ,0)) for all µ V h. he MDG method can be interpreted as a discontinuous stabilization method for standard finite elements. he relation between MDG and bubble type stabilization deserves further investigation. A. Buffa (IMAI-CNR Italy) MDG method 25 / 27

47 Multiscale interpretation and comparison with RFB Multiscale local problem Given ν = h ( ν, f ), with ν V h, then ν = ν + ν, ν = fast scale When k = 1 (to fix ideas), the fast scale ν solves the local problems: b (ν, µ) = (f L ν)µ, µ V h () hus, ν is a DG discretization for the Residual Free Bubble : L ν bubble = f L ν on, ν bubble Γ = 0. Multiscale global problem Compute φ V h B DG ( φ+φ ( φ,f ), µ+µ ( φ,0)) = L DG (g, f ; µ+µ ( φ,0)) for all µ V h. he MDG method can be interpreted as a discontinuous stabilization method for standard finite elements. he relation between MDG and bubble type stabilization deserves further investigation. A. Buffa (IMAI-CNR Italy) MDG method 25 / 27

48 Multiscale interpretation and comparison with RFB Multiscale local problem Given ν = h ( ν, f ), with ν V h, then ν = ν + ν, ν = fast scale When k = 1 (to fix ideas), the fast scale ν solves the local problems: b (ν, µ) = (f L ν)µ, µ V h () hus, ν is a DG discretization for the Residual Free Bubble : L ν bubble = f L ν on, ν bubble Γ = 0. Multiscale global problem Compute φ V h B DG ( φ+φ ( φ,f ), µ+µ ( φ,0)) = L DG (g, f ; µ+µ ( φ,0)) for all µ V h. he MDG method can be interpreted as a discontinuous stabilization method for standard finite elements. he relation between MDG and bubble type stabilization deserves further investigation. A. Buffa (IMAI-CNR Italy) MDG method 25 / 27

49 Multiscale interpretation and comparison with RFB Multiscale local problem Given ν = h ( ν, f ), with ν V h, then ν = ν + ν, ν = fast scale When k = 1 (to fix ideas), the fast scale ν solves the local problems: b (ν, µ) = (f L ν)µ, µ V h () hus, ν is a DG discretization for the Residual Free Bubble : L ν bubble = f L ν on, ν bubble Γ = 0. Multiscale global problem Compute φ V h B DG ( φ+φ ( φ,f ), µ+µ ( φ,0)) = L DG (g, f ; µ+µ ( φ,0)) for all µ V h. he MDG method can be interpreted as a discontinuous stabilization method for standard finite elements. he relation between MDG and bubble type stabilization deserves further investigation. A. Buffa (IMAI-CNR Italy) MDG method 25 / 27

50 Conclusions We presented the first results on the analysis of a Multiscale DG method. he method can be interpreted as: a strategy to reduce the number of d.o.f for DG methods while preserving the main features such flexibility and stability; a multiscale stabilization mechanism for convection dominated problems & standard conforming finite elements. Our approach is related to the hybridization technique of DG methods recently introduced by Cockburn et al: further investigations are due. hanks for the attention! A. Buffa (IMAI-CNR Italy) MDG method 26 / 27

51 Conclusions We presented the first results on the analysis of a Multiscale DG method. he method can be interpreted as: a strategy to reduce the number of d.o.f for DG methods while preserving the main features such flexibility and stability; a multiscale stabilization mechanism for convection dominated problems & standard conforming finite elements. Our approach is related to the hybridization technique of DG methods recently introduced by Cockburn et al: further investigations are due. hanks for the attention! A. Buffa (IMAI-CNR Italy) MDG method 26 / 27

52 Conclusions We presented the first results on the analysis of a Multiscale DG method. he method can be interpreted as: a strategy to reduce the number of d.o.f for DG methods while preserving the main features such flexibility and stability; a multiscale stabilization mechanism for convection dominated problems & standard conforming finite elements. Our approach is related to the hybridization technique of DG methods recently introduced by Cockburn et al: further investigations are due. hanks for the attention! A. Buffa (IMAI-CNR Italy) MDG method 26 / 27

53 SUPG stabilization Recalling τ = τ min { h a, h2 κ B SDG (ν, µ) = B DG (ν, µ) + h τ L SDG (g, f ; µ) = L DG (g, f ; µ) + h τ and solve: Find φ SMDG h (V h, f ) such that }, we set: back (L ν)(a µ), f (a µ), B SDG (φ SMDG, µ) = L SDG (g, f ; µ) for all µ h (V h, 0). (SMDG) Coercivity: : B SDG (ν, ν) α ν 2 SDG ν h (V h, 0). Error estimate : for τ τ: φ φ SMDG 2 SDG h ( a h 2k+1 ) + κ h 2k φ 2 H k+1 () A. Buffa (IMAI-CNR Italy) MDG method 27 / 27

54 SUPG stabilization Recalling τ = τ min { h a, h2 κ B SDG (ν, µ) = B DG (ν, µ) + h τ L SDG (g, f ; µ) = L DG (g, f ; µ) + h τ and solve: Find φ SMDG h (V h, f ) such that }, we set: back (L ν)(a µ), f (a µ), B SDG (φ SMDG, µ) = L SDG (g, f ; µ) for all µ h (V h, 0). (SMDG) Coercivity: : B SDG (ν, ν) α ν 2 SDG ν h (V h, 0). Error estimate : for τ τ: φ φ SMDG 2 SDG h ( a h 2k+1 ) + κ h 2k φ 2 H k+1 () A. Buffa (IMAI-CNR Italy) MDG method 27 / 27

55 SUPG stabilization Recalling τ = τ min { h a, h2 κ B SDG (ν, µ) = B DG (ν, µ) + h τ L SDG (g, f ; µ) = L DG (g, f ; µ) + h τ and solve: Find φ SMDG h (V h, f ) such that }, we set: back (L ν)(a µ), f (a µ), B SDG (φ SMDG, µ) = L SDG (g, f ; µ) for all µ h (V h, 0). (SMDG) Coercivity: : B SDG (ν, ν) α ν 2 SDG ν h (V h, 0). Error estimate : for τ τ: φ φ SMDG 2 SDG h ( a h 2k+1 ) + κ h 2k φ 2 H k+1 () A. Buffa (IMAI-CNR Italy) MDG method 27 / 27