Convergence of the CEM-GMsFEM for Stokes flows in heterogeneous perforated domains
CConvergence of the CEM-GMsFEM for Stokes flows inheterogeneous perforated domains
Eric Chung ∗ , Jiuhua Hu † , and Sai-Mang Pun ‡ July 8, 2020
Abstract
In this paper, we consider the incompressible Stokes flow problem in a perforated do-main and employ the constraint energy minimizing generalized multiscale finite elementmethod (CEM-GMsFEM) to solve this problem. The proposed method provides a flexibleand systematical approach to construct crucial divergence-free multiscale basis functions forapproximating the displacement field. These basis functions are constructed by solving aclass of local energy minimization problems over the eigenspaces that contain local informa-tion on the heterogeneities. These multiscale basis functions are shown to have the propertyof exponential decay outside the corresponding local oversampling regions. By adapting thetechnique of oversampling, the spectral convergence of the method with error bounds relatedto the coarse mesh size is proved.
In physics and structural mechanics there is a wide range of applications involving perforateddomains (see Figure 1 for an example of perforated domain). The perforated domain is char-acterized by partitioning a material into a solid portion and a pore space, referred as “matrix”and “pores”, respectively. In the model of differential equations over porous media, the stateequation is built in the matrix and the boundary conditions are imposed on the boundary of thematrix, including the boundary of the pores. A direct numerical treatment of solving differentialequations on such a domain is challenging because a fine mesh discretization is needed near thepores and this will result in a large computation.Many model reduction techniques for problems with perforation have been well developed in theexisting literature to improve the computational efficiency. For example, in numerical upscalingmethods [2, 14, 22, 23, 29, 30], one typically derives upscaled media or upscaled models and solvesthe resulting upscaled problem globally on a coarse grid. The dimensions of the correspondinglinear systems are much smaller, giving a guaranteed saving of computational cost. In addition,various multiscale methods for simulating multiscale problems with perforations are presentedin the literature. For instance, multiscale finite element methods (MsFEM) of Crouzeix-Raviarttype have been developed for elliptic problem [25] and Stokes flows [18, 24, 27]. In [21], the ∗ Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong. (
E-mail:[email protected] ) † Department of Mathematics, Texas A&M University, College Station, TX 77843, USA. (
E-mail:[email protected] ) ‡ Department of Mathematics, Texas A&M University, College Station, TX 77843, USA. (
E-mail:[email protected] ) a r X i v : . [ m a t h . NA ] J u l eterogeneous multiscale method (HMM) is proposed to discretize the elliptic problem withperforations in a coarse grid. Recently, a class of generalized finite element methods for theelliptic problem in perforated domain [3] has been proposed. This type of methods is basedon the idea of localized orthogonal decomposition (LOD) [16, 26] and generalize the traditionalfinite element method to accurately resolve the multiscale problems with a cheaper cost.In this research, we focus on the recently-developed generalized multiscale finite element method(GMsFEM) [4, 13]. The GMsFEM is a generalization of the classical MsFEM [15] in the sensethat multiple basis functions can be systematically constructed for each coarse block. TheGMsFEM consists of two stages: the offline and online stages. In the offline stage, a set of(local supported) snapshot functions are constructed, which can be used to essentially captureall fine-scale features of the solution. Then, a model reduction is performed by the use of a well-designed local spectral decomposition, and the dominant modes are chosen to be the multiscalebasis functions. All these computations are done before the actual simulations of the model.In the online stage, with a given source term and boundary conditions, the multiscale basisfunctions obtained in the offline stage are used to approximate the solution. There are someprevious works using GMsFEM for the Darcy’s flow model in perforated domain [7, 8], the Stokesequation with perforation [10] as well as coupled flow and transport in perforated domains [11].In this paper, we will develop and analyze a novel multiscale method for incompressible Stokesflows in perforated domains. Our idea is motivated by the recently-developed Constraint En-ergy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM), which hasachieved great success in solving elliptic problems with multiscale features [5, 6]. This methodhas been applied successfully in dealing with many problems, e.g., embedded fracture model forcoupled flow and mechanics problem [28], poroelasticity problems [19, 20], and wave equation[9]. CEM-GMsFEM is based on the framework of GMsFEM to design multiscale basis functionssuch that the convergence of the method is independent of the contrast from the heterogeneities;and the error linearly decreases with respect to coarse mesh size if oversampling parameter isappropriately chosen. Our approach of solving velocity has two ingredients. Firstly, we con-struct auxiliary multiscale basis functions by solving a local eigenvalue problem on each coarseblock. The global auxiliary space is formed by extending these auxiliary basis and the auxiliaryspace contains the information related to the pores. Secondly, the multiscale basis is sought ina weakly divergence free space by solving a minimization problem in an oversampling domain.The impose of weakly divergence free condition on the multiscale basis enables us solving veloc-ity solitarily. We prove in Lemma 4.7 that the multiscale basis decay exponentially outside thelocal oversampling domain. This exponential decay property plays a vital role in the convergenceanalysis of the proposed method and justifies the use of local multiscale basis functions.We organize the paper as follows. In Section 2, we state the model problem and its variationalformulation. In Section 3, we introduce auxiliary space and the construction of multiscalebasis functions for pressure using relaxed constraint energy minimization. The multiscale basisfunctions are constructed by solving a local spectral problem. We analyze convergence resultsin Section 4. Concluding remarks will be drawn in Section 5. In this section, we start with stating the Stokes flow in heterogenous perforated domains. Thensome notations and function spaces are introduced. We also introduce its corresponding varia-tional formulation. 2igure 1: Illustration of a perforated domain
Let Ω ⊆ R d be a bounded domain and B (cid:15) be a set of perforations within this domain. Theperforations are supposedly small and of a large number. We denote by Ω (cid:15) := Ω \ B (cid:15) theperforated domain. Then, we consider the basic linear model for incompressible fluid mechanics,i.e, Stokes equations. Stokes problem consists of finding vector function u : Ω (cid:15) → R d and scalarfunction p : Ω (cid:15) → R satisfying − µ ∆ u + ∇ p = f in Ω (cid:15) , ∇ · u = 0 in Ω (cid:15) , u = g on ∂ Ω (cid:15) , (1)where the vector filed f : Ω (cid:15) → R d is the body force acting on the fluid, u can be interpreted asthe velocity of an incompressible fluid motion, p is the associated pressure, and the constant µ isthe viscosity coefficient fluid. For the sake of simplicity, we only consider homogeneous Dirichletboundary for the velocity, i.e., g = 0, and the viscosity constant µ = 1. The extension to thegeneral viscosity constant and other types of boundary conditions is straightforward. Since thepressure p is uniquely defined up to a constant, we assume that (cid:82) Ω (cid:15) p dx = 0 so that the problemhas a unique solution. In this model, the primary source of the heterogeneity comes from theperforations in the computational domain; model reduction is necessary for practical simulationin this case. In this subsection, we clarify the notations used throughout the article. We write ( · , · ) to denotethe inner product in L (Ω (cid:15) ) and (cid:107)·(cid:107) for the corresponding norm. We denote L (Ω (cid:15) ) the subspaceof L (Ω (cid:15) ) containing functions with zero mean. Let H (Ω (cid:15) ) be the classical Sobolev space withthe norm (cid:107) v (cid:107) := (cid:16) (cid:107) v (cid:107) + (cid:107)∇ v (cid:107) (cid:17) / for any v ∈ H (Ω (cid:15) ) and H (Ω (cid:15) ) the subspace of functionshaving a vanishing trace. For vector-valued functions, we denote L (Ω (cid:15) ) := (cid:0) L (Ω (cid:15) ) (cid:1) d and H (Ω (cid:15) ) := (cid:0) H (Ω (cid:15) ) (cid:1) d . We write (cid:104)· , ·(cid:105) to denote the inner product in L (Ω (cid:15) ). We also denote (cid:107)·(cid:107) the norm induced by the inner product (cid:104)· , ·(cid:105) . To shorten notations, we define the spaces for thevelocity field u and the pressure p by V := H (Ω (cid:15) ) and Q := L (Ω (cid:15) ) . .3 Variational formulation and fine-grid discretization In this subsection, we provide the variational formulation corresponding to the system (1). Wemultiply the first equation and the second one with test functions from V and Q , respectively.Then, applying Green’s formula and making use of the boundary condition, the associatedvariational formulation of Stokes equation reads: Find ( u , p ) ∈ V × Q such that a ( u , v ) − b ( v , p ) = (cid:104) f , v (cid:105) for all v ∈ V ,b ( u , q ) = 0 for all q ∈ Q , (2)where a ( u , v ) := (cid:90) Ω (cid:15) ∇ u : ∇ v dx, and b ( u , q ) := (cid:90) Ω (cid:15) q ∇ · u dx. The well-posedness of (2) can be proved (see, for example [17, Chapter 4]). Throughout thiswork, we denote (cid:107)·(cid:107) a := (cid:112) a ( · , · ) the energy norm. In particular, for v = ( v , ..., v d ) T ∈ V , (cid:107) v (cid:107) a = (cid:80) di =1 (cid:107) v i (cid:107) a .To discretize the variational problem (2), let T h be a conforming partition for the computationaldomain Ω (cid:15) with (local) grid sizes h K := diam( K ) for K ∈ T h and h := max K ∈T h h K . We remarkthat T h is referred to as the fine grid . Next, let V h and Q h be any conforming stable pair offinite element spaces with respect to the fine grid T h . For the coupling numerical scheme, onemay use continuous Galerkin (CG) formulation: Find ( u h , p h ) ∈ V h × Q h such that a ( u h , v h ) − b ( v h , p h ) = (cid:104) f , v h (cid:105) for all v h ∈ V h ,b ( u h , q h ) = 0 for all q h ∈ Q h . (3)We remark that this classical approach will serve as a reference solution. The aim of this researchis to construct a reduced system based on (3). To this end, we introduce finite-dimensional mul-tiscale spaces V ms ⊆ V and Q ms ⊆ Q , whose dimensions are much smaller, for approximatingthe solution on some feasible coarse grid. In this section, we construct multiscale spaces on a coarse grid. Let T H be a conforming partitionof the computational domain Ω (cid:15) such that T h is a refinement of T H . We call T H the coarsegrid and each element of T H a coarse block. We denote H := max K ∈T H diam( K ) the coarsegrid size. Let N c be the total number of (interior) vertices of T H and N be the total number ofcoarse elements. We remark that the coarse element K ∈ T H is a closed subset (of the domainΩ (cid:15) ) with nonempty interior and piecewise smooth boundary. Let { x i } N c i =1 be the set of nodes in T H . Figure 2 illustrates the fine grid and a coarse element K i .The construction of the multiscale spaces consists of two steps. The first step is to constructauxiliary multiscale spaces using the concept of GMsFEM. Based on the auxiliary spaces, wecan then construct multiscale spaces containing basis functions whose energy are minimizedin some subregions of the domain. These energy-minimized basis functions will be shown todecay exponentially outside the oversampling domain, and can be used to construct a multiscalesolution. In this section, we begin with the construction of the auxiliary multiscale basis functions. Let V ( S ) be the restriction of V on S ⊂ Ω (cid:15) and V ( S ) be the subspace of V ( S ), whose element4igure 2: Illustration of the coarse grid, the fine grid, and the oversampling domain.is of zero trace on ∂S . We also define Q ( S ) := L ( S ). Consider the following local spectralproblem: Find ( φ ij , λ ij ) ∈ V ( K i ) × R such that a i ( φ ij , v ) = λ ij s i ( φ ij , v ) for all v ∈ V ( K i ) , (4)where a i ( · , · ) and s i ( · , · ) are defined as follows: a i ( u , v ) := (cid:90) K i ∇ u : ∇ v dx and s i ( u , v ) := (cid:90) K i ˜ κ u · v dx (5)for any u , v ∈ V ( K i ). Here, we define ˜ κ := (cid:80) N c j =1 |∇ χ ms j | , where { χ ms j } N c j =1 is a set of neighborhood-wise defined partition of unity functions [1] on the coarse grid. In particular, the function χ ms j satisfies H |∇ χ ms j | = O (1) and 0 ≤ χ ms j ≤ ≤ λ i ≤ · · · ≤ λ i(cid:96) i ≤ · · · for each i ∈ { , · · · , N } . Also, we assume that the eigenfunctions satisfy the normalizationcondition s i ( φ ij , φ ij ) = 1. Then, we choose the first (cid:96) i ∈ N + eigenfunctions and define V i aux :=span { φ ij : j = 1 , · · · , (cid:96) i } . Based on these local spaces, the global auxiliary space V aux is definedto be V aux := N (cid:77) i =1 V i aux with inner product s ( u , v ) := N (cid:88) i =1 s i ( u , v )for any u , v ∈ V aux . Further, we define an orthogonal projection π : V → V aux such that π ( v ) := N (cid:88) i =1 π i ( v ) , where π i ( v ) := (cid:96) i (cid:88) j =1 s i ( v , φ ij ) φ ij for all v ∈ V . 5 .2 Multiscale space In this section, we construct multiscale basis functions based on constraint energy minimization.For each coarse element K i , we define the oversampled region K i,k i ⊆ Ω (cid:15) by enlarging K i by k i ∈ N layer(s), i.e., K i, := K i , K i,k i := (cid:91) { K ∈ T H : K ∩ K i,k i − (cid:54) = ∅} for k i = 1 , , · · · . We call k i a parameter of oversampling related to the coarse element K i . See Figure 2 for anillustration of K i, . For simplicity, we denote K + i a generic oversampling region related to thecoarse element K i with a specific oversampling parameter k i . Next, we define multiscale basisfunction possessing the property of constraint energy minimization [5]. In particular, for eachauxiliary function φ ij ∈ V aux , we solve the following minimization problem: Find ψ ij, ms ∈ V ( K + i )such that ψ ij, ms := argmin (cid:8) a ( ψ, ψ ) + s (cid:0) π ( ψ ) − φ ij , π ( ψ ) − φ ij (cid:1) : ψ ∈ V ( K + i ) and ∇ · ψ = 0 (cid:9) . (6)Note that problem (6) is equivalent to the local problem: Find ( ψ ij, ms , ξ ij, ms ) ∈ V ( K + i ) × Q ( K + i )such that a ( ψ ij, ms , v ) + s (cid:0) π ( ψ ij, ms ) , π ( v ) (cid:1) + b ( v, ξ ij, ms ) = s (cid:0) φ ij , π ( v ) (cid:1) for all v ∈ V ( K + i ) ,b ( ψ ij, ms , q ) = 0 for all q ∈ Q ( K + i ) . (7)Finally, for fixed parameters k i and (cid:96) i , the multiscale space V ms is defined by V ms := span (cid:8) ψ ij, ms : 1 ≤ j ≤ (cid:96) i , ≤ i ≤ N (cid:9) . The multiscale basis functions can be interpreted as approximations to global multiscale basisfunctions ψ ij ∈ V defined by ψ ij := argmin (cid:8) a ( ψ, ψ ) + s (cid:0) π ( ψ ) − φ ij , π ( ψ ) − φ ij (cid:1) : ψ ∈ V and ∇ · ψ = 0 (cid:9) , which is equivalent to the following variational formulation: Find ( ψ ij , ξ ij ) ∈ V × Q such that a ( ψ ij , v ) + s (cid:0) π ( ψ ij ) , π ( v ) (cid:1) + b ( v, ξ ij ) = s (cid:0) φ ij , π ( v ) (cid:1) for all v ∈ V ,b ( ψ ij , q ) = 0 for all q ∈ Q . (8)These basis functions have global support in the domain Ω (cid:15) , but, as shown in Lemma 4.7, decayexponentially outside some local (oversampled) region. This property plays a vital role in theconvergence analysis of the proposed method and justifies the use of local basis functions in V ms .Furthermore, we define V glo := span (cid:110) ψ ij : 1 ≤ j ≤ (cid:96) i , ≤ i ≤ N (cid:111) and ˜ V := { v ∈ V div0 : π ( v ) =0 } , where V div0 is the closed subspace of V containing divergence-free vector fields. Then, onecan show that V div0 = V glo (cid:76) a ˜ V . Remark.
Suppose that S ⊂ Ω (cid:15) is any non-empty connected union of coarse elements K i ∈ T H .Denote D S : H ( S ) → L ( S ) the divergence operator corresponding to the set S . We have thefollowing auxiliary result from functional analysis. Lemma 3.1 (cf. Theorem 6.14-1 in [12]) . Suppose that S is any non-empty connected union ofcoarse elements. Restricting the domain of D S on the orthogonal complement (with respect tostandard L inner product) of its kernel, the divergence operator D S is injective and surjective.Moreover, it has a continuous inverse and there is a generic constant β S > β S (cid:13)(cid:13) D − S µ (cid:13)(cid:13) a ≤ (cid:107) µ (cid:107) for any µ ∈ L ( S ) . S be thewhole domain Ω (cid:15) or an oversampled region K + i . Then, for any non-zero element v ∈ H ( S ), wehave sup v ∈ H ( S ) ,v (cid:54) =0 | b ( v, µ ) |(cid:107) v (cid:107) a ≥ | b ( D − S µ, µ ) | (cid:13)(cid:13) D − S µ (cid:13)(cid:13) a = (cid:107) µ (cid:107) (cid:13)(cid:13) D − S µ (cid:13)(cid:13) a ≥ β S (cid:107) µ (cid:107) (9)for any µ ∈ L ( S ), which shows that the inf-sup condition holds for (8). Similarly, we can provethe inf-sup condition holds for (7). From the above, we have the multiscale space V ms for the approximation of velocity field. Themultiscale solution u ms ∈ V ms is obtained by solving the following equation: a ( u ms , v ) = (cid:104) f , v (cid:105) for all v ∈ V ms . (10)To approximate the pressure based on coarse grid, we will construct a specific solution space offinite dimension. Let W ( K i ) := { v ∈ H ( K i ) : (cid:82) K i v dx = 0 , b ( w , v ) = 0 for all w ∈ ( I − π ) V } .We consider the following spectral problem: Find ( q ij , ζ ij ) ∈ W ( K i ) × R such that A i ( q ij , v ) = ζ ij S i ( q ij , v ) for all v ∈ W ( K i ) , (11)where A i ( · , · ) and S i ( · , · ) are defined as follows: A i ( u, v ) := (cid:90) K i ∇ u · ∇ v dx and S i ( u, v ) := (cid:90) K i ˜ κuv dx (12)for any u, v ∈ H ( K i ). Assume that for each i ∈ { , · · · , N } the eigenvalues ζ ij are arranged inascending order such that 0 ≤ ζ i ≤ ζ i ≤ · · · . We then define a finite dimensional solution space Q H as follows: Q H := span (cid:8) q ij : i = 1 , · · · , N, j = 1 , · · · , (cid:96) i (cid:9) . Then, we solve the following variational problem over the domain Ω (cid:15) : Find p ms ∈ Q H such that b ( v , p ms ) = a ( u ms , v ) − (cid:104) f , v (cid:105) for all v ∈ V aux . (13)Note that dim( Q H ) = dim( V aux ). To prove the well-posedness of (13), it suffices to verify inf-supcondition for the bilinear form b ( · , · ) over V aux and Q H . Recall that the variational formulation(2) is well-posed and inf-sup condition holds for b ( · , · ) under spaces V and Q . Hence, for any q ∈ Q H , there exists w ∈ V such that b ( w , q ) ≥ C (cid:107) w (cid:107) a (cid:107) q (cid:107) for some constant C >
0. Choosing v := π w , we have v ∈ V aux and b ( v , q ) = b ( π w , q ) = b ( w , q ) ≥ C (cid:107) w (cid:107) a (cid:107) q (cid:107) ≥ C (cid:107) v (cid:107) a (cid:107) q (cid:107) . Therefore, the problem (13) is well-posed. Note that the pressure p solves the following equation: b ( v , p ) = a ( u , v ) − (cid:104) f , v (cid:105) for all v ∈ V . Then, we have b ( v , p − p ms ) = a ( u − u ms , v ) ≤ (cid:107) u − u ms (cid:107) a (cid:107) v (cid:107) a , for all v ∈ V aux . It implies thatsup v ∈ V aux b ( v , p − p ms ) (cid:107) v (cid:107) a ≤ (cid:107) u − u ms (cid:107) a . The multiscale solution p ms serves as an approximation of the solution p and (cid:107) p − p ms (cid:107) (cid:46) (cid:107) u − u ms (cid:107) a . 7 Convergence analysis
In this section, we analyze the proposed method. We denote (cid:107)·(cid:107) s := (cid:112) s ( · , · ) the s -norm. Inparticular, (cid:107) v (cid:107) s = (cid:80) di =1 (cid:107) v i (cid:107) s for any v = ( v , · · · , v d ) T . We also denote spt( v ) the support of agiven function or vector field. We write a (cid:46) b if there exists a generic constant C > a ≤ Cb . Define Λ := min ≤ i ≤ N λ i(cid:96) i +1 and Γ := max ≤ i ≤ N λ i(cid:96) i . For a given subregion S ⊂ Ω (cid:15) , we definelocal norms (cid:107) v (cid:107) a ( S ) := (cid:0)(cid:82) S |∇ v | dx (cid:1) / and (cid:107) v (cid:107) s ( S ) := (cid:0)(cid:82) S ˜ κ | v | dx (cid:1) / for any v ∈ V .Before estimating the error between global and local multiscale basis functions, we introducesome notions that will be used in the analysis. First, we introduce cutoff function with respectto oversampling region. Given a coarse block K i ∈ T H and a parameter of oversampling m ∈ N ,we recall that K i,m ⊂ Ω (cid:15) is an m -layer oversampling region corresponding to K i . Definition 4.1.
For two positive integers M and m with M > m ≥
1, we define cutoff function χ M,mi ∈ span { χ ms j } N c j =1 such that 0 ≤ χ M,mi ≤ χ M,mi = (cid:26) K i,m , (cid:15) \ K i,M . Note that, we have K i,m ⊂ K i,M ⊂ Ω (cid:15) and spt( χ M,mi ) ⊂ K i,M .First, we establish the following auxiliary results for later use in the analysis. Lemma 4.2.
Let v ∈ V and k ≥ (cid:107) v (cid:107) a ≤ Γ / (cid:107) v (cid:107) s if v ∈ V aux ;(ii) (cid:107) v (cid:107) s ≤ Λ − / (cid:107) v (cid:107) a if v / ∈ V aux ;(iii) (cid:107) v (cid:107) s (cid:46) Λ − (cid:107) ( I − π ) v (cid:107) a + (cid:107) π v (cid:107) s ;(iv) (cid:13)(cid:13)(cid:13) (1 − χ k,k − i ) v (cid:13)(cid:13)(cid:13) a ≤ − ) (cid:107) v (cid:107) a (Ω (cid:15) \ K i,k − ) + 2 (cid:107) π v (cid:107) s (Ω (cid:15) \ K i,k − ) (v) (cid:13)(cid:13)(cid:13) (1 − χ k,k − i ) v (cid:13)(cid:13)(cid:13) s ≤ Λ − (cid:107) v (cid:107) a (Ω (cid:15) \ K i,k − ) + (cid:107) π v (cid:107) s (Ω (cid:15) \ K i,k − ) . Proof.
Note that one can write v = (cid:80) Ni =1 (cid:80) j ≥ α ij φ ij with α ij ∈ R for any v ∈ V .(i) Since v ∈ V aux , then α ij = 0 for j ≥ (cid:96) i + 1. Using the local spectral problem (11), weobtain (cid:107) v (cid:107) a = N (cid:88) i =1 (cid:96) i (cid:88) j =1 α ij a ( φ ij , v ) = N (cid:88) i =1 (cid:96) i (cid:88) j =1 α ij λ ij s ( φ ij , v ) ≤ Γ N (cid:88) i =1 (cid:96) i (cid:88) j =1 α ij s ( φ ij , v ) = Γ (cid:107) v (cid:107) s . (ii) For any v / ∈ V aux , one can write v = (cid:80) Ni =1 (cid:80) j ≥ (cid:96) i +1 α ij φ ij . Then, we have (cid:107) v (cid:107) a = N (cid:88) i =1 (cid:88) j ≥ (cid:96) i +1 α ij a ( φ ij , v ) = N (cid:88) i =1 (cid:88) j ≥ (cid:96) i +1 α ij λ ij s ( φ ij , v ) ≥ Λ N (cid:88) i =1 (cid:88) j ≥ (cid:96) i +1 α ij s ( φ ij , v ) = Λ (cid:107) v (cid:107) s . (iii) The result follows from (i), (ii), and the triangle inequality.8iv) By using the property of cutoff function χ k,k − i and (iii), we have (cid:13)(cid:13)(cid:13) (1 − χ k,k − i ) v (cid:13)(cid:13)(cid:13) a ≤ (cid:32)(cid:90) Ω (cid:15) \ K i,k − (1 − χ k,k − i ) |∇ v | + | v ∇ χ k,k − i | dx (cid:33) ≤ (cid:16) (cid:107) v (cid:107) a (Ω (cid:15) \ K i,k − ) + (cid:107) v (cid:107) s (Ω (cid:15) \ K i,k − ) (cid:17) ≤ − ) (cid:107) v (cid:107) a (Ω (cid:15) \ K i,k − ) + 2 (cid:107) π v (cid:107) s (Ω (cid:15) \ K i,k − ) . (v) For any k ≥
2, we have (cid:13)(cid:13)(cid:13) (1 − χ k,k − i ) v (cid:13)(cid:13)(cid:13) s ≤ (cid:107) v (cid:107) s (Ω (cid:15) \ K i,k − ) ≤ Λ − (cid:107) v (cid:107) a (Ω (cid:15) \ K i,k − ) + (cid:107) π v (cid:107) s (Ω (cid:15) \ K i,k − ) . This completes the proof.First, we present the convergence of using global basis functions constructed in (8). We define u glo ∈ V glo as the global multiscale solution satisfying a ( u glo , v ) = (cid:104) f , v (cid:105) for all v ∈ V glo . (14) Theorem 4.3.
Let u be the solution of (2) and u glo be the solution of (14). We have (cid:107) u − u glo (cid:107) a (cid:46) Λ − / (cid:13)(cid:13)(cid:13) ˜ κ − / f (cid:13)(cid:13)(cid:13) . Moreover, if { χ ms j } N c j =1 is a set of bilinear partition of unity, we have (cid:107) u − u glo (cid:107) a (cid:46) H Λ − (cid:107) f (cid:107) . Proof.
By the definition of u and u glo , we have a ( u − u glo , v ) = 0 for all v ∈ V glo . (15)Hence, we have u − u glo ∈ ˜ V and a ( u − u glo , u − u glo ) = a ( u − u glo , u ) = ( f , u − u glo ) ≤ (cid:13)(cid:13)(cid:13) ˜ κ − f (cid:13)(cid:13)(cid:13) (cid:107) u − u glo (cid:107) s . (16)Since u − u glo ∈ V − V aux , it follows from Lemma (4.2) (ii) that (cid:107) u − u glo (cid:107) s ≤ Λ − (cid:107) u − u glo (cid:107) a . (17)The result follows by combining (16) and (17). The second part follows from the fact that |∇ χ ms j | = O ( H − ) when { χ ms j } N c j =1 is a set of bilinear partition of unity functions.Next we analyze the convergence of the proposed multiscale method. We first recall ProjectionTheorem, which can be found in many functional analysis literature, e.g., [12, Section 4.3]. Theorem 4.4 (Projection Theorem) . Let V be a closed subspace of the Hilbert space H equipped with an inner product ( · , · ) H . Then, for any given element f ∈ H , there exists aunique element p ∈ V such that (cid:107) f − p (cid:107) = min v ∈V (cid:107) f − v (cid:107) . Here, (cid:107)·(cid:107) is the norm induced by the inner product ( · , · ) H . Moreover, the mapping P : f (cid:55)→ p islinear and satisfies the inequality (cid:107) P f (cid:107) ≤ (cid:107) f (cid:107) for any f ∈ H .9n the following lemma, we show the existence of a projection from V ( D ) to V div0 ( D ) using theProjection Theorem. Lemma 4.5.
Let
D ⊆ Ω (cid:15) . Then, there exists a divergence-free projection P D : V ( D ) → V div0 ( D ), where V div0 ( D ) := { v ∈ V ( D ) : b ( v , q ) = 0 for all q ∈ L ( D ) } . Proof.
Define a bilinear form on V ( D ) as follows: ( u , v ) as ( D ) := a D ( u , v ) + s D ( u , v ), where a D ( · , · ) and s D ( · , · ) are the restriction of a ( · , · ) and s ( · , · ) on the subregion D . One can easilyshow that ( · , · ) as is an inner product defined on V ( D ).Next, we show that V div0 ( D ) is a closed subspace of V ( D ) with respect to the inner product( · , · ) as . Let { f n } be a sequence in V div0 ( D ) that converges to f in V ( D ). Since f n ∈ V div0 ( D ), thenwe have b ( f n , g ) = 0 for all g ∈ L ( D ). Then lim n →∞ b ( f n , g ) = b ( f , g ) = 0 for all g ∈ L ( D ). Itimplies that f ∈ V div0 ( D ). Consequently, V div0 ( D ) is a closed subspace of V ( D ). An applicationof Projection Theorem proves the desired result. Remark.
We denote (cid:107)·(cid:107) as ( D ) the norm induced by the inner product ( · , · ) as ( D ) . Then, we have (cid:107) P D ( v ) (cid:107) as ( D ) ≤ (cid:107) v (cid:107) as ( D ) for any v ∈ V ( D ). We simply write (cid:107)·(cid:107) as in short for (cid:107)·(cid:107) as ( D ) when D = Ω (cid:15) . Moreover, the subscript D will be dropped from P D when there is no ambiguity. Lemma 4.6.
For any auxiliary function v aux ∈ V aux , there exists a function z ∈ V div0 such that π ( z ) = v aux , (cid:107) z (cid:107) a ≤ D (cid:107) v aux (cid:107) s , and spt( z ) ⊆ spt( v aux ) . Here, D is a generic constant depending only on the coarse mesh, the partition of unity, and theeigenvalues obtained in (11). Proof.
Without loss of generality, we can assume that v aux ∈ V i aux . Consider the followingvariational problem: Find z ∈ V div0 ( K i ) and µ ∈ V i aux such that a i ( z, v ) + s i ( v, µ ) = 0 for all v ∈ V div0 ( K i ) ,s i ( z, q ) = s i ( v aux , q ) for all q ∈ V i aux . (18)Here, the bilinear forms a i ( · , · ) and s i ( · , · ) are defined in (12). We will show the well-posednessof the problem (18). It suffices to show that there is a function z ∈ V div0 ( K i ) such that s i ( z, v aux ) ≥ C (cid:107) v aux (cid:107) s ( K i ) and (cid:107) z (cid:107) a ( K i ) ≤ C (cid:107) v aux (cid:107) s ( K i ) for some generic constants C and C . We denote I K i := { j : x j is a coarse vertex of K i } anddefine B := (cid:81) j ∈I Ki χ ms j . Taking z = P ( Bv aux ), we have s i ( z, v aux ) = s i ( P ( Bv aux ) , v aux ) = (cid:90) K i ˜ κ P ( Bv aux ) v aux dx ≥ C − π (cid:107) v aux (cid:107) s ( K i ) . Here, the constant C π is defined to be C π := sup K ∈T H , µ ∈ V aux (cid:82) K ˜ κµ dx (cid:82) K ˜ κ P ( Bµ ) µ dx > . Note that | B | ≤ |∇ B | ≤ C T (cid:80) j ∈I Ki | χ ms j | with C T := max K ∈T H |I K | . The followinginequalities hold (cid:107) z (cid:107) a ( K i ) ≤ (cid:107) P ( Bv aux ) (cid:107) as ( K i ) ≤ (cid:107) Bv aux (cid:107) as ( K i ) (cid:46) (cid:107) Bv aux (cid:107) a ( K i ) . (cid:107) z (cid:107) a ( K i ) (cid:46) (cid:107) Bv aux (cid:107) a ( K i ) ≤ C T C π (1 + Γ) (cid:107) v aux (cid:107) s ( K i ) . It shows the existence and uniqueness of the function z for a given auxiliary function v aux ∈ V i aux .From the second equality in (18), we see that π i ( z ) = v aux . This completes the proof.The following lemma shows that the global multiscale basis functions have a decay property. Lemma 4.7.
Let φ ij ∈ V aux be a given auxiliary function. Suppose that ψ ij, ms is a multiscalebasis function obtained in (7) over the oversampling domain K i,k with k ≥ ψ ij is thecorresponding global basis function obtained in (8). Then, the following estimate holds: (cid:13)(cid:13) ψ ij − ψ ij, ms (cid:13)(cid:13) a + (cid:13)(cid:13) π ( ψ ij − ψ ij, ms ) (cid:13)(cid:13) s ≤ E (cid:16)(cid:13)(cid:13) ψ ij (cid:13)(cid:13) a + (cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13) s (cid:17) , where E = 3 (cid:0) − (cid:1) (cid:16) (cid:2) − ) (cid:3) − / (cid:17) − k is a factor of exponential decay. Proof.
Subtracting the first equation of (7) from that of (8), we obtain a ( ψ ij − ψ ij, ms , v ) + s ( π ( ψ ij − ψ ij, ms ) , π ( v )) + b ( v, ξ ij − ξ ij, ms ) = 0 for all v ∈ V ( K i,k ) . Taking v = w − ψ ij, ms with w ∈ V div0 ( K i,k ), then we have a ( ψ ij − ψ ij, ms , ψ ij, ms )+ s ( π ( ψ ij − ψ ij, ms ) , π ( ψ ij, ms )) = a ( ψ ij − ψ ij, ms , w )+ s ( π ( ψ ij − ψ ij, ms ) , π ( w )) . (19)Utilizing (19) and Cauchy-Schwarz inequality, one can show that (cid:13)(cid:13) ψ ij − ψ ij, ms (cid:13)(cid:13) a + (cid:13)(cid:13) π ( ψ ij − ψ ij, ms ) (cid:13)(cid:13) s ≤ (cid:13)(cid:13) ψ ij − w (cid:13)(cid:13) a + (cid:13)(cid:13) π ( ψ ij − w ) (cid:13)(cid:13) s for any w ∈ V div0 ( K i,k ). Let w = P ( χ k,k − i ψ ij ). Note that ψ ij = P ( ψ ij ). Then, we have (cid:13)(cid:13) ψ ij − ψ ij, ms (cid:13)(cid:13) a + (cid:13)(cid:13) π ( ψ ij − ψ ij, ms ) (cid:13)(cid:13) s ≤ (cid:13)(cid:13)(cid:13) ψ ij − P ( χ k,k − i ψ ij ) (cid:13)(cid:13)(cid:13) a + (cid:13)(cid:13)(cid:13) π ( ψ ij − P ( χ k,k − i ψ ij )) (cid:13)(cid:13)(cid:13) s ≤ (cid:13)(cid:13)(cid:13) (1 − χ k,k − i ) ψ ij (cid:13)(cid:13)(cid:13) a + (cid:13)(cid:13)(cid:13) (1 − χ k,k − i ) ψ ij (cid:13)(cid:13)(cid:13) s . (20)Using (iv) and (v) of Lemma 4.2, we have (cid:13)(cid:13) ψ ij − ψ ij, ms (cid:13)(cid:13) a + (cid:13)(cid:13) π ( ψ ij − ψ ij, ms ) (cid:13)(cid:13) s ≤ − ) (cid:16)(cid:13)(cid:13) ψ ij (cid:13)(cid:13) a (Ω (cid:15) \ K i,k − ) + (cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13) s (Ω (cid:15) \ K i,k − ) (cid:17) . (21)Next, we estimate the term (cid:13)(cid:13)(cid:13) ψ ij (cid:13)(cid:13)(cid:13) a (Ω (cid:15) \ K i,k − ) + (cid:13)(cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13)(cid:13) s (Ω (cid:15) \ K i,k − ) . We claim that it can bebounded by the term F := (cid:13)(cid:13)(cid:13) ψ ij (cid:13)(cid:13)(cid:13) a ( K i,k − \ K i,k − ) + (cid:13)(cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13)(cid:13) s ( K i,k − \ K i,k − ) . This recursive propertyis crucial in our convergence estimate.Note that spt(1 − χ k − ,k − i ) ⊆ Ω (cid:15) \ K i,k − and spt( φ ij ) ⊆ K i . So s ( φ ij , π P ((1 − χ k − ,k − i ) ψ ij )) = 0.Choosing test function v = P ((1 − χ k − ,k − i ) ψ ij ) in the variational formulation (8), we have a (cid:16) ψ ij , P ((1 − χ k − ,k − i ) ψ ij ) (cid:17) + s (cid:16) π ( ψ ij ) , π ( P ((1 − χ k − ,k − i ) ψ ij )) (cid:17) = 0 . (22)Note that a (cid:16) ψ ij , P ((1 − χ k − ,k − i ) ψ ij ) (cid:17) = (cid:90) Ω (cid:15) \ K i,k − ∇ ψ ij : ∇ (cid:16) P ((1 − χ k − ,k − i ) ψ ij ) (cid:17) dx = (cid:90) Ω (cid:15) \ K i,k − |∇ ψ ij | dx − (cid:90) Ω (cid:15) \ K i,k − ∇ ψ ij : ∇ (cid:16) P ( χ k − ,k − i ψ ij ) (cid:17) dx. (cid:13)(cid:13) ψ ij (cid:13)(cid:13) a (Ω (cid:15) \ K i,k − ) ≤ (cid:90) Ω (cid:15) \ K i,k − |∇ ψ ij | dx = a (cid:16) ψ ij , P ((1 − χ k − ,k − i ) ψ ij ) (cid:17) + (cid:90) Ω (cid:15) \ K i,k − ∇ ψ ij : ∇ (cid:16) P ( χ k − ,k − i ψ ij ) (cid:17) dx ≤ a (cid:16) ψ ij , P ((1 − χ k − ,k − i ) ψ ij ) (cid:17) + (cid:13)(cid:13) ψ ij (cid:13)(cid:13) a ( K i,k − \ K i,k − ) (cid:13)(cid:13)(cid:13) P ( χ k − ,k − i ψ ij ) (cid:13)(cid:13)(cid:13) as ( K i,k − \ K i,k − ) . (23)Note that χ k − ,k − i ≡ (cid:15) \ K i,k − . Thus, we have s (cid:16) π ( ψ ij ) , π ( P ((1 − χ k − ,k − i ) ψ ij )) (cid:17) = (cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13) s (Ω (cid:15) \ K i,k − ) + (cid:90) K i,k − \ K i,k − ˜ κπ ( ψ ij ) π (cid:16) P ((1 − χ k − ,k − i ) ψ ij ) (cid:17) dx and (cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13) s (Ω (cid:15) \ K i,k − ) = s (cid:16) π ( ψ ij ) , π ( P ((1 − χ k − ,k − i ) ψ ij )) (cid:17) − (cid:90) K i,k − \ K i,k − ˜ κπ ( ψ ij ) π (cid:16) P ((1 − χ k − ,k − i ) ψ ij ) (cid:17) dx ≤ s (cid:16) π ( ψ ij ) , π ( P ((1 − χ k − ,k − i ) ψ ij )) (cid:17) + (cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13) s ( K i,k − \ K i,k − ) (cid:13)(cid:13)(cid:13) P ((1 − χ k − ,k − i ) ψ ij ) (cid:13)(cid:13)(cid:13) as ( K i,k − \ K i,k − ) . (24)Using (iv) and (v) of Lemma 4.2, one can show that (cid:13)(cid:13)(cid:13) P ((1 − χ k − ,k − i ) ψ ij ) (cid:13)(cid:13)(cid:13) as ( K i,k − \ K i,k − ) ≤ − ) F , (cid:13)(cid:13)(cid:13) P ( χ k − ,k − i ψ ij ) (cid:13)(cid:13)(cid:13) as ( K i,k − \ K i,k − ) ≤ − ) F . (25)Combining (22) and the inequalities (23) – (25), we have (cid:13)(cid:13) ψ ij (cid:13)(cid:13) a (Ω (cid:15) \ K i,k − ) + (cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13) s (Ω (cid:15) \ K i,k − ) ≤ (cid:16)(cid:13)(cid:13) ψ ij (cid:13)(cid:13) a ( K i,k − \ K i,k − ) + (cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13) s ( K i,k − \ K i,k − ) (cid:17) / (cid:2) − ) (cid:3) / F = (cid:2) − ) (cid:3) / F . (26)Notice that, using the inequality (26), we have (cid:13)(cid:13) ψ ij (cid:13)(cid:13) a (Ω (cid:15) \ K i,k − ) + (cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13) s (Ω (cid:15) \ K i,k − ) = (cid:13)(cid:13) ψ ij (cid:13)(cid:13) a (Ω (cid:15) \ K i,k − ) + (cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13) s (Ω (cid:15) \ K i,k − ) + (cid:13)(cid:13) ψ ij (cid:13)(cid:13) a ( K i,k − \ K i,k − ) + (cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13) s ( K i,k − \ K i,k − ) ≥ (cid:16) (cid:2) − ) (cid:3) − / (cid:17) (cid:16)(cid:13)(cid:13) ψ ij (cid:13)(cid:13) a (Ω (cid:15) \ K i,k − ) + (cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13) s (Ω (cid:15) \ K i,k − ) (cid:17) . Using the above inequality recursively, we obtain (cid:13)(cid:13) ψ ij (cid:13)(cid:13) a (Ω (cid:15) \ K i,k − ) + (cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13) s (Ω (cid:15) \ K i,k − ) ≤ (cid:16) (cid:2) − ) (cid:3) − / (cid:17) − k (cid:16)(cid:13)(cid:13) ψ ij (cid:13)(cid:13) a + (cid:13)(cid:13) π ( ψ ij ) (cid:13)(cid:13) s (cid:17) . This completes the proof. 12he above lemma shows that the global multiscale basis is localizable. We need the followingresult to show the convergence estimate.
Lemma 4.8.
With the same notations in Lemma 4.7, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) i =1 ( ψ ij − ψ ij, ms ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) i =1 π ( ψ ij − ψ ij, ms ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) s (cid:46) ( k + 1) d N (cid:88) i =1 (cid:104)(cid:13)(cid:13) ψ ij − ψ ij, ms (cid:13)(cid:13) a + (cid:13)(cid:13) π ( ψ ij − ψ ij, ms ) (cid:13)(cid:13) s (cid:105) . Proof.
Denote w := N (cid:88) i =1 ( ψ ij − ψ ij, ms ). Noice that, for any i ∈ { , · · · , N } , it holds that a ( ψ ij − ψ ij, ms , v ) + s ( π ( ψ ij − ψ ij, ms ) , π ( v )) = 0 , for all v ∈ V div0 ( K i,k ) . (27)Choosing v = P ((1 − χ k +1 ,ki ) w ) in (27), we have a (cid:16) ψ ij − ψ ij, ms , P ((1 − χ k +1 ,ki ) w ) (cid:17) + s (cid:16) π ( ψ ij − ψ ij, ms ) , π ( P ((1 − χ k +1 ,ki ) w )) (cid:17) = 0 . Note that P ( w ) = w . Hence, we have (cid:107) w (cid:107) a + (cid:107) πw (cid:107) s = N (cid:88) i =1 a ( ψ ij − ψ ij, ms , w ) + s ( π ( ψ ij − ψ ij, ms ) , π ( w ))= N (cid:88) i =1 a (cid:16) ψ ij − ψ ij, ms , P ( χ k +1 ,ki w ) (cid:17) + s (cid:16) π ( ψ ij − ψ ij, ms ) , π ( P ( χ k +1 ,ki w )) (cid:17) . For each i ∈ { , · · · , N } , using the properties of the cutoff function χ k +1 ,ki and (ii) of Lemma4.2, we have the following estimates: (cid:13)(cid:13)(cid:13) χ k +1 ,ki w (cid:13)(cid:13)(cid:13) a (cid:46) (cid:107) w (cid:107) s ( K i,k +1 ) + (cid:107) w (cid:107) a ( K i,k +1 ) ≤ (1 + Λ − ) (cid:16) (cid:107) w (cid:107) a ( K i,k +1 ) + (cid:107) π ( w ) (cid:107) s ( K i,k +1 ) (cid:17) , (cid:13)(cid:13)(cid:13) π ( χ k +1 ,ki w ) (cid:13)(cid:13)(cid:13) s ≤ (cid:13)(cid:13)(cid:13) χ k +1 ,ki w (cid:13)(cid:13)(cid:13) s ( K i,k +1 ) ≤ Λ − (cid:107) w (cid:107) a ( K i,k +1 ) + (cid:107) π ( w ) (cid:107) s ( K i,k +1 ) . (28)Furthermore, an application of (28) we arrive at the following estimate: (cid:13)(cid:13)(cid:13) χ k +1 ,ki w (cid:13)(cid:13)(cid:13) as = (cid:13)(cid:13)(cid:13) χ k +1 ,ki w (cid:13)(cid:13)(cid:13) a + (cid:13)(cid:13)(cid:13) π ( χ k +1 ,ki w ) (cid:13)(cid:13)(cid:13) s (cid:46) (cid:107) w (cid:107) a ( K i,k +1 ) + (cid:107) π ( w ) (cid:107) s ( K i,k +1 ) (29)Combining (28) and (29), we have (cid:107) w (cid:107) a + (cid:107) π ( w ) (cid:107) s ≤ N (cid:88) i =1 (cid:13)(cid:13) ψ ij − ψ ij, ms (cid:13)(cid:13) a · (cid:13)(cid:13)(cid:13) χ k +1 ,ki w (cid:13)(cid:13)(cid:13) as + (cid:13)(cid:13) π ( ψ ij − ψ ij, ms ) (cid:13)(cid:13) s · (cid:13)(cid:13)(cid:13) χ k +1 ,ki w (cid:13)(cid:13)(cid:13) as (cid:46) N (cid:88) i =1 (cid:16)(cid:13)(cid:13) ψ ij − ψ ij, ms (cid:13)(cid:13) a + (cid:13)(cid:13) π ( ψ ij − ψ ij, ms ) (cid:13)(cid:13) s (cid:17) / · (cid:16) (cid:107) w (cid:107) a ( K i,k +1 ) + (cid:107) π ( w ) (cid:107) s ( K i,k +1 ) (cid:17) / (cid:46) (cid:32) N (cid:88) i =1 (cid:13)(cid:13) ψ ij − ψ ij, ms (cid:13)(cid:13) a + (cid:13)(cid:13) π ( ψ ij − ψ ij, ms ) (cid:13)(cid:13) s (cid:33) / (cid:32) N (cid:88) i =1 (cid:107) w (cid:107) a ( K i,k +1 ) + (cid:107) π ( w ) (cid:107) s ( K i,k +1 ) (cid:33) / (cid:46) ( k + 1) d/ (cid:32) N (cid:88) i =1 (cid:13)(cid:13) ψ ij − ψ ij, ms (cid:13)(cid:13) a + (cid:13)(cid:13) π ( ψ ij − ψ ij, ms ) (cid:13)(cid:13) s (cid:33) / (cid:16) (cid:107) w (cid:107) a + (cid:107) π ( w ) (cid:107) s (cid:17) / . (cid:107) w (cid:107) a + (cid:107) π ( w ) (cid:107) s (cid:46) ( k + 1) d N (cid:88) i =1 (cid:104)(cid:13)(cid:13) ψ ij − ψ ij, ms (cid:13)(cid:13) a + (cid:13)(cid:13) π ( ψ ij − ψ ij, ms ) (cid:13)(cid:13) s (cid:105) . This completes the proof.Finally, we state and prove the main result of this work. It reads as follows.
Theorem 4.9.
Let u be the solution of (2) and u ms be the solution of (10). Then, we have (cid:107) u − u ms (cid:107) a (cid:46) Λ − (cid:13)(cid:13)(cid:13) ˜ κ − / f (cid:13)(cid:13)(cid:13) + max { ˜ κ } ( k + 1) d/ E / (1 + D ) (cid:107) u glo (cid:107) s , where u glo is the solution of (14). Moreover, if the oversampling parameter k is sufficiently largeand { χ ms i } N c i =1 is a set of bilinear partition of unity, we have (cid:107) u − u ms (cid:107) a (cid:46) H Λ − (cid:107) f (cid:107) . Proof.
It follows from Galerkin orthogonality that (cid:107) u − u ms (cid:107) a ≤ (cid:107) u − v (cid:107) a for any v ∈ V ms . Wewrite u glo := N (cid:88) i =1 (cid:96) i (cid:88) j =1 c ij ψ ij and define a function v such that v := N (cid:88) i =1 (cid:96) i (cid:88) j =1 c ij ψ ij, ms . Then, we have (cid:107) u − u ms (cid:107) a ≤ (cid:107) u − v (cid:107) a ≤ (cid:107) u − u glo (cid:107) a + (cid:107) u glo − v (cid:107) a . The first term of the right-hand side can be estimated by the result of (17). It suffices to estimatethe second term. By Lemmas 4.7 and 4.8, we have (cid:107) u glo − v (cid:107) a = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N (cid:88) i =1 (cid:96) i (cid:88) j =1 c ij ( ψ ij − ψ ij, ms ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a ≤ C ( k + 1) d N (cid:88) i =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) i (cid:88) j =1 c ij ( ψ ij − ψ ij, ms ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) i (cid:88) j =1 c ij π ( ψ ij − ψ ij, ms ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) s ≤ C ( k + 1) d E N (cid:88) i =1 (cid:96) i (cid:88) j =1 ( c ij ) (cid:16)(cid:13)(cid:13) ψ ij (cid:13)(cid:13) a + (cid:13)(cid:13) πψ ij (cid:13)(cid:13) s (cid:17) . Choosing test function v = ψ ij in (8), we obtain that (cid:13)(cid:13)(cid:13) ψ ij (cid:13)(cid:13)(cid:13) a + (cid:13)(cid:13)(cid:13) πψ ij (cid:13)(cid:13)(cid:13) s ≤ (cid:13)(cid:13)(cid:13) φ ij (cid:13)(cid:13)(cid:13) s . Therefore, (cid:107) u glo − v (cid:107) a (cid:46) E ( k + 1) d N (cid:88) i =1 (cid:96) i (cid:88) j =1 ( c ij ) (cid:13)(cid:13) φ ij (cid:13)(cid:13) s = E ( k + 1) d N (cid:88) i =1 (cid:96) i (cid:88) j =1 ( c ij ) . (30)Next, we estimate the term N (cid:88) i =1 (cid:96) i (cid:88) j =1 ( c ij ) . Note that π ( u glo ) = N (cid:88) i =1 (cid:96) i (cid:88) j =1 c ij π ( ψ ij ). Using the vari-ational formulation (8) with test function v = ψ ij , we obtain b (cid:96)k := s (cid:16) π ( u glo ) , φ (cid:96)k (cid:17) = N (cid:88) i =1 (cid:96) i (cid:88) j =1 c ij s ( π ( ψ ij ) , φ (cid:96)k ) = N (cid:88) i =1 (cid:96) i (cid:88) j =1 c ij a ( ψ ij , ψ (cid:96)k ) + s ( π ( ψ ij ) , π ( ψ (cid:96)k )) (cid:124) (cid:123)(cid:122) (cid:125) =: a ij,(cid:96)k .
14f we denote b = ( b (cid:96)k ) ∈ R N and c = ( c ij ) ∈ R N with N := N (cid:88) i =1 (cid:96) i , then we have b = A c and (cid:107) c (cid:107) ≤ (cid:13)(cid:13) A − (cid:13)(cid:13) (cid:107) b (cid:107) , where A := ( a ij,(cid:96)k ) ∈ R N ×N and (cid:107)·(cid:107) denotes the standard Euclidean norm for vectors in R N and its induced matrix norm in R N ×N . By the definition of π : V → V aux , we have π ( u glo ) = π ( π ( u glo )) = N (cid:88) i =1 (cid:96) i (cid:88) j =1 s ( π ( u glo ) , φ ij ) φ ij = N (cid:88) i =1 (cid:96) i (cid:88) j =1 b ij φ ij . Thus, we have (cid:107) b (cid:107) = (cid:107) π ( u glo ) (cid:107) s . We define φ := N (cid:88) i =1 (cid:96) i (cid:88) j =1 c ij φ ij . Note that (cid:107) φ (cid:107) s = (cid:107) c (cid:107) .Consequently, by Lemma 4.6, there exists a function z ∈ V div0 such that π ( z ) = φ and (cid:107) z (cid:107) a ≤ D (cid:107) φ (cid:107) s . Since the multiscale basis ψ ij satisfies (8) and u glo is a linear combination of ψ ij ’s, wehave a ( u glo , v ) + s ( π ( u glo ) , πv ) = s ( φ, πv ) for all v ∈ V div0 (Ω (cid:15) ) . (31)Picking v = z in (31), we arrive at (cid:107) φ (cid:107) s = a ( u glo , z ) + s ( π ( u glo ) , πz ) ≤ (cid:107) u glo (cid:107) a · D / (cid:107) φ (cid:107) s + (cid:107) π u glo (cid:107) s · (cid:107) φ (cid:107) s ≤ (1 + D ) / (cid:107) φ (cid:107) s (cid:16) (cid:107) u glo (cid:107) a + (cid:107) π u glo (cid:107) s (cid:17) / . Therefore, we have (cid:107) c (cid:107) = (cid:107) φ (cid:107) s ≤ (1 + D ) (cid:16) (cid:107) u glo (cid:107) a + (cid:107) π u glo (cid:107) s (cid:17) = (1 + D ) c T A c . From the above, we see that the largest eigenvalue of A − is bounded by (1 + D ) and we havethe following estimate (cid:107) c (cid:107) ≤ (1 + D ) (cid:107) b (cid:107) = (1 + D ) (cid:107) u glo (cid:107) s . As a result, we have (cid:107) u glo − v (cid:107) a ≤ ( k + 1) d E (1 + D ) (cid:107) u glo (cid:107) s . It remains to estimate the term (cid:107) u glo (cid:107) s . In particular, we have (cid:107) u glo (cid:107) s (cid:46) max { ˜ κ } (cid:107) u glo (cid:107) a = max { ˜ κ } (cid:104) f , u glo (cid:105) ≤ max { ˜ κ } (cid:13)(cid:13)(cid:13) ˜ κ − / f (cid:13)(cid:13)(cid:13) (cid:107) u glo (cid:107) s . Therefore, we have (cid:107) u − u ms (cid:107) a (cid:46) Λ − (cid:13)(cid:13)(cid:13) ˜ κ − / f (cid:13)(cid:13)(cid:13) + max { ˜ κ } ( k + 1) d/ E / (1 + D ) (cid:13)(cid:13)(cid:13) ˜ κ − / f (cid:13)(cid:13)(cid:13) . If we take k = O (log( H − )) and assume that { χ ms i } N c i =1 is a set of bilinear partition of unity,then we have (cid:107) u − u ms (cid:107) a (cid:46) H Λ − (cid:107) f (cid:107) . This completes the proof. 15
Conclusion
In this work, we have proposed and analyzed the constraint energy minimizing generalized mul-tiscale finite element method for solving the incompressible Stokes flows in perforated domain.The proposed method started with a local spectral decomposition of the continuous Sobolevspace. Based on the concepts of constraint energy minimization and oversampling, we constructdivergence-free multiscale basis functions for displacement variable satisfying the property ofleast energy. The pressure variable is thus recovered on the coarse-grid based on the multiscaleapproximation of displacement. The method is shown to have spectral convergence with errorbound proportional to the coarse mesh size.
Acknowledgement
The research of Eric Chung is partially supported by the Hong Kong RGC General ResearchFund (Project numbers 14304217 and 14302018) and CUHK Faculty of Science Direct Grant2019-20.
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