Xingye Yue. Soochow University, Suzhou, China.

Transcription

1 Relations between the multiscale methods p. 1/30 Relationships beteween multiscale methods for elliptic homogenization problems Xingye Yue Soochow University, Suzhou, China

2 Relations between the multiscale methods p. 2/30 Outline 1. A short Overview of Numerical Methods for Elliptic Homogenization Problems Generalized Finite Element Method (GFEM) by Babuska and Osborn 83 Residual Free Bubble Method (RFB) by F. Brezzi et al 94 Variational Multiscale Methods (VMS) by T. Hughes 95, 98 Multiscale Finite Element Method(MsFEM) by T. Hou et al 97 Heterogenous Multiscale Methods (HMM) by W. E and B. Engquist Intrinsic links between these multiscale methods

3 Relations between the multiscale methods p. 3/30 Model Problem Transport in strongly heterogeneous media Lu ε (k ε (x) u ε (x)) = f(x), x Ω. (1) u ε = 0 on Ω Assume that 0 < ε 1 and k ε (x) = k(x, x ) microscopic structure, ε oscillated everywhere.

4 Relations between the multiscale methods p. 4/30 Model Problem -2 For standard piecewise linear finite element method (FEM), the following error estimate holds u ε U FEM 1 Ch u ε 2 C h ε. If h ε, everything is OK. If we can not afford or do not want to afford h ε,? We maybe only care about the macro behavior. e.g. We care about the total production of oil. Don t care how the oil flows in the pore. So we need a multiscale algorithm. We only want to work on a coarse mesh h ε.

5 Relations between the multiscale methods p. 5/30 GFEM by Babuska and Osborn 83 Standard FEM variational form piecewise polynomial Generalized adapting the finite element space to the particular small scale features of the problem For one-dimensional problem, Find uε V = H 1 0 (0,1), a(u ε,v) (k ε u ε,v ) = (f,v), v V Standard FEM, Find uh S h = {v H 1 0 (0,1) v P 1(I j ),j = 1,2,...,N}, s.t. a(u h,v h ) = (f,v h ), v h S h GFEM: Find uh S h, s.t. a(au h, Av h ) = (f, Av h ), v h S h. For v h S h, w = Av h S h = A(S h ) is the solution of the local problems w H 1 0(0,1), Lw Ij = 0, w(x j ) = v h (x j ), w(x j+1 ) = v h (x j+1 ), j = 0,1,2,. A: reconstruction operator from coarse scale to micro scale.

6 Relations between the multiscale methods p. 6/30 GFEM by Babuska and Osborn 83 2 Standard FEM, Find uh S h = {v H 1 0 (0,1) v P 1(I j ),j = 1,2,...,N}, s.t. a(u h,v h ) = (f,v h ), v h S h GFEM: Find uh S h = A(S h ), s.t. a(u h,v h ) = (f,v h ), v h S h. Theorem For GFEM, if 0 < k0 k ε k 2, u ǫ u h 1 Ch f 0, no matter whether the solution u ε is smooth or not! But this is not true for multi-dimensional problems. Remark 1 : GFEM is equivalent to standard FEM, if k ε constant or piecewise constant. Remark 2 : Parallel implementation of local reconstruction.

7 Relations between the multiscale methods p. 7/30 MsFEM by Hou et al 97, 99, 00 Key idea : Construct the multiscale finite element basis instead of piecewise linear basis. For each triangle element T T h, the standard linear finite element space VL T = span{ϕ 1,ϕ 2,ϕ 3 } is replaced by VMS T = span{ψ 1,ψ 2,ψ 3 }: (k( x ε ) ψ j(x)) = 0, x T; (2) ψ j = ϕ j on x T Reconstruction operator: ψ j = R(ϕ j ). MsFEM (Simplest version) Find u ε h V MS = R(V h L ) s.t. a(uε h,v) (kε u ε h, v h) = (f,v h ) v h V MS Error estimate for periodic meida: u ε u ε h 1 C(h+ ε+ ε/h) First attempt for multi-dimension homogenization problems. Other versions of MsFEM, Over-sampling MsFEM A extension of GFEM from one-d to higher dimensional problem

8 Relations between the multiscale methods p. 8/30 Variational multiscale method (VMS) by Hughes 95 Scale Splitting: u = ū+u (Space splitting: H 1 0 (Ω) = V V ). where ū V coarse scale part and u V fine scale part. Weak form: Find u = ū+u such that a(u,v) (k ε u, v) = (f,v) v = v +v V V Split into two problems: Coarse Scale: a(ū+u, v) = (f, v) v V (3) Fine Scale: a(ū+u,v ) = (f,v ) v V (4) (4) yields the adjustment for the residual f Lū: (Lu,v ) = (f Lū,v ) v V Solve it to obtain u = M (f Lū). Substitute it into (3):

9 Relations between the multiscale methods p. 9/30 VMS 2nd The right model at the coarse scale: Find ū V, a(ū, v)+a(m (f Lū), v) = (f, v) v V Compare with the standard Galerkin model. Find ū V, a(ū, v) = (f, v) v V The operator M is global and needs to be approximated (localized) to obtain a practical coarse model. Bubble FEM is one of the choices.

10 Relations between the multiscale methods p. 10/30 Residaul-free bubble (RFB) FEM by Brezzi et al 94 Let Th be a triangulation of Ω: Ω = T T h T. Coarse space: VL h the conforming piecewise linear finite element space on T h. Bubble space : V B = T V b T, with V b T = H0 1(T). RFB FEM scheme (Fully analyzed at 99, 00 for convection-dominated problem) Find u h = u L +u b V h = V h L VB such that a(u h,v h ) = (f,v h ) v h = v L +v b V h = V h L VB Split into two problems: Coarse Scale: a(u L +u b,v L ) = (f,v L ) v L V h L (5) Fine Scale: a(u b,v b ) = (f Lu L,v b ) v b V b (6) For each element T T h, (6) is exactly the problem (parallel computing): Lu b = f Lu L, in T, u b = 0, on T Residual Free L(u L +u b ) f = 0, in T.

11 Relations between the multiscale methods p. 11/30 Heterogeneous Multiscale Methods (HMM) by W. E and B. Engquist 03 There are two main components in the heterogeneous multiscale method: An overall macroscopic scheme for the macro-scale variables on a macroscale grid Estimating the missing macroscopic data from the microscopic model. A simple version HMM scheme (E, Ming & Zhang 05): Find ul VL h such that a HMM (u L,v L ) T T T h K HMM u L v L dx = (f,v L ), v L V h L, (K HMM ) u L v L T = 1 I δ I δ k ε R(u L ) R(v L )dx, (7) where for any T T h, I δ T is the sampling window with size of δ, and for each v L V h L, the reconstruction operator R(v L) is defined by LR(v L ) (k ε R(v L )) = 0, in I δ, R(v L ) = v L, on I δ. (8)

12 Relations between the multiscale methods p. 12/30 HMM 2nd Adaptive sampling The size of δ can be chosen adaptively to take the full advantage of the scale separation (if it exists). It is possible that ε δ h. Almost Equivalent to MsFEM if one chooses Iδ = T. Find u L V h L such that T T T h k ε R(u L ) R(v L )dx = (f,v L ), v L V h L, Other choices of the reconstruction operator R. (E, Yue 07) Dirichlet Form, Neumann Form, Periodic Form, Multiscale principle: Information exchanges between macro and micro scales reconstruction vs compression

13 Relations between the multiscale methods p. 13/30 Classification of these multiscale methods 1. GFEM, MsFEM : modifying the finite element space to make the information in micro scale be involved. 2. VMS, RFB : scale splitting, V = V V 3. HMM : choosing proper macro model and macro solver, estimating the missing data from local micro solver. - The philosophy is very general and totally different. Relations between them??

14 Relations between the multiscale methods p. 14/30 Relations between these multiscle methods Generalized FEM (GFEM) 83 vs Multiscale FEM (MsFEM) 97 GFEM: Find uh S h, s.t. a(u h,v h ) = (f,v h ), v h S h. where S h = A(S h ),S h is the standard FEM space. A reconstruction operator: for v h S h, LA(v h ) Ij = 0, A(v h )(x j ) = v h (x j ), A(v h )(x j+1 ) = v h (x j+1 ), j = 0,1,2,,. MsFEM (Simplest version) Find u ε h V MS s.t. a(u ε h,v) (kε u ε h, v h) = (f,v h ) v h V MS We can also write V MS = A(S h ). MsFEM is a extension of GFEM from one-d to higher dimensional preblem.

15 Relations between the multiscale methods p. 15/30 VMS 95 vs RFB 94 Variational multiscale methods (VMS): Find u = ū+u H 1 0 (Ω) = V V such that a(u,v) (k ε u, v) = (f,v) v = v +v V V Residual-free bubble (RFB) FEM: Find uh = u L +u b V h = V h L VB such that a(u h,v h ) = (f,v h ) v h = v L +v b V h = V h L VB Bubble space : V B = T V T b, with V T b = H 1 0 (T). RFB is a localization strategy for VMS.

16 Relations between the multiscale methods p. 16/30 RFB vs MsFEM For MsFEM solution u ε h V MS, Lu ε h = 0, in T. For RFB solution uh = u L +u b, L(u L +u b ) = f, in T. There is a clear difference!! But for u ε h V MS, u ε h = u L +(u ε h u L), and u ε h u L V B = T V b T, with V b T = H0 1 (T) is exactly a local bubble. Split the bubble into two parts ub = M 0 (u L )+M(f), such that intrinsic part: M 0 (u L ) V B, L(u L +M 0 (u L )) = 0, in T T h. external part: M(f) V B, L(M(f)) = f, in T T h. Key point: The effect of M(f) can be canceled in the coarse scale model.

17 Relations between the multiscale methods p. 17/30 RFB vs MsFEM 2nd Recall RFB FEM scheme a(u L +M 0 (u L )+M(f),v L ) = (f,v L ) v L V h L a(u L +M 0 (u L )+M(f),v b ) = (f,v b ) v b V b (9) Setting v b = M 0 (v L ) in the second formula, then taking the summation, a(u L +M 0 (u L )+M(f),v L +M 0 (v L )) = (f,v L +M 0 (v L )) v L V h L Noting that if the coefficient tensor k( x ) is symmetric, ε a(m(f),v L +M 0 (v L )) = a(v L +M 0 (v L ),M(f)) = 0. The coarse scale model is (Nolen, Papanicolaou, Pironneau 07) a(u L +M 0 (u L ),v L +M 0 (v L )) = (f,v L +M 0 (v L )) v L V h L.

18 Relations between the multiscale methods p. 18/30 RFB vs MsFEM 3nd The reconstruction operator is now : R(vL ) v L +M 0 (v L ). And the multiscale basis functions such that ψ i = R(φ i ) = φ i +M 0 (φ i ). Then the coarse scale model of RFB: Find u L VL h such that T T T h k ε R(u L ) R(v L )dx = (f,r(v L )), v L V h L, The coarse scale model of RFB is equivalent to MsFEM (simplest version). In another form: Find ũh R(VL h ) such that T T T h k ε ũ h v h dx = (f,v h ), v h R(V h L ), The solution of RFB uh satisfies that u h = u L +M 0 (u L )+M(f) = u ε h +M(f), where u ε h is the solution of MsFEM.

19 Relations between the multiscale methods p. 19/30 RFB vs MsFEM 4nd Local reconstruction: Lu h = f, in T, u h = u ε h, on T. Proposition. Under the assumption that the coefficient k ε is symmetric, let u h = u C +M 0 (u C )+M(f) be the RFB solution and u ε h be the MsFEM solution. Then we have u ε h = u C +M 0 (u C ), and the induced coarse scale model of RFB scheme is equivalent to the MsFEM scheme. Furthermore, the RFB solution u h can be obtained from the MsFEM solution u ε h by the local reconstruction. Remark. Local reconstruction is a common strategy for multiscale simulation: one first solves the macro model on the coarse scale mesh, then solves the original problems locally on the fine scale, over some local regions where the details in fine scale are interesting, using the coarse scale solution as the boundary condition (Oden, Vemaganti 00).

20 Relations between the multiscale methods p. 20/30 Adaptive sampling RFB vs HMM Revisit the Bubble FEM Set R(u L ) = u L +M 0 (u L ). The coarse scale model of RFB is as : Find u L V h L, a(r(u L),R(v L )) = (f,r(v L )) v L V h L. i.e. T T K h u L v L dx = Ω f R(v L )dx, v L V h L where (K h ) ij T = 1 T T k ε R(x i ) R(x j )dx. Remark: G. Sangalli (MMS 03) obtained this result under the assumption that the source term f is piecewise constant. This means to take the whole element T as the sampling window. If there exists scale separation, instead of the whole element, can we take a smaller region as the sampling window?

21 Relations between the multiscale methods p. 21/30 Adaptive sampling 2 Let s reformulate the RFB FEM scheme First set the bubble space as: V B = T V T b, with V T b = H 1 0 (I δ) (I δ T with size of δ) for each element T. For v L V h L, M0 (v L ) is defined now as M 0 (v L ) V B, a(v L +M 0 (v L ),v) = 0, v H 1 0(I δ ), T T h, then extending to the whole element T by 0. Denote by the reconstruction operator R(v L ) = v L +M 0 (v L ) for all v L VL h. Then define a new RFB Scheme as: Find ul VL h such that a(r(u L ),R(v L )) = (f,r(v L )) v L V h L The coarse scale equation can be rewritten now as: Find ul V h L such that a (u L,v L ): = T T H T K h u L v L dx = (f,r(v L )) v L V h L, with (K h ) ij T = 1 T T k ε R(x i ) R(x j )dx.

22 Relations between the multiscale methods p. 22/30 Adaptive sampling 3 Does this still give a right macro-scale model? First let s go back to original RFB (or MsFEM) scheme and standard Galerkin scheme. RFB (or MsFEM) scheme : Iδ = T. It does work. a(u L +M 0 (u L ),v L ) = (f,v L +M 0 (v L )) v L V h L. If we neglect the effect of M 0 (u L ) and M 0 (v L ), we get the standard Galerkin finite element scheme on the coarse scale of h(> ε). Standard Galerkin : Iδ = φ It fails. a(u L,v L ) = (f,v L ) v L V h L. This is a wrong coarse scale model. In fact, in this case the effective coefficient is defined as K T =< k ε > T = 1 k ε (x)dx. T T

23 Relations between the multiscale methods p. 23/30 Adaptive sampling 4 The reason is that although the fine scale bubble M 0 (u L ) is negligible when compared to the coarse scale function u L, this is not the case for M 0 (u L ), which is comparable to u L. On the other hand, we denote by a(u L +M 0 (u L ),v L ) = (k ε u L +k ε M 0 (u L ))dx v L T T T (q L +q F )dx v L T. T T The macro mass or energy flux q L and micro flux q F are comparable. Neglecting of the micro mass flux q F will lead to the failure of mass balance. For an effective media with coefficient K, we must have the balance: K < u ε > T =< k ε u ε > T for any proper micro state u ε.

24 Relations between the multiscale methods p. 24/30 Adaptive sampling 5 From these points of view, we will see that in the new scheme, only part of micro mass flux on I δ is included. So we can not expect the balance of mass. We should modify the bubble function to meet the balance. If the size of the sample I δ is properly chosen, the average flux (or flux density ) < q F > δ = 1 I δ I k ε M 0 δ δ (u L)dx would be accurate enough. Therefore the modified bubble function M 0 δ (u L) should satisfy that 1 T T k ε M 0 δ (u L)dx = 1 I δ I δ k ε M 0 δ (u L)dx. A simplest choice is M 0 δ (v L) = T I δ M0 δ (v L), for any v L V h L. (10) Then the coarse scale model for adaptive RFB scheme reads: Find u L VL h, such that a(u L + M 0 δ (u L),v L ) = (f,v L + M 0 δ (v L)), v L V h L. (11)

25 Relations between the multiscale methods p. 25/30 Adaptive sampling 6 Equivalently, we rewritten (11) as : Find ul VL h, such that a (u L,v L ) T T H T K h u L v L dx = (f, R(v L )), v L V h L,(12) with (K h ) ij T = 1 T where in each element T, R(vL ) = v L V h L. T k ε R(x i ) x j dx, (13) v L, x I δ, v L + M 0 δ (v L), x I δ, for any

26 Relations between the multiscale methods p. 26/30 Adaptive sampling 7 Modeling error for periodic media between this scheme and the HMM K h K HMM = < kε > T < k ε > Iδ C(ε/h+ε/δ) Modeling error for HMM (E,Ming, Zhang 05) K KHMM C(ε/δ +δ), where K is the homogenized coefficient. Remark The scheme is slightly different with the HMM: they test the right hand side in different ways. Precisely, in the right hand side of HMM scheme, only coarse scale test functions are used, while in adaptive RFB scheme, multiscale test functions are used. If the right hand side term f does contain micro information: to which I mean that we only have f f 0 weakly in H 1, as ε 0, for example f = f ε (x) with f ε = f(x/ε) R n and f(y) be periodic, the multiscale test functions at the right hand side will play a key role to capture the micro scale information. (Yu, Yue 2011)

27 Relations between the multiscale methods p. 27/30 HMM vs VMS Nordbotten 09 HMM (original form. E, Engquist 03) Variational problem : minv V J(v) = 1 2 a(v,v) (f,v). HMM components: Selecting a macro model and a macro solver, estimating the missing data from micro model. Macro solver: min v D V D J D (v D ), V D is the coarse scale FEM space. If we solve directly min v D V D J(v D ), that is the standard Ritz FEM equivalent to Galerkin FEM. It doss not work on coarse scale. How to define the macroscopic energy density JD (v D ) for any macro displacement v D V D?

28 Relations between the multiscale methods p. 28/30 HMM vs VMS 2 Let Q : V VD to be the compression operator, then the coarse scale energy functional: J D (v D ) = min v:qv=v D J(v), its solution (the minimizer) is denoted by Rv D reconstruction operator. RuD is the exact solution, if u D is the minimizer of J D (v D ). So, HMM scheme is Or, min v V min v D V D J D (v D ) = J(Rv D ). J(v) = min v D V D min v:qv=v D J(v).

29 Relations between the multiscale methods p. 29/30 HMM vs VMS 3 Recall the VMS scheme. Scale Splitting: u = ū+u (Space splitting: V = V V ). Original weak form: Find u = ū+u such that Coarse Scale: a(ū+u, v) = (f, v) v V (14) Fine Scale: a(ū+u,v ) = (f,v ) v V (15) Reconstruction operator : R1 ū = ū+u, then equivalently, Coarse Scale: Fine Scale: min v V J(R 1 v), (16) R 1 v = arg min J(v), (17) v:pv= v P : V V is the projection operator. If we set VD = V, Q = P, then R = R 1. VMS is a special case of HMM.

30 Relations between the multiscale methods p. 30/30 Thank you! Thank you!