Multiscale discontinuous Petrov--Galerkin method for the multiscale elliptic problems
aa r X i v : . [ m a t h . NA ] F e b MULTISCALE DISCONTINUOUS PETROV–GALERKIN METHODFOR THE MULTISCALE ELLIPTIC PROBLEMS
FEI SONG ∗ AND
WEIBING DENG † Abstract.
In this paper we present a new multiscale discontinuous Petrov–Galerkin method(MsDPGM) for multiscale elliptic problems. This method utilizes the classical oversampling mul-tiscale basis in the framework of Petrov–Galerkin version of discontinuous Galerkin finite elementmethod, allowing us to better cope with multiscale features in the solution. The introduced Ms-DPGM takes advantages of the multiscale Petrov–Galerkin method (MsPGM) and discontinuousGalerkin method (DGM), which can eliminate the resonance error completely, and can decrease thecomputational complexity, allowing for more efficient solution algorithms. Upon the H norm errorestimate between the multiscale solution and the homogenized solution with the first order correc-tor, we give a detailed multiscale convergence analysis under the assumption that the oscillatingcoefficient is periodic. We also investigate the corresponding multiscale discontinuous finite elementmethod (MsDFEM) which coupling the classical oversampling multiscale basis with DGM since ithas not been studied detailedly in both aspects of error analysis and numerical tests in the litera-ture. Numerical experiments are carried out for the multiscale elliptic problems with periodic andrandomly generated log-normal coefficients to demonstrate the proposed method. Key words.
Multiscale discontinuous Petrov–Galerkin method, multiscale problems, error es-timate.
AMS subject classifications.
1. Introduction.
This paper considers the numerical approximation of secondorder elliptic problems with heterogeneous and highly oscillating coefficients. Theseproblems arise in many applications such as flows in porous media or composite mate-rials. The numerical simulation of such problems in heterogeneous media poses majormathematical and computational challenges. Standard numerical methods such asthe finite element method (FEM) or the finite volume method (FVM) usually require ∗ Department of Mathematics, Nanjing University, Jiangsu, 210093, P.R. China. Cur-rent address: College of Science, Nanjing Forestry University, Jiangsu, 210037, P.R. China.( [email protected] ). The work of this author was partially supported by the UniversityPostgraduate Research and Innovation Project of Jiangsu Province 2014 under Grant KYZZ 0021. † Department of Mathematics, Nanjing University, Jiangsu, 210093, P.R. China.( [email protected] ). The work of this author was partially supported by the NSF of Chinagrant 10971096 and the Project Funded by the Priority Academic Program Development of JiangsuHigher Education Institutions. 1
F. Song and W. Deng the mesh size very fine. This leads to tremendous amount of computer memory andCPU time. In the past several decades, a number of multiscale numerical methodshave been proposed to solve these problems, see e.g., multiscale finite element method(MsFEM) [41, 51, 52], heterogeneous multiscale method (HMM) [33, 34, 35], upscalingor numerical homogenization method [18, 32, 46, 47], variational multiscale method(or the residual-free bubble method) [14, 16, 55, 54, 63], wavelet homogenization tech-niques [30, 44], and multigrid numerical homogenization techniques [48, 61]. Most ofthem are presented on meshes that are coarser than the scale of oscillations. Thesmall scale effect on the coarse scale is either captured by localized multiscale basisfunctions or modeled into the coarse scale equations with prescribed analytical forms.In this paper, the framework of the MsFEM is used to propose a new method.Two main ingredients of the MsFEM are the global formulation of the method suchas various finite element methods and the construction of basis functions. The keypoint of MsFEM is to construct the multiscale basis from the local solutions of theelliptic operator for finite element formulation. There have been many extensions andapplications of the method in the past fifteen years (cf. [1, 2, 3, 19, 22, 24, 25, 29, 38,39, 53, 56]). We refer the reader to the book [40] for more discussions on the theoryand applications of MsFEMs.It is shown that the oversampling MsFEM is a nonconforming FEM where thenumerical solution has certain continuity across the inner-element boundaries, whileits basis functions are discontinuous at the inner-element boundaries (see [51, 41]).Note that the discontinuous Garlerkin (DG) FEMs do not ask for any continuity, whichcomes the natural idea that using the DG FEM as the global formulation coupling withthe oversampling multiscale bases (see [40]). DG FEMs for elliptic boundary valueproblems have been studied since late 1970s, and it is now an active research area (see[31, 7, 8]). Examples of the DG methods include the Local Discontinuous Galerkin(LDG) method [4, 17, 62], and the interior penalty discontinuous Galerkin (IPDG)methods [7, 8, 9, 10, 11, 15, 23]. In this paper we are concerned with the IPDGmethod, still named DGM. DG methods admit good local conservation propertiesof the state variable and also offer the use of very general meshes due to the lackof inter-element continuity requirement, e.g., meshes that contain several differenttypes of elements or hanging nodes. These features are crucial in many multiscaleapplications (see [28, 65]). sDPGM for Multiscale Problems H norm error estimate betweenthe multiscale solution and the homogenized solution with the first order corrector,which plays an important role in the error estimate. The MsDPGM takes advantagesof the MsPGM and DGM, which is expected to better approximate the multiscale F. Song and W. Deng solution than the standard MsPGM.The proposed method is related with a combined finite element and oversamplingmultiscale Petrov–Galerkin method (FE-OMsPGM) [65]. The idea of FE-OMsPGMis to utilize the traditional FEM directly on a fine mesh of the problematic part of thedomain and use the OMsPGM on a coarse mesh of the other part. The transmissioncondition across the FE-OMsPGM interface is treated by the penalty technique ofDGM. In [65], they deal with the transmission condition by penalizing the jumpsfrom linear function values as well as the fluxes of the finite element solution onthe fine mesh to those of the oversampling multiscale solution on the coarse mesh.Compared to [65], in this paper, we develop and analyze MsDPGM for the multiscaleelliptic problems. The basic idea is to use PG formulation based on the discontinuousmultiscale approximation space. The jump terms across each inter-element are dealtwith penalty technique. The penalty term of linear function values is taken as thatof the FE-OMsPGM, while the penalty term of the fluxes is not needed here.Although the error analysis is given under the assumption that the oscillatingcoefficient is periodic, our method is not restrict to the periodic case. The numericalresults show that the introduced MsDPGM is very efficient for randomly generatedcoefficients. Recently, the multiscale methods on localization of the elliptic multiscaleproblems with highly varying (non-periodic) coefficients are studied in some papers.For instance, the new variational multiscale method is presented in [59]; a new over-sampling strategy for the MsFEM is presented in [50]. In the future work, we willgive more extensions and developments on our method with the new oversamplingstrategy.The outline of this paper is as follows. In Section 2, we give the model problemand recall the DG variational formulation of the model problem in the broken Sobolevspaces. Section 3 is devoted to derive the MsDPGM. It includes the introduction ofdiscontinuous oversampling multiscale approximation space and the derivation of theformulations of MsDFEM and MsDPGM. In Section 4, we review the homogenizationresults and give some preliminaries for the error analysis. In Section 5, we give themain results of our method. It includes the stability and a priori error estimate ofthe proposed method. In Section 6, we first give several numerical examples withperiodic coefficients to demonstrate the accuracy of the method. Then we do theexperiment to study how the size of oversampling elements affects the errors. Finally, sDPGM for Multiscale Problems
2. Model problem and DG variational formulation.
In this section weintroduce the multiscale model problem and give the DG variational formulation ofthe model problem. First we state some notations and conventions. Throughoutthis paper, the Einstein summation convention is used: summation is taken overrepeated indices. Standard notation on Lebesgue and Sobolev spaces is employed.Subsequently
C, C , C , C , · · · denote generic constants, which are independent of ε, h , unless otherwise stated. We also use the shorthand notation A . B and B . A for the inequality A ≤ CB and B ≤ CA . The notation A h B is equivalent to thestatement A . B and B . A . Let Ω ⊂ R n , n = 2 , −∇ · ( a ε ( x ) ∇ u ε ( x )) = f ( x ) in Ω ,u ε ( x ) = 0 on ∂ Ω , where ǫ ≪ f ∈ L (Ω), and a ε ( x ) = ( a εij ( x )) is a symmetric, positive definite matrix:(2.2) λ | ξ | ≤ a εij ( x ) ξ i ξ j ≤ Λ | ξ | ∀ ξ ∈ R n , x ∈ ¯Ωfor some positive constants λ and Λ. In this subsection, we derive the DG vari-ational formulation of the model problem in the broken Sobolev spaces. Let T h be aquasi-uniform triangulation of the domain Ω. We define h K as diam ( K ) and denoteby h = max K ∈T h h K .We introduce the broken Sobolev spaces for any real number s , H s ( T h ) = { v ∈ L (Ω) : ∀ K ∈ T h , v | K ∈ H s ( K ) } , equipped with the broken Sobolev norm: ||| v ||| H s ( T h ) = (cid:16) X K ∈T h || v || H s ( K ) (cid:17) / . Denote by Γ h the set of interior edges/faces of the T h . With each edge/face e , weassociate a unit normal vector n . If e is on the boundary ∂ Ω, then n is taken to bethe unit outward vector normal to ∂ Ω. F. Song and W. Deng If v ∈ H ( T h ), the trace of v along any side of one element K is well defined.If two elements K e and K e are neighbors and share one common side e , there aretwo traces of v along e . We define the average and jump for v . We assume that thenormal vector n is oriented from K e to K e :(2.3) { v } := v | K e + v | K e , [ v ] := v | K e − v | K e ∀ e = ∂K e ∩ ∂K e . We extend the definition of jump and average to sides that belong to the boundary ∂ Ω: { v } = [ v ] = v | K e ∀ e = ∂K e ∩ ∂ Ω . In the following, we assume that s = 2. Multiplying (2.1) by any v ∈ H s ( T h ),integrating on each element K , and using integration by parts, we obtain Z K a ε ∇ u ε · ∇ v d x − Z ∂K a ε ∇ u ε · n K v d s = Z K f v. We recall that n K is the outward normal to K . Summing over all elements, andswitching to the normal vectors n , we have X K ∈T h Z ∂K a ε ∇ u ε · n K v d s = X e ∈ Γ h ∪ ∂ Ω Z e [ a ε ∇ u ε · n v ] d s. From the regularity of the solution u ε , it follows that X K ∈T h Z K a ε ∇ u ε · ∇ v d x − X e ∈ Γ h ∪ ∂ Ω Z e { a ε ∇ u ε · n } [ v ] d s = Z Ω f v, where we have used the formula [ vw ] = { v } [ w ] + [ v ] { w } and the fact that[ a ε ∇ u ε · n ] = 0.We now define the DG bilinear form a ( · , · ) : H s ( T h ) × H s ( T h ) → R : a ( u, v ) : = X K ∈T h Z K a ε ∇ u · ∇ v d x − X e ∈ Γ h ∪ ∂ Ω Z e { a ε ∇ u · n } [ v ] d s + β X e ∈ Γ h ∪ ∂ Ω Z e [ u ] { a ε ∇ v · n } d s + X e ∈ Γ h ∪ ∂ Ω γ ρ Z e [ u ] [ v ] d s, where β is a real number such as − , , γ is called penalty parameter, and ρ > u ε ∈ H s ( T h ), such that(2.4) a ( u ε , v ) = ( f, v ) ∀ v ∈ H s ( T h ) . sDPGM for Multiscale Problems Remark 2.1.
It is easy to check that if the solution u ε of problem (2.1) belongsto H (Ω) , then u ε satisfies the variational formulation (2.4) . Conversely, if u ε ∈ H (Ω) ∩ H s ( T h ) satisfies (2.4) , then u ε is the solution of problem (2.1) .
3. Multiscale discontinuous Petrov-Galerkin method.
This section is de-voted to the formulations of multiscale discontinuous methods for solving (2.1). Insubsection 3.1, we introduce the oversampling multiscale approximation space definedon the triangulation T h . The formulations of the MsDFE and MsDPG methods arepresented in subsection 3.2. In this subsection, weintroduce the oversampling multiscale approximation space defined on the triangu-lation T h (cf. [20, 40, 51]). Here we only consider the case where n = 2. Forany K ∈ T h with nodes { x Ki } i =1 , let { ϕ Ki } i =1 be the basis of P ( K ) satisfying ϕ Ki ( x Kj ) = δ ij , where δ ij stands for the Kroneckers symbol. For any K ∈ T h , wedenote by S = S ( K ) a macro-element which contains K and d K = dist( ∂S, K ). Weassume that d K ≥ δ h K for some positive constant δ independent of h K . The mini-mum angle of S ( K ) is bounded from below by some positive constant θ independentof h K . In our later numerical experiments, for any coarse-grid element K ∈ T h we putits macro-element S ( K ) in such a way that their barycenters are coincide and theircorresponding edges are parallel. See Figure 6.1 for an illustration.Let ψ Si , i = 1 , , , with ψ Si ∈ H ( S ), be the solution of the problem:(3.1) −∇ · ( a ε ∇ ψ Si ) = 0 in S, ψ Si | ∂S = ϕ Si . Here { ϕ Si } i =1 is the nodal basis of P ( S ) such that ϕ Si ( x Sj ) = δ ij , i, j = 1 , , . The oversampling multiscale basis functions on K are defined by(3.2) ¯ ψ iK = X j =1 c Kij ψ Sj | K in K, with the constants so chosen that(3.3) ϕ Ki = X j =1 c Kij ϕ Sj | K in K. The existence of the constants c Kij is guaranteed because { ϕ Sj } j =1 also forms thebasis of P ( K ). To illustrate the basis functions, we depict two examples of them inFigure 3.1 (cf. [65]) . F. Song and W. Deng
Fig. 3.1 . Example of oversampling basis functions. Left: basis function for periodic media.Right: basis function for random media.
Let OMS( K ) = span { ¯ ψ iK } i =1 be the set of space functions on K . Define theprojection Π K : OMS( K ) → P ( K ) as follows:Π K ψ = c i ϕ Ki if ψ = c i ¯ ψ Ki ∈ OMS( K ) . Further, we introduce the discontinuous piecewise “OMS” approximation spaceand the discontinuous piecewise linear space: V msh,dc = { ψ h ∈ L (Ω) : ψ h | K ∈ OMS( K ) ∀ K ∈ T h } ,V h,dc = { v h ∈ L (Ω) : v h | K ∈ P ( K ) ∀ K ∈ T h } . Here we use the abbreviated indexes ‘ms’, ‘dc’ for multiscale, discontinuous, respec-tively.
In this subsection we presentthe formulations of the MsDFEM and MsDPGM.By use of the DG variational formulation (2.4) and the discontinuous piecewise“OMS” approximation space, we are now ready to define the MsDFE method: Find e u h ∈ V msh,dc such that(3.4) a ( e u h , v h ) = ( f, v h ) ∀ v h ∈ V msh,dc . To define the discrete bilinear form for MsDPGM, we need the transfer operatorΠ h : V msh,dc → V h,dc as following:Π h ψ h | K = Π K ψ h for any K ∈ T h , ψ h ∈ V msh,dc . sDPGM for Multiscale Problems Remark 3.1.
In general, the trial and test functions of PGM are not in the samespace. For example, here we might use V msh,dc and V h,dc as the trial function and testfunction spaces respectively. However, it may result in a difficulty to prove the inf-supcondition of the corresponding bilinear form. Hence, in this paper we introduce thetransfer operator Π h to connect the two spaces, and use it in the bilinear form whichcauses an easy way to establish the stability of the MsDPGM. The idea of connectingthe trial function and test function spaces in the Petrov-Galerkin method through anoperator was introduced in [26] (see also [60]). The discrete bilinear form of MsDPGM on V msh,dc × V msh,dc is defined as: a h ( u h , v h ) := X K ∈T h Z K a ε ∇ u h · ∇ Π h v h d x − X e ∈ Γ h ∪ ∂ Ω Z e { a ε ∇ u h · n } [Π h v h ] d s + β X e ∈ Γ h ∪ ∂ Ω Z e [Π h u h ] { a ε ∇ v h · n } d s + J ( u h , v h ) ,J ( u h , v h ) := X e ∈ Γ h ∪ ∂ Ω γ ρ Z e [Π h u h ] [Π h v h ] d s, where β is a real number such as − , , γ is the penalty parameter, and ρ will bespecified later. Remark 3.2.
It is well known that DG methods utilize discontinuous piecewisepolynomial functions and numerical fluxes, which implies that the weak formulationsubject to discretization must include jump terms across interfaces and that somepenalty terms must be added to control the jump terms. Therefore, the methods needthe restriction on the penalty parameter to ensure stability and convergence in somesense. In fact, the optional convergence is related with the penalty parameter (see[45]). In this paper, our MsDPGM takes advantage of the penalty technique, which isinevitable to consider the choice of the penalty parameter. In our theoretical analysis,the penalty parameter γ is constrained by a large constant from below to ensure thecoercivity of a h . However, in practice, the penalty parameter is chosen through ourexperience. In later numerical tests, we try different choice of the penalty parameterto study its affection on the error. Remark 3.3.
The parameter ρ is chosen as ǫ in our later error analysis, whilein practical computation, it may be chosen as the mesh size h . F. Song and W. Deng
Then, our MsDPG method is: Find u h ∈ V msh,dc such that(3.5) a h ( u h , v h ) = ( f, Π h v h ) ∀ v h ∈ V msh,dc . Remark 3.4.
The design of the last two terms in a h is tricky. As a matter offact, we have tried numerically different possibilities of using Π h ( or not before each u h or v h ) before we found that the current form of a h is the best one and, mostimportantly, the corresponding MsDPGM can be analyzed theoretically. Indeed, ourMsDPGM is some kind of pseudo Petrov-Galerkin formulation of the method that thetest function space is formally the same as the solution space, however some termsinvolve a projection of the multiscale test function into a piecewise linear functionspace. We denote the discrete norm for MsDPGM on V msh,dc , k v h k h, Ω := (cid:16) X K ∈T h Z K a ε ∇ v h · ∇ v h d x + X e ∈ Γ h ∪ ∂ Ω ργ Z e { a ε ∇ v h · n } d s + X e ∈ Γ h ∪ ∂ Ω γ ρ Z e [Π h v h ] d s (cid:17) / . Noting that the operator Π h is not defined for the exact solution u ε , we introducethe following function to measure the error of the discrete solution: E ( v, v h ) := (cid:18) X K ∈T h k ( a ε ) / ∇ ( v − v h ) k L ( K ) + X e ∈ Γ h ∪ ∂ Ω ργ k{ a ε ∇ ( v − v h ) · n }k L ( e ) + X e ∈ Γ h ∪ ∂ Ω γ ρ k [ v − Π h v h ] k L ( e ) (cid:19) / ∀ v ∈ H (Ω) , v h ∈ V msh,dc . (3.6)From the triangle inequality, it is clear that, for any v ∈ H (Ω) , v h , w h ∈ V msh,dc , E ( v, v h ) . E ( v, w h ) + k w h − v h k h, Ω . (3.7)
4. Homogenization results and preliminaries.
In this section we first reviewthe results of classical homogenization theory, and give an important H norm errorestimate between the multiscale solution and the homogenized solution with the firstorder corrector. Then we recall some preliminaries for our following analysis. Hereafter, we assume that a ε ( x ) has theform a ( x/ε ) and a ij ( y ) are sufficiently smooth periodic functions in y with respect to sDPGM for Multiscale Problems
11a unit cube Y . For our analysis it is sufficient to assume that a ij ( y ) ∈ W ,p ( Y ) with p > n .For convenience sake, we take u = u + εχ j ( x/ε ) ∂u ∂x j , where u is the homogenized solution, χ j is the periodic solution of the following cellproblem (cf. [12, 57]):(4.1) −∇ y · ( a ( y ) ∇ y χ j ( y )) = ∇ y · ( a ( y ) e j ) , j = 1 , · · · , n with zero mean, i.e., R Y χ j dy = 0, and e j is the unit vector in the j th direction.The following theorem gives the H semi-norm estimate of the error u ε − u , whichplays a key role in the later error analysis.We arrange the proof in the Appendix A. Theorem 4.1.
Assume that u ∈ H (Ω) . Then the following estimate is valid: (4.2) | u ε − u | H (Ω) . | u | H (Ω) + 1 √ ε | u | W , ∞ (Ω) + ε | u | H (Ω) . In this subsection, we give some preliminaries for our lateranalysis. We first recall the definition of ψ Si , i = 1 , , ψ Si = ϕ Si + εχ j ( x/ε ) ∂ϕ Si ∂x j + εη j ( x ) ∂ϕ Si ∂x j , with η j being the solution of(4.4) −∇ · ( a ε ∇ η j ) = 0 in S, η j | ∂S = − χ j ( x/ε ) . Substituting (4.3) to (3.2), we see that ¯ ψ iK can be expanded as follows:(4.5) ¯ ψ iK = ϕ Ki + εχ j ( x/ε ) ∂ϕ Ki ∂x j + εη j ( x ) ∂ϕ Ki ∂x j . Recall that d K = dist( ∂S, K ), which satisfies: d K ≥ δ h K . Denote by d = min K ∈T h d K . Remark 4.1.
It has been shown in [51, 53] that the distance d K is determinedby the thickness of the boundary layer of η j . Numerically, it has been observed thatthe boundary layer of η j is about O ( ǫ ) thick (see [51]). It was also observed that d K = h K ( > ǫ ) is usually sufficient for eliminating the boundary layer effect. Therefore F. Song and W. Deng in our numerical tests we choose h K as the oversampling size in this paper. To studyhow the size of oversampling elements affects the errors, in Section 6 we include anumerical test which use a series of d K with different δ to compare the correspondingerrors. By the Maximum Principle we have(4.6) (cid:13)(cid:13) η j (cid:13)(cid:13) L ∞ ( S ) ≤ (cid:12)(cid:12) χ j (cid:12)(cid:12) L ∞ ( S ) . , which together with the interior gradient estimate (see [28, Lemma 3.6] or [41, Propo-sition C.1]) imply that(4.7) (cid:13)(cid:13) ∇ η j (cid:13)(cid:13) L ∞ ( K ) . d K . Next, we give a trace inequality which will be used in this paper frequently (see[13, Theorem 1.6.6], [27]).
Lemma 4.1.
Let K be an element of the triangulation T h . Then, for any v ∈ H ( K ) , we have (4.8) k v k L ( ∂K ) ≤ C (cid:16) diam( K ) − / k v k L ( K ) + k v k / L ( K ) k∇ v k / L ( K ) (cid:17) . The following lemma gives some approximation properties of the space OMS(K) (cf.[28, Lemma 4.1]).
Lemma 4.2.
Take φ Kh = P x Ki node of K u ( x Ki ) ¯ ψ Ki ( x ) , ∀ K ∈ T h . Then, the fol-lowing estimates hold: (cid:12)(cid:12) u − φ Kh (cid:12)(cid:12) H ( K ) . h K | u | H ( K ) + εh n/ K d − K | u | W , ∞ ( K ) , (4.9) (cid:13)(cid:13) u − φ Kh (cid:13)(cid:13) L ( K ) . h K | u | H ( K ) + εh n/ K | u | W , ∞ ( K ) , (4.10) (cid:12)(cid:12) u − φ Kh (cid:12)(cid:12) H ( K ) . ε − h K | u | H ( K ) + h n/ K d − K | u | W , ∞ ( K ) + ε | u | H ( K ) . (4.11)Moreover, we recall the stability estimate for Π K , which will be used in our lateranalysis (cf. Lemma 3.2 in [65]). Lemma 4.3.
There exist positive constants γ , α and α which are independentof h and ε such that if ε/h K ≤ γ for all K ∈ T h , then the following estimates arevalid for any v h ∈ OMS ( K ) , k∇ v h k L ( K ) h k∇ Π K v h k L ( K ) , (4.12) α k∇ v h k L ( K ) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z K a ε ∇ v h · ∇ Π K v h d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ α k∇ v h k L ( K ) . (4.13) sDPGM for Multiscale Problems K ) [28,Lemma 5.2]. Lemma 4.4.
Under the assumptions of Lemma 4.3, and assuming ε . h K . d K ,we have (4.14) | v h | H ( K ) . ε k∇ v h k L ( K ) ∀ v h ∈ OM S ( K ) .
5. Main results.
In this section we only carry out the convergence analysis ofMsDPGM. For the case of MsDFEM, similar results can be obtained by the sameargument and are arranged in the Appendix B for convenience of the reader. ForMsDPGM, we first show the stability of the bilinear form guaranteeing the existenceand uniqueness of the solution, and then prove the error estimate with β = − , ρ = ε .For other cases such as β = 0 ,
1, the analysis is similar and is omitted here.
Westart by establishing the stability of the bilinear form of the MsDPGM.
Theorem 5.1.
We have (5.1) | a h ( u h , v h ) | ≤ C k u h k h, Ω k v h k h, Ω ∀ u h , v h ∈ V msh,dc . Further, let the assumptions of Lemma 4.4 be fulfilled and γ is large enough, then (5.2) a h ( v h , v h ) ≥ κ k v h k h, Ω ∀ v h ∈ V msh,dc , where κ > is a constant independent of h, ε, γ .Proof . From the definition of the norms, the Cauchy-Schwarz inequality andLemma 4.3, it follows (5.1) immediately.Next we prove (5.2). From (4.13), we get a h ( v h , v h ) ≥ C X K ∈T h (cid:13)(cid:13)(cid:13) ( a ε ) / ∇ v h (cid:13)(cid:13)(cid:13) L ( K ) − X e ∈ Γ h ∪ ∂ Ω Z e { a ε ∇ v h · n } [Π h v h ] d s + X e ∈ Γ h ∪ ∂ Ω γ ε k [Π h v h ] k L ( e ) . It is easy to see that,2 X e ∈ Γ h ∪ ∂ Ω Z e { a ε ∇ v h · n } [Π h v h ] d s ≤ X e ∈ Γ h ∪ ∂ Ω k{ a ε ∇ v h · n }k L ( e ) k [Π h v h ] k L ( e ) ≤ X e ∈ Γ h ∪ ∂ Ω γ ε k [Π h v h ] k L ( e ) + X e ∈ Γ h ∪ ∂ Ω εγ k{ a ε ∇ v h · n }k L ( e ) . F. Song and W. Deng
Hence, we obtain a h ( v h , v h ) ≥ C X K ∈T h (cid:13)(cid:13)(cid:13) ( a ε ) / ∇ v h (cid:13)(cid:13)(cid:13) L ( K ) − X e ∈ Γ h ∪ ∂ Ω γ ε k [Π h v h ] k L ( e ) − X e ∈ Γ h ∪ ∂ Ω εγ k{ a ε ∇ v h · n }k L ( e ) + X e ∈ Γ h ∪ ∂ Ω γ ε k [Π h v h ] k L ( e ) = C X K ∈T h (cid:13)(cid:13)(cid:13) ( a ε ) / ∇ v h (cid:13)(cid:13)(cid:13) L ( K ) + 12 X e ∈ Γ h ∪ ∂ Ω γ ε k [Π h v h ] k L ( e ) + 12 X e ∈ Γ h ∪ ∂ Ω εγ k{ a ε ∇ v h · n }k L ( e ) − X e ∈ Γ h ∪ ∂ Ω εγ k{ a ε ∇ v h · n }k L ( e ) . (5.3)By use of Lemmas 4.1, 4.4 and ε . h , we have(5.4) εγ k{ a ε ∇ v h · n }k L ( e ) ≤ C γ (cid:13)(cid:13)(cid:13) ( a ε ) / ∇ v h (cid:13)(cid:13)(cid:13) L ( K ) . Therefore, from (5.3) and (5.4), we have a h ( v h , v h ) ≥ (cid:18) C − C γ (cid:19) X K ∈T h (cid:13)(cid:13)(cid:13) ( a ε ) / ∇ v h (cid:13)(cid:13)(cid:13) L ( K ) + 12 X e ∈ Γ h ∪ ∂ Ω εγ k{ a ε ∇ v h · n }k L ( e ) + 12 X e ∈ Γ h ∪ ∂ Ω γ ε k [Π h v h ] k L ( e ) , where γ is large enough such that C γ < C . Then by choosing κ = min( C , ), itfollows (5.2). This completes the proof.Theorem 5.1 guarantees that there exists a unique solution to our MsDPGM. Nowwe establish an analogue of C´ea lemma written in the following theorem: Theorem 5.2.
For large enough γ , the following inequality holds: (5.5) E ( u ε , u h ) . inf v h ∈ V msh,dc E ( u ε , v h ) , where the error function E is defined in (3.6) .Proof . It is clear that by Theorem 5.1 we have k u h − v h k h, Ω . a h ( u h − v h , u h − v h )= a h ( u h , u h − v h ) − a h ( v h , u h − v h )=( f, Π h ( u h − v h )) − a h ( v h , u h − v h ) . sDPGM for Multiscale Problems f, Π h ( u h − v h )) = X K ∈T h Z K a ε ∇ u ε · ∇ Π h ( u h − v h ) d x − X e ∈ Γ h ∪ ∂ Ω Z e { a ε ∇ u ε · n } [Π h ( u h − v h )] d s. Then, using the facts that [ u ε ] = 0 and [ a ε ∇ u ε · n ] = 0, we have( f, Π h ( u h − v h )) − a h ( v h , u h − v h )= X K ∈T h Z K a ε ∇ ( u ε − v h ) · ∇ Π h ( u h − v h ) d x − X e ∈ Γ h ∪ ∂ Ω Z e { a ε ∇ ( u ε − v h ) · n } [Π h ( u h − v h )] d s + X e ∈ Γ h ∪ ∂ Ω Z e { a ε ∇ ( u h − v h ) · n } [Π h v h − u ε ] d s − X e ∈ Γ h ∪ ∂ Ω γ ε Z e [Π h v h − u ε ] [Π h ( u h − v h )] d s . E ( u ε , v h ) k u h − v h k h, Ω . Therefore, we obtain k u h − v h k h, Ω . E ( u ε , v h ) , which together with (3.7) yield E ( u ε , u h ) . E ( u ε , v h ) + k u h − v h k h, Ω . E ( u ε , v h ) . The proof is completed.
We present the main result ofthe paper which gives the error estimate of the MsDPGM.
Theorem 5.3.
Let u ε be the solution of (2.1) , and let u h be the numericalsolution computed using MsDPGM defined by (3.5) . Assume that u ∈ H (Ω) , f ∈ L (Ω) , and that ε . h . d , and that the penalty parameter γ is large enough. Thenthere exits a constant γ independent of h and ε such that if ε/h K ≤ γ for all K ∈ T h ,the following error estimate holds: (5.6) E ( u ε , u h ) . √ ε + εd + h + h / √ ε , where d = min K ∈T h d K . F. Song and W. Deng
Proof . According to Theorem 5.2, the proof is devoted to estimating the interpo-lation error. To do this, we define ψ h by(5.7) ψ h | K = φ Kh = X x Ki node of K u ( x Ki ) ¯ ψ Ki ( x ) ∀ K ∈ T h . Clearly, ψ h ∈ V msh,dc . It is easy to see thatΠ K φ Kh = I h u | K , where I h : H s ( T h ) → V h,dc is the Lagrange interpolation operator. Then we set v h as ψ h . It is shown that in [28], (cid:18) X K ∈T h k ( a ε ) / ∇ ( u ε − v h ) k L ( K ) (cid:19) / . h | u | H (Ω) + √ ε | u | W , ∞ (Ω) + εd | u | W , ∞ (Ω) . (5.8)Next, we estimate the term X e ∈ Γ h ∪ ∂ Ω εγ k{ a ε ∇ ( u ε − v h ) · n }k L ( e ) := I . From (4.8), we haveI . εh − k∇ ( u ε − u ) k L (Ω) + εh − X K ∈T h k∇ ( u − ψ h ) k L ( K ) + ε k∇ ( u ε − u ) k L (Ω) (cid:13)(cid:13) ∇ ( u ε − u ) (cid:13)(cid:13) L (Ω) + ε (cid:16) X K ∈T h k∇ ( u − ψ h ) k L ( K ) (cid:17) / (cid:16) X K ∈T h (cid:13)(cid:13) ∇ ( u − ψ h ) (cid:13)(cid:13) L ( K ) (cid:17) / . Therefore, it follows from Theorem 4.1, Lemma 4.2 and the assumption ε . h . d that,(5.9) I . h | u | H (Ω) + ε | u | W , ∞ (Ω) + ε | u | H (Ω) , where we have used ε √ h < √ ε and the Young’s inequality to derive the above inequal-ity. It remains to consider the term P e ∈ Γ h ∪ ∂ Ω γ ε k [ u ε − Π h v h ] k L ( e ) . Noting thatboth u ε and u are continuous functions, we have X e ∈ Γ h ∪ ∂ Ω γ ε k [ u ε − Π h v h ] k L ( e ) = X e ∈ Γ h ∪ ∂ Ω γ ε Z e [ u − Π h v h ] d s . X e ∈ Γ h ∪ ∂ Ω γ ε Z e ( u − Π h ψ h ) d s := II . sDPGM for Multiscale Problems Z e ( u − Π h ψ h ) d s = Z e ( u − I h u ) d s . h − k u − I h u k L ( K ) + k u − I h u k L ( K ) k∇ ( u − I h u ) k L ( K ) . h | u | H ( K ) , which yields(5.10) II . h ε | u | H (Ω) . Hence, from (5.8)–(5.10), it follows (5.6) immediately.
6. Numerical experiments.
In this section, we present numerical experimentsto confirm the theoretical results in Section 5. We show the numerical results ofMsDPGM defined in (3.5), and also results of MsDFEM defined in (3.4) which showgood performance as well as MsDPGM. In order to illustrate the accuracy of ourmethods, we also implement the standard MsFEM in Petrov–Galerkin formulationwhich is denoted as MsPGM, and the MsPGM which uses the classical oversamplingmultiscale basis (OMsPGM). We also show the results of the traditional linear finiteelement method (FEM) and discontinuous finite element method (DFEM) on thecorresponding coarse grid to get a feeling for the accuracy of the multiscale methods.All numerical experiments are designed to show better performance of MsDPGM thanthe other MsPGMs.In all tests, for simplicity, we use the standard triangulation which is constructedby first dividing the domain Ω into sub-squares of equal length h and then connectingthe lower-left and the upper-right vertices of each sub-square. For any coarse-gridelement K ∈ T h we put its macro-element S ( K ) in such a way that their barycentersare coincide and their corresponding edges are parallel. The length of the horizontaland vertical edges of S ( K ) is four times of the corresponding length of the edges of K .We assume that all right-angle sides of S ( K ) , K ∈ T h have the same length denotedby h S . Recall the definition of the d = min K ∈T h d K , define˜ d = ( h S − h ) / . (6.1)It is clear that d h ˜ d . See Figure 6.1 for an illustration.In all of these computations, we have used finely resolved numerical solutionsobtained using the traditional linear finite element method with mesh size h f = 1 / F. Song and W. Deng (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)
K S ( K ) KS ( K )˜ dh S h K Fig. 6.1 . The element K and its oversampling element S ( K ) : lower-right elements (left) andupper-left elements (right). as the “exact” solutions which are denoted as u e . Denoting u h as the numericalsolutions computed by the methods considered in this section, we measure the relativeerror in L , L ∞ and energy norms as following: k u h − u e k L k u e k L , k u h − u e k L ∞ k u e k L ∞ , k u h − u e k ,h k u e k ,h , where || v || ,h := X K ∈T h k ( a ε ) / ∇ v k L ( K ) + k v k L (Ω) ! / . In all tests, the coefficient a ε is chosen as the form a ε = a ε I where a ε is a scalarfunction and I is the 2 by 2 identity matrix. We first consider the model problem (2.1) in the squared domain Ω = [0 , × [0 , f = 1 and the coefficient a ε ( x , x ) has the following periodic form:(6.2) a ε ( x , x ) = 2 + 1 . πx /ǫ )2 + 1 . πx /ǫ ) + 2 + 1 . πx /ǫ )2 + 1 . πx /ǫ ) , where we fix ε = 1 / h = 1 /
32 and report errors in the L , L ∞ and energy normsin Table 6.1. We can see that the MsDPGM and MsDFEM give more accurate resultsthan the other multiscale methods considered here, while the FEM and DFEM giveworse approximations to the gradient of solution. We also compare the CPU time T and T spent by the MsDFEM and MsDPGM to show the good performance ofMsDPGM in computational complexity, where T is the CPU time of assembling thestiffness matrix, and T is the CPU time of solving the discrete system of algebraicequations. We can observe that the CPU time T of our MsDPGM for assembling the sDPGM for Multiscale Problems Table 6.1
Compare different methods to show the accuracy of MsDPGM in periodic case given by (6.2) . ρ = ε = 1 / , ˜ d = h = 1 / , γ = 20 . Relative error L L ∞ Energy norm CPU time(s) T T FEM 0.1150e-00 0.2311e-00 0.8790e-00 – –DFEM 0.2667e-00 0.2634e-00 0.5498e-00 – –MsPGM 0.7448e-01 0.7342e-01 0.2929e-00 – –OMsPGM 0.1430e-01 0.1521e-01 0.1641e-00 – –MsDFEM 0.1007e-01 0.1029e-01 0.1629e-00 1.300 0.028MsDPGM 0.1266e-01 0.1395e-01 0.1631e-00 1.119 0.027Secondly, we do an experiment to study how the penalty parameter γ affects theerrors. We fix ρ = ε = 1 / , ˜ d = h = 1 /
32 and choose a series of γ in the test. Theresult is shown in Table 6.2. We observe that as γ goes larger, the relative error isclose to the error of the OMsPGM. It seems that MsDPGM converges to OMsPGMas the penalty parameter γ goes to infinity (cf. [58]). Table 6.2
Convergence with respect to γ . ρ = ε = 1 / , ˜ d = h = 1 / . Relative Error L L ∞ Energy norm γ = 10 0.1100e-01 0.1266e-01 0.1637e-00 γ = 20 0.1266e-01 0.1395e-01 0.1631e-00 γ = 100 0.1397e-01 0.1496e-01 0.1638e-00 γ = 1000 0.1426e-01 0.1519e-01 0.1641e-00 γ = 10000 0.1429e-01 0.1521e-01 0.1641e-00The third numerical experiment is to show the mesh size h plays a role as thatdescribing in Theorem 5.3. We fix ˜ d = 1 /
32 and ε = 1 / h = 1 / , / , / , /
8. The results are shown in Table 6.3. Relativeerror in energy norm against the mesh size h is clearly shown in Figure 6.2. It is easyto see that as h goes larger, the relative error in energy norm goes larger, which is in0 F. Song and W. Deng agreement with the theoretical results in Theorem 5.3. We remark that the classicalMsFEM suffers from the resonance error since the H –error estimate has the term ǫ/h due to the nonconforming error (see [52]). But for MsDPGM, the error estimatein Theorem 5.3, and the numerical results in Table 6.3 and Figure 6.2 show that theresonance error has been removed completely. Table 6.3
Error with respect to h . ρ = ε = 1 / , ˜ d = 1 / , γ = 20 . Relative error L L ∞ Energy norm h = 1 /
64 0.1371e-01 0.1474e-01 0.1593e-00 h = 1 /
32 0.1266e-01 0.1395e-01 0.1631e-00 h = 1 /
16 0.1948e-01 0.2532e-01 0.1870e-00 h = 1 / R e l a t i v e E rr o r i n E ne r g y N o r m Fig. 6.2 . Relative error with respect to h . In this subsectionwe study how the size of oversampling elements affects the error. The experimentto verify the inequality (4.7) for the model example with coefficient (6.2) has beendone in [65]. The figures have been shown that (cid:13)(cid:13) ∇ η j (cid:13)(cid:13) L ∞ ( K ) · d K are bounded by aconstant which is consistent with (4.7) (see Figure 5 in [65]).The following numerical experiment is to show how the oversampling size affectsthe error. Recalling the requirement of the oversampling size d K ≥ δ h K , we show therelative oversampling size δ against the error. Note the distance d = min K ∈T h d K .Here it is equivalent to d ≥ δ h . We set ρ = ε = 1 / , h = 1 /
32. The result isshown in Table 6.4. We can see that as δ (equivalently ˜ d ) goes larger, the relative sDPGM for Multiscale Problems d is close to √ ε , the errors begin to decreasevery slowly. Recall that there is a homogenization error √ ε in the error estimate (5.6).We think that when d is large enough, √ ε becomes the dominated error instead of ε/d . Table 6.4
Error with respect to δ . ρ = ε = 1 / , h = 1 / , γ = 20 . Relative error L L ∞ Energy norm δ = 1 /
32 0.4304e-01 0.4423e-01 0.2184e-00 δ = 1 /
16 0.2924e-01 0.3172e-01 0.1893e-00 δ = 1 / δ = 1 / δ = 1 / δ = 1 0.1266e-01 0.1295e-01 0.1631e-00 δ = 2 0.1197e-01 0.1398e-01 0.1631e-00 We con-sider the multiscale problem on the L–shaped domain of Figure 6.3 with Dirichletboundary condition so chosen that the true solution is u = r sin(2 θ/
3) in polar co-ordinates. It is known that the solution has the singular behavior around reentrantcorners. So the classical finite element method fails to provide satisfactory result. −0.5 0 0.5−0.500.5
Fig. 6.3 . The L–shape domain.
Firstly, we simulate the problem with coefficient given by (6.2). We fix ε = 1 / F. Song and W. Deng and choose h = 1 /
16. The relative error is shown in Table 6.5. We observe that bothMsDPGM and MsDFEM give better approximation than the other MsPG methods.
Table 6.5
Relative errors in the L , L ∞ and energy norm for the L–shaped problem with periodic coef-ficient (6.2). ρ = ε = 1 / , ˜ d = h = 1 / , γ = 20 . Relative error L L ∞ Energy normMsPGM 0.7765e-02 0.3635e-01 0.2014e-00OMsPGM 0.6285e-02 0.3277e-01 0.1035e-00MsDFEM 0.3903e-02 0.2244e-01 0.9260e-01MsDPGM 0.4654e-02 0.2299e-01 0.9275e-01Secondly, we simulate the problem with the random log-normal permeability field a ( x ), which is generated by using the moving ellipse average [32] with the variance ofthe logarithm of the permeability σ = 1 .
0, and the correlation lengths l = l = 0 . x and x directions, respectively. One realization of the resulting permeabilityfield is depicted in Figure 6.4, where a max ( x ) a min ( x ) = 2 . e + 003. In this test, we set ρ = h = 1 /
16 since there is no explicit ε in the example. The result is shown in Table6.6. We can see that MsDPGM gives a better approximation than the other MsPGmethods, while standard MsPGM gives the wrong approximation to the gradient ofsolution. −0.5 0 0.5−0.500.5 510152025 Fig. 6.4 . The random log-normal permeability field a ( x ) . a max ( x ) a min ( x ) = 2 . e + 003 .
7. Conclusion.
In this paper, we have proposed a new Petrov–Galerkin methodbased on the discontinuous multiscale approximation space for the multiscale elliptic sDPGM for Multiscale Problems Table 6.6
Relative errors in the L , L ∞ and energy norm for the L–shaped problem with random coeffi-cient σ = 1 . and l = l = 0 . . ˜ d = ρ = h = 1 / , γ = 20 . Relative error L L ∞ Energy normMsPGM 0.9074e-00 0.1290e+01 0.6601e+02OMsPGM 0.9307e-02 0.3851e-01 0.1428e-00MsDFEM 0.6504e-02 0.3718e-01 0.9931e-01MsDPGM 0.8587e-02 0.3810e-01 0.1013e-00problems. Under some assumptions on the coefficients, we give the error analysis ofour method. The H –error is of the order O (cid:16) √ ε + εd + h + h / √ ε (cid:17) , which consists of the oversampling multiscale approximation error and the error con-tributed by the penalty. Note that the unpleasant resonance error does not appear.The reason is that our method uses discontinuous piecewise linear functions as testfunctions, which is only needed to estimate the interpolation error. Several numer-ical experiments have demonstrated the efficiency of MsDPGM. We also study thecorresponding MsDFEM which coupling the classical oversampling multiscale basiswith DGM. Our convergence analysis shows that MsDFEM can also eliminate theresonance error completely. That is the reason why MsDFEM is working as well asMsDPGM (even a little better). Furthermore, we can see that the CPU-time cost ofMsDPGM for assembling the stiffness matrix is shorter than that of the MsDFEMdue to its PG version. Therefore, we think that MsDPGM is a good choice whenwe need to take into consideration of the computational accuracy and the computerresource at the same time.We emphasize that the proposed method is not restrict to the periodic case. Thenumerical experiments show that it is applicable to the random coefficient case verywell. However, with the classical oversampling multiscale basis function space intro-duced in [51], the error estimate method is based on the classical homogenizationtheory, which needs the assumption that the oscillating coefficient is periodic. Inthe future work, we would like to combine the Petrov–Galerkin method with the newoversampling multiscale space [50] to consider the elliptic multiscale problems withoutany assumption on scale separation or periodicity. Besides, the introduced method4 F. Song and W. Deng may be inefficient for the multiscale problems which have some singularities, such as,the Dirac function singularities which stems from the simulation of steady flow trans-port through highly heterogeneous porous media driven by extraction wells [21], orhigh-conductivity channels that connect the boundaries of coarse-grid blocks [37]. Tosolve these problems, it needs some special definition of the multiscale basis functionsaround the channels such as the local spectral basis functions (see [37]), or local re-finement of the elements near the channels (see [28]). We will couple these techniqueswith the introduced method in our future work. Finally, we remark that GeneralizedMultiscale Finite Element method coupling DGM was explored in [36]. The computa-tion is divided into two stages: offline and online. In the offline stage, they construct areduced dimensional multiscale space to be used for rapid computations in the onlinestage. In the online stage, they use the basis functions computed offline to solve theproblem for current realization of the parameters. Similar to MsDPGM, in the onlinestage we can use the Petrov–Galerkin version of DGM to solve the problem with thebasis functions computed offline, which leads to a kind of Generalized Multiscale Dis-continuous Petrov-Galerkin method. The difficulty is the choice of the test functionspace and the proof of inf-sup condition, which is worth studying.
Acknowledgments.
The authors would like to thank the referees for their care-fully reading and constructive comments that improved the paper.
Appendix A. Proof of Theorem 4.1.
The following theorem plays an important role in our analysis (cf. [19, 20]).
Theorem A.1.
Assume that u ∈ H (Ω) ∩ W , ∞ (Ω) . There exists a constant C independent of u , ε, Ω such that || u ε − u − εθ ε || H (Ω) ≤ Cε | u | H (Ω) , || εθ ε || H (Ω) ≤ C √ ε | u | W , ∞ (Ω) + Cε | u | H (Ω) , where θ ε denote the boundary corrector defined by (A.1) −∇ · ( a ε ∇ θ ε ) = 0 in Ω ,θ ε = − χ j ( x/ε ) ∂u ( x ) ∂x j on ∂ Ω . We first estimate | εθ ε | H (Ω) . sDPGM for Multiscale Problems Lemma A.1.
Assume that u ∈ H (Ω) ∩ W , ∞ (Ω) . Then the following estimateholds: (A.2) | εθ ε | H (Ω) . √ ε | u | W , ∞ (Ω) + | u | H (Ω) . Proof . We only consider the case where n = 2. For n = 3, the proof is similar.Let ξ ∈ C ∞ ( R ) be the cut-off function such that 0 ≤ ξ ≤ , ξ = 1 in Ω \ Ω ε/ , ξ = 0in Ω ε , and |∇ ξ | ≤ C/ε, (cid:12)(cid:12) ∇ ξ (cid:12)(cid:12) ≤ C/ε in Ω, where Ω ε := { x : dist { x, ∂ Ω } ≥ ε } . Then v = θ ε + ξ ( χ j ∂u ∂x j ) ∈ H (Ω)satisfies(A.3) −∇ · ( a ε ∇ v ) = −∇ · ( a ε ∇ ( ξχ j ∂u ∂x j )) in Ω , v | ∂ Ω = 0 . By use of Theorem 4.3.1.4 in [49] , and together with Theorem A.1, we have | v | H (Ω) . ε k v k L (Ω) + (cid:13)(cid:13)(cid:13)(cid:13) ∇ · ( a ε ∇ ( ξχ j ∂u ∂x j )) (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) . √ εε | u | W , ∞ (Ω) + 1 ε | u | H (Ω) , which implies(A.4) | θ ε | H (Ω) . √ εε | u | W , ∞ (Ω) + 1 ε | u | H (Ω) . This completes the proof.
Proof of Theorem 4.1.
It is shown that, for any ϕ ∈ H (Ω) (see [19, p.550] or [20,p.125]), ( a ( x/ε ) ∇ ( u ε − u ) , ∇ ϕ ) Ω = ( a ∗ ∇ u , ∇ ϕ ) Ω − (cid:18) a ( x/ε ) ∇ (cid:18) u + εχ k ∂u ∂x k (cid:19) , ∇ ϕ (cid:19) Ω = ε Z Ω a ij ( x/ε ) χ k ∂ u ∂x j ∂x k ∂ϕ∂x i d x − ε Z Ω α kij ( x/ε ) ∂ u ∂x j ∂x k ∂ϕ∂x i d x, (A.5)where α k ( x/ε ) = ( α kij ( x/ε )) are skew-symmetric matrices which satisfy that (see [57,p.6]) G ki ( y ) = ∂∂y j ( α kij ( y )) , Z Y α kij ( y ) d y = 06 F. Song and W. Deng with G ki = a ∗ ik − a ij (cid:18) δ kj + ∂χ k ∂y j (cid:19) . From (A.5), it follows that, ∇ · ( a ( x/ε ) ∇ ( u ε − u )) = ε ∂∂x i (cid:18) a ij ( x/ε ) χ k ∂ u ∂x j ∂x k − α kij ( x/ε ) ∂ u ∂x j ∂x k (cid:19) , which combines the definition of θ ε yield ∇ · ( a ( x/ε ) ∇ ( u ε − u − εθ ε )) = ε ∂∂x i (cid:18) a ij ( x/ε ) χ k ∂ u ∂x j ∂x k − α kij ( x/ε ) ∂ u ∂x j ∂x k (cid:19) . Thus, from Theorem 4.3.1.4 in [49], it follows that(A.6) | u ε − u − εθ ε | H (Ω) . ε k u ε − u k L (Ω) + | u | H (Ω) + ε | u | H (Ω) , which combing (A.2) and Theorem A.1, yield (4.2) immediately. Appendix B. Theoretical Results of MsDFEM.
We give some theoretical results of MsDFEM here for convenience of the reader.Detailed analysis can be found in the first author’s PHD thesis [64].
Lemma B.1.
We have (B.1) | a (˜ u h , v h ) | ≤ C k ˜ u h k E k v h k E ∀ ˜ u h , v h ∈ V msh,dc . Further, let the assumptions of Lemma 4.4 be fulfilled and γ is large enough, then (B.2) a ( v h , v h ) ≥ k v h k E ∀ v h ∈ V msh,dc . Here k v k E := (cid:16) X K ∈T h Z K a ε |∇ v | d x + X e ∈ Γ h ∪ ∂ Ω ργ Z e { a ε ∇ v · n } d s + X e ∈ Γ h ∪ ∂ Ω γ ρ Z e [ v ] d s (cid:17) / ∀ v ∈ V msh,dc . Using the definition of the above norm, the Cauchy-Schwarz inequality and (4.8),Lemma 4.4, we can obtain (B.1) and (B.2) immediately. The proof is similar toTheorem 5.1 and is omitted here.
Theorem B.1.
Let u ε be the solution of (2.1) , and let ˜ u h be the numericalsolution computed by MsDFEM defined in (3.4). Assume that u ∈ H (Ω) , f ∈ L (Ω) , ε . h . d , and that the penalty parameter γ is large enough. Then there exits a sDPGM for Multiscale Problems constant γ independent of h and ε such that if ε/h K ≤ γ for all K ∈ T h , the followingerror estimate holds: (B.3) k u ε − ˜ u h k E . h + h / √ ε + √ ε + εd , where d = min K ∈T h d K .Proof . By use of the Galerkin orthogonality of a ( · , · ), we only need to estimatethe interpolation error.Take v h as ψ h (see (5.7)). The following two estimates of the error have beenshown in the proof of Theorem 5.3:(B.4) (cid:16) X K ∈T h (cid:13)(cid:13)(cid:13) ( a ε ) / ∇ ( u ε − v h ) (cid:13)(cid:13)(cid:13) L ( K ) (cid:17) . h | u | H (Ω) + (cid:16) √ ε + εd (cid:17) | u | W , ∞ (Ω) , and X e ∈ Γ h ∪ ∂ Ω εγ k{ a ε ∇ ( u ε − v h ) · n }k L ( e ) . h | u | H (Ω) + ε | u | W , ∞ (Ω) + ε | u | H (Ω) . (B.5)It remains to consider the term P e ∈ Γ h ∪ ∂ Ω γ ε k [ u ε − v h ] k L ( e ) . Noting that [ u ε ] =[ u ] = 0, then by use of the trace inequality (4.8) and Lemma 4.2, we have X e ∈ Γ h ∪ ∂ Ω γ ε k [ u ε − v h ] k L ( e ) . X e ∈ Γ h ∪ ∂ Ω γ ε k [ u − v h ] k L ( e ) . ε − h − X K ∈T h k u − v h k L ( K ) + ε − (cid:16) X K ∈T h k u − v h k L ( K ) (cid:17) / (cid:16) X K ∈T h k∇ ( u − v h ) k L ( K ) (cid:17) / . h ε | u | H (Ω) + ε | u | W , ∞ (Ω) , (B.6)where we have used the assumption ε . h . d and the Young’s inequality to derivethe above inequality.Hence, from (B.4), (B.5) and (B.6), it follows (B.3) immediately. This completesthe proof. REFERENCES[1]
J. E. Aarnes , On the use of a mixed multiscale finite element method for greater flexibilityand increased speed or improved accuracy in reservoir simulation , SIAM MMS, 2 (2004),pp. 421–439. F. Song and W. Deng[2]
J. E. Aarnes and Y. Efendiev , Mixed multiscale finite element methods for stochastic porousmedia flows , SIAM J. Sci. Comput., 30 (2008), pp. 2319–2339.[3]
J. E. Aarnes, Y. Efendiev, and L. Jiang , Mixed multiscale finite element methods usinglimited global information , SIAM Multiscale Model. Simul., 7 (2008), pp. 655–676.[4]
J. E. Aarnes and B.-O. Heimsund , Multiscale discontinuous Galerkin methods for ellipticproblems with multiple scales , Multiscale methods in science and engineering, Lect. NotesComput. Sci. Eng., Springer, Berlin, (2005), pp. 1–20.[5]
A. Abdulle , Multiscale method based on discontinuous Galerkin methods for homogenizationproblems , C. R. Math. Acad. Sci. Paris, 346 (2008), pp. 97–102.[6]
A. Abdulle and M. E. Huber , Discontinuous Galerkin finite element heterogeneous multiscalemethod for advection-diffusion problems with multiple scales , Numer. Math., 126 (2014),pp. 589–633.[7]
D. Arnold , An interior penalty finite element method with discontinuous elements , SIAM J.Numer. Anal., 19 (1982), pp. 742–760.[8]
D. Arnold, F. Brezzi, B. Cockburn, and D. Marini , Unified analysis of discontinuousGalerkin methods for elliptic problems , SIAM J. Numer. Anal., 39 (2001), pp. 1749–1779.[9]
I. Babuˇska , The finite element method with penalty , Math. Comp, 27 (1973), pp. 221–228.[10]
I. Babuˇska and M. Zl´amal , Nonconforming elements in the finite element method withpenalty , SIAM J. Numer. Anal., 10 (1973), pp. 863–875.[11]
P. Bastian and C. Engwer , An unfitted finite element method using discontinuous Galerkin ,Int. J. Numer. Meth. Engng, 79 (2009), pp. 1557–1576.[12]
A. Bensoussan, J. L. Lions, and G. Papanicolaou , Asymptotic Analysis for Periodic Struc-ture , vol. 5 of Studies in Mathematics and Its Application, North-Holland Publ., 1978.[13]
S. C. Brenner and L. R. Scott , The mathematical theory of finite element methods , Springer-Verlag, New York, 2002.[14]
F. Brezzi, L. P. Franca, T. J. R. Hughes, and A. Russo , b = R g , Comput. Methods Appl.Mech. Engrg., 145 (1997), pp. 329–339.[15] F. Brezzi, G. Manzini, D. Marini, P. Pietra, and A. Russo , Discontinuous Galerkin approx-imations for elliptic problems , Numer. Methods Partial Differential Equations, 4 (2000),pp. 365–378.[16]
F. Brezzi, D. Marini, and E. S ¨ u li , Residual-free bubbles for advection-diffusion problems:the general error analysis , Numer.Math., (2000), pp. 31–47.[17]
P. Castillo , A review of the Local Discontinuous Galerkin (LDG) method applied to ellipticproblems , Applied Numerical Mathematics, 56 (2006), pp. 1307 – 1313.[18]
Z. Chen, W. B. Deng, and H. Ye , A new upscaling method for the solute transport equations ,Discrete and Continuous Dynamical Systems-series A, (2005), pp. 941–960.[19]
Z. Chen and T. Y. Hou , A mixed multiscale finite method for elliptic problems with oscillatingcoefficients , Math. Comp., 72 (2002), pp. 541–576.[20]
Z. Chen and H. Wu , Selected topics in finite element method , Science Press, Beijing, 2010.[21]
Z. Chen and X. Y. Yue , Numerical homogenization of well singularities in the flow transportthrough heterogeneous porous media , SIAM MMS, 1 (2003), pp. 260–303.[22]
Z. X. Chen , Multiscale methods for elliptic homogenization problems , Numer. Methods PartialsDPGM for Multiscale Problems Differential Equations, 2 (2006), pp. 317–360.[23]
Z. X. Chen and H. Chen , Pointwise error estimates of discontinuous Galerkin methods withpenalty for second-order elliptic problems , SIAM J. Numer. Anal., 3 (2004), pp. 1146–1166.[24]
Z. X. Chen, M. Cui, T. Y. Savchuk, and X. Yu , The multiscale finite element method withnonconforming elements for elliptic homogenization problems , Multiscale Model. Simul.,2 (2008), pp. 517–538.[25]
Z. X. Chen and T. Y. Savchuk , Analysis of the multiscale finite element method for nonlinearand random homogenization problems , SIAM J. Numer. Anal., 1 (2007/08), pp. 260–279.[26]
S. Chou , Analysis and convergence of a covolume method for the generalized stokes problem ,Mathematics of Computation, 66 (1997), pp. 85–104.[27]
P. G. Ciarlet , The finite element method for elliptic problems , North Holland, Amsterdam,1978.[28]
W. Deng and H. Wu , A combined finite element and multiscale finite element method for themultiscale elliptic problems , SIAM Multiscale Model. Simul., 12 (2014), pp. 1424–1457.[29]
W. Deng, X. Yun, and C. Xie , Convergence analysis of the multiscale method for a class ofconvection-diffusion equations with highly oscillating coefficients , Appl. Numer. Math., 59(2009), pp. 1549–1567.[30]
M. Dorobantu and B. Engquist , Wavelet-based numerical homogenization , SIAM J. Numer.Anal., 35 (1998), pp. 540–559.[31]
J. Douglas Jr and T. Dupont , Interior Penalty Procedures for Elliptic and ParabolicGalerkin methods , Lecture Notes in Phys. 58, Springer-Verlag, Berlin, 1976.[32]
L. Durlofsky , Numerical calculation of equivalent grid block permeability tensors for hetero-geneous porous media , Water Resources Research, 27 (1991), pp. 699–708.[33]
W. E and B. Engquist , The heterogeneous multiscale methods , Commun. Math. Sci., 1 (2003),pp. 87–132.[34] ,
Multiscale modeling and computation , Notice Amer. Math. Soc., 50 (2003), pp. 1062–1070.[35]
W. E, P. Ming, and P. Zhang , Analysis of the heterogeneous multiscale method for elliptichomogenization problems , J. Am. Math. Soc., 18 (2005), pp. 121–156.[36]
Y. Efendiev, J. Galvis, R. Lazarov, and M. Moon , Generalized multiscale finite elementmethod. Symmetric interior penalty coupling , J. Comput. Phys., 255 (2013), pp. 1–15.[37]
Y. Efendiev, J. Galvis, and X. H. Wu , Multiscale finite element methods for high-contrastproblems using local spectral basis functions , J. Comput. Phys., 230 (2011), pp. 937–955.[38]
Y. Efendiev, V. Ginting, T. Y. Hou, and R. Ewing , Accurate multiscale finite elementmethods for two-phase flow simulations , J. Comput. Phys., 220 (2006), pp. 155–174.[39]
Y. Efendiev and T. Hou , Multiscale finite element methods for porous media flows and theirapplications , Appl. Numer. Math., 57 (2007), pp. 577–596.[40]
Y. Efendiev and T. Y. Hou , Multiscale finite element methods theory and applications ,Springer, Lexington, KY, 2009.[41]
Y. Efendiev, T. Y. Hou, and X. H. Wu , Convergence of a nonconforming multiscale finiteelement method , SIAM J. Numer. Anal., 37 (2000), pp. 888–910.[42]
D. Elfverson, E. H. Georgoulis, and A. M˚alqvist , An adaptive discontinuous galerkin F. Song and W. Deng multiscale method for elliptic problems , Multiscale Model. Simul., 11 (2013), pp. 747–765.[43]
D. Elfverson, E. H. Georgoulis, A. M˚alqvist, and D. Peterseim , Convergence of a dis-continuous Galerkin multiscale method , SIAM J. Numer. Anal., 51 (2013), pp. 3351–3372.[44]
B. Engquist and O. Runborg , Wavelet-based numerical homogenization with applications ,in Multiscale and Multiresolution Methods: Theory and Applications, T. Barth, T. Chan,and R. Heimes, eds., vol. 20 of Lecture Notes in Computational Sciences and Engineering,Springer-Verlag, Berlin, 2002, pp. 97–148.[45]
Y. Epshteyn and B. Rivire , Estimation of penalty parameters for symmetric interiorpenalty galerkin methods , Journal of Computational and Applied Mathematics, 206 (2007),pp. 843–872.[46]
R. E. Ewing , Aspects of upscaling in simulation of flow in porous media , Advance in WaterResources, 20 (1997), pp. 349–358.[47]
C. L. Farmer , Upscaling: A review , in Proceedings of the Institute of Computational FluidDynamics Conference on Numerical Methods for Fluid Dynamics, Oxford, UK, 2001.[48]
J. Fish and V. Belsky , Multigrid method for a periodic heterogeneous medium, part i: Mul-tiscale modeling and quality in multidimensional case , Comput. Meth. Appl. Mech. Eng.,126 (1995), pp. 17–38.[49]
P. Grisvard , Elliptic problems on nonsmooth domains , Pitman, Boston, 1985.[50]
P. Henning and D. Peterseim , Oversampling for the multiscale finite element method , Mul-tiscale Model. Simul., 11 (2013), pp. 1149–1175.[51]
T. Y. Hou and X. H. Wu , A multiscale finite element method for elliptic problems in compositematerials and porous media , J. Comput. Phys., 134 (1997), pp. 169–189.[52]
T. Y. Hou, X. H. Wu, and Z. Cai , Convergence of a multiscale finite element method forelliptic problems with rapidly oscillation coefficients , Math. Comp., 68 (1999), pp. 913–943.[53]
T. Y. Hou, X. H. Wu, and Y. Zhang , Removing the cell resonance error in the multiscalefinite element method via a Petrov-Galerkin formulation , Commun. Math. Sci., 2 (2004),pp. 185–205.[54]
T. Hughes , Multiscale phenomena: Green’s functions, the Dirichlet to Neumann formulation,subgrid scale models, bubbles and the origin of stabilized methods , Comput. Meth. Appl.Mech. Eng., 127 (1995), pp. 387–401.[55]
T. Hughes, G. R. Feij ¨ o o, L. Mazzei, and J.-B. Quincy , The variational multiscale methodaparadigm for computational mechanics , Comput. Meth. Appl. Mech. Eng., 1–2 (1998),pp. 3–24.[56]
P. Jenny, S. Lee, and H. Tchelepi , Multi-scale finite-volume method for elliptic problems insubsurface flow simulation , J. Comput. Phys., 187 (2003), pp. 47–67.[57]
V. V. Jikov, S. M. Kozlov, and O. A. Oleinik , Homogenization of differential operators andintegral functionals , Springer-Verlag, Berlin, 1994.[58]
M. G. Larson and A. J. Niklasson , Conservation properties for the continuous and discon-tinuous galerkin methods , Tech. Rep. 2000-08, Chalmers University of Technology, (2000).[59]
A. M˚alqvist and D. Peterseim , Localization of elliptic multiscale problems , Math. Comp.,83 (2014), pp. 2583–2603.[60]
K. W. Morton , Petrov-Galerkin methods for non-self-adjoint problems , Springer Berlin Hei-sDPGM for Multiscale Problems delberg, 1980.[61] J. D. Moulton, J. E. Dendy, and J. M. Hyman , The black box multigrid numerical homog-enization algorithm , J. Comput. Phys., 141 (1998), pp. 1–29.[62]
B. C. P. Castillo , An a priori error analysis of the Local Discontinuous Galerkin method forelliptic problems , SIAM J. Numer. Anal., 38 (2000), pp. 1676–1706.[63]
G. Sangalli , Capturing small scales in elliptic problems using a residual-free bubbles finiteelement method , SIAM MMS, 1 (2003), pp. 485–503.[64]
F. Song , Analysis and calculation of discontinuous and combined multiscale finite elementmethods , PHD Thesis, (2016).[65]
F. Song, W. Deng, and H. Wu , A combined finite element and oversampling Petrov-Galerkinmethod for the multiscale elliptic problems with singularities , J. Comput. Phys., 305 (2016),pp. 722–743.[66]
W. Wang, J. Guzm´an, and C.-W. Shu , The multiscale discontinuous Galerkin method forsolving a class of second order elliptic problems with rough coefficients , Int. J. Numer.Anal. Model., 8 (2011), pp. 28–47.[67]
L. Yuan and C.-W. Shu , Discontinuous Galerkin method for a class of elliptic multi-scaleproblems , Internat. J. Numer. Methods Fluids, 56 (2008), pp. 1017–1032.[68]