A conservative local multiscale model reduction technique for Stokes flows in heterogeneous perforated domains
AA conservative local multiscale model reduction technique forStokes flows in heterogeneous perforated domains
Eric T. Chung ∗ Maria Vasilyeva † Yating Wang ‡ October 12, 2018
Abstract
In this paper, we present a new multiscale model reduction technique for the Stokes flows in hetero-geneous perforated domains. The challenge in the numerical simulations of this problem lies in the factthat the solution contains many multiscale features and requires a very fine mesh to resolve all details.In order to efficiently compute the solutions, some model reductions are necessary. To obtain a reducedmodel, we apply the generalized multiscale finite element approach, which is a framework allowing sys-tematic construction of reduced models. Based on this general framework, we will first construct a localsnapshot space, which contains many possible multiscale features of the solution. Using the snapshotspace and a local spectral problem, we identify dominant modes in the snapshot space and use them asthe multiscale basis functions. Our basis functions are constructed locally with non-overlapping supports,which enhances the sparsity of the resulting linear system. In order to enforce the mass conservation,we propose a hybridized technique, and uses a Lagrange multiplier to achieve mass conservation. Wewill mathematically analyze the stability and the convergence of the proposed method. In addition, wewill present some numerical examples to show the performance of the scheme. We show that, with a fewbasis functions per coarse region, one can obtain a solution with excellent accuracy.
Many application problems, such as fluid flow in heterogeneous porous media, involve perforated domains (seeFigure 1 for an example of perforated domain) where the perforations can have various sizes and geometries.Due to these features, the solutions of differential equations posed in perforated domains have multiscaleproperties. Numerical simulations for these problems are prohibitively expensive, because the computationalcost to recover the fine scale properties between perforations is extremely high. Similar to other types ofmultiscale problems, some model reduction methods are necessary in order to improve the computationalefficiency. There are in literature many model reduction techniques that are performed on a coarse grid whichhas much larger length scale compared with the size of perforations, such as numerical homogenization ([1,38, 34, 40, 27, 41, 3, 6, 29, 39, 30, 28, 42]) and multiscale methods ([31, 32, 11, 23, 20, 25, 33, 10, 2, 5, 37, 8]).In these approaches, macroscopic equations are formulated on a coarse grid with mesh size independent ofthe size of perforations. While these approaches are excellent in some cases, they are lack of systematicenrichment strategies in order to tackle problems with more complicated structures.The recently developed Generalized multiscale finite element method (GMsFEM) [23, 13] is a frameworkthat allows systematic enrichment of the coarse spaces and take into account fine scale information for the ∗ Department of Mathematics, The Chinese University of Hong Kong (CUHK), Hong Kong SAR. Email: [email protected] . † Department of Computational Technologies, Institute of Mathematics and Informatics, North-Eastern Federal University,Yakutsk, 677980, Republic of Sakha (Yakutia), Russia & Institute for Scientific Computation, Texas A&M University, CollegeStation, TX 77843. Email: [email protected] . ‡ Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA. Email: [email protected] . a r X i v : . [ m a t h . NA ] A ug onstruction of these spaces. The framework therefore provides a convincing approach to solve problemsposed in heterogeneous perforated domains, whose solutions have multiscale features and require sophisti-cated enrichment techniques. The main idea of GMsFEM is to employ local snapshots to approximate thefine scale solution space, and then identify local multiscale spaces by performing some carefully selectedlocal spectral problems defined in the snapshot spaces. The spectral problems give a systematic strategy toidentify the dominant modes in the snapshot spaces, and the dominant modes are selected to form the localmultiscale spaces. By appropriately choosing the snapshot space and the spectral problem, the GMsFEMrequires only a few basis functions per coarse region in order to obtain solutions with excellent accuracy. In[20, 18], we have developed and analyzed a GMsFEM for elliptic problem, elastic problem and the Stokesproblem in perforated domains using the continuous Galerkin (CG) framework. For this CG approach, wepartition the computational domain as a union of overlapping coarse neighborhoods, and construct a set oflocal multiscale basis functions for each coarse neighborhood. We also developed an adaptivity procedurebased on local residuals to enrich the coarse space by adaptively adding new basis functions. However, onedrawback of the CG approach is the need to multiply each basis function by a partition of unity function.This step may modify the local heterogeneity and cause some difficulties.In this paper, we propose a new GMsFEM for problems in perforated domains using a discontinuousGalerkin (DG) approach. The use of the DG approach in GMsFEM has been successfully developed formany problems, such as the elliptic equations and the wave equations with heterogeneous coefficients ([17,14, 12, 22, 16]). The main feature of the DG approach is that the basis functions are constructed locally foreach non-overlapping coarse region. This fact allows much more flexibility in the design of the coarse meshand in the choice of the local multiscale space. Another advantage of the DG approach is that there is no needto construct and use any partition of unity functions. We will, in this paper, consider a GMsFEM based ona DG approach for the Stokes flows in heterogeneous perforated domains. To construct the multiscale basisfunctions, we will obtain the local snapshots by solving the Stokes equations for each non-overlapping coarseregion with some suitable boundary conditions. Then, we will construct local spectral problems and identifydominant modes in the snapshot space. The multiscale space is obtained by the span of all these dominantmodes. Furthermore, it is important to note that the mass conservation is a crucial property for the Stokesflow. By the construction of the basis functions, the multiscale solution satisfies some local mass conservationproperty within coarse regions. However, mass conservation does not in general hold globally in the coarsegrid level. To tackle this issue, we construct a hybridized scheme and introduce additional pressure variableson the coarse grid edges. This additional pressure variable serves as a Lagrange multiplier to enforce themass conservation property in the coarse grid level. Thus, our new GMsFEM provides solutions using onlyfew basis functions per coarse regions, and having both local and global mass conservation.To investigate the performance of our proposed method, we will numerically study the Stokes problemin various perforated domains (see Figure 3) with various choices of boundary conditions and forcing terms.We will present the construction of the snapshot space using both the standard and the oversamplingapproaches ([24, 9]). Local spectral decompositions are also proposed for various approaches of snapshotscorrespondingly. Moreover, when constructing multiscale basis, we will test the use of different shapes ofcoarse blocks for different types of perforated domains. Numerical results are presented and convergence ofthe method is analyzed. Moreover, we will numerically show that the local mass conservation property issatisfied by the multiscale solution. Our numerical results show that we can approximate the solution usinga fairly small degrees of freedom. In addition, the oversampling technique can be particularly helpful andimprove the accuracy and the convergence.We organize the paper as follows. In Section 2, we state the model problem and define the fine and coarsescale discretizations. We present the detailed constructions of the snapshot space and the offline space inSection 3. Section 4 presents the numerical results for various examples. We analyze the stability and theconvergence of our method in Section 5. A conclusion is given at the end of the paper.2 Problem settings
In this section, we state the Stokes flow in heterogeneous perforated domains and introduce some notations.Let Ω ⊂ R n ( n = 2 ,
3) be a bounded domain. We define a perforated domain Ω (cid:15) ⊂ Ω with a set ofperforations denoted by B (cid:15) , that is, Ω (cid:15) = Ω \B (cid:15) . We assume that the set B (cid:15) contains circular perforationswith various sizes and positions. An illustration of a perforated domain is shown in Figure 1. We notice thatthe variable sizes and positions of these perforations lead to some multiscale features in the solutions of theproblems posed in perforated domains. Given the source function f and two boundary functions g D , g N , weconsider the following Stokes flow in the perforated domain Ω (cid:15) : − ∆ u + ∇ p = f, in Ω (cid:15) div u = 0 , in Ω (cid:15) (1)subject to boundary condition u = g D on Γ D , and ( ∇ u − pI ) n = g N on Γ N , where Γ D ∪ Γ N = ∂ Ω (cid:15) , n is theunit outward normal vector on ∂ Ω (cid:15) and I is the n × n identity matrix. The unknown variable u denotes thefluid velocity and p denotes the fluid pressure. Since p is uniquely defined up to a constant, we assume that (cid:82) Ω (cid:15) p = 0, so that the problem (1) has a unique solution.Figure 1: An illustration of a perforated domain.Let V (Ω (cid:15) ) = H (Ω (cid:15) ) n and Q (Ω (cid:15) ) = L (Ω (cid:15) ), where L (Ω (cid:15) ) is the set of L functions defined in Ω (cid:15) withzero mean. The variational formulation of (1) is given by: find u ∈ V (Ω (cid:15) ) and p ∈ Q (Ω (cid:15) ) such that a ( u, v ) + b ( v, p ) = ( f, v ) , for all v ∈ V (Ω (cid:15) ) b ( u, q ) = 0 , for all q ∈ Q (Ω (cid:15) ) (2)where a ( u, v ) = (cid:90) Ω (cid:15) ∇ u : ∇ v, b ( v, q ) = − (cid:90) Ω (cid:15) q div v and ( f, v ) = (cid:90) Ω (cid:15) f v. It is well known that there is a unique weak solution to (2) (see for example [7]).For the numerical approximation of the above problem, we first introduce the notations of fine and coarsegrids. Let T H be a coarse-grid partition of the domain Ω (cid:15) with mesh size H . We assume that this coarsemesh does not necessarily resolve the full details of the perforations. By using a conforming refinementof the coarse mesh T H , we can obtain a fine mesh T h of Ω (cid:15) with mesh size h . Typically, we assume that0 < h (cid:28) H <
1, and that the fine-scale mesh T h is sufficiently fine to fully resolve the small-scale information3f the domain, and T H is a coarse mesh containing many fine-scale feature. We use the notations K and E to denote a coarse element and a coarse edge in the coarse grid T H .We let E H be the set of edges in T H . We write E H = E Hint ∪ E
Hout , where E Hint is the set of interior edgesand E Hout is the set of boundary edges. For each interior edge E ∈ E Hint , we define the jump [ u ] and the average { u } of a function u by [ u ] E = u | K + − u | K − , { u } E = u | K + + u | K − , where K + and K − are the two coarse elements sharing the edge E , and the unit normal vector n on E isdefined so that n points from K + to K − . For E ∈ E Hout , we define[ u ] E = u | E , { u } E = u | E . Next we introduce our DG scheme. Similar to the standard derivation of DG formulations [4, 26, 35, 36],the main idea is to consider the problem in each element K in the coarse mesh, and impose boundaryconditions weakly on ∂K using the value of the velocity function in the neighboring elements. In addition, apenalizing term which penalize the jump of velocity will be introduced. After obtaining the local problemsin each element, one can sum over all elements to get the global DG scheme. Remark that, in our approach,we will only assume discontinuity across the coarse edges, but use the standard continuous element insidecoarse blocks. In this work, we also add an additional Lagrange multiplier in order to impose local massconservation on the coarse elements. The details are given as follows.We start with the definitions of the approximation spaces. We let Q H be the piecewise constant functionspace for the approximation of the pressure p . That is, the restriction of the functions of Q H in each coarseelement is a constant. In addition, we will define a piecewise constant space (cid:98) Q H for the approximation ofthe pressure (cid:98) p , which is defined on the set of coarse edges E H . That is, the functions in (cid:98) Q H are defined onlyin E H and the restriction of the functions of (cid:98) Q H in each coarse edge is a constant. We remark that thisadditional pressure space is used to enforce local mass conservation in the coarse grid level. Moreover, wedefine V H as the multiscale velocity space, which contains a set of basis functions supported in each coarseblock K . To obtain these basis functions, we will solve some local problems in each coarse block with variousDirichlet boundary conditions to form a snapshot space and use a spectral problem to perform a dimensionreduction. The details for the construction of this space will be presented in the next section.For our GMsFEM using a DG approach, we define the bilinear forms a DG ( u, v ) = (cid:90) Ω (cid:15) ∇ u : ∇ v − (cid:88) E ∈E H (cid:16) (cid:90) E { ( ∇ u ) n } · [ v ] + { ( ∇ v ) n } · [ u ] (cid:17) + γh (cid:88) E ∈E H (cid:90) E [ u ] · [ v ] , (3) b DG ( v, q, (cid:98) q ) = − (cid:88) K ∈T H (cid:90) K q div v + (cid:88) E ∈E H (cid:90) E (cid:98) q ([ v ] · n ) . (4)Then, we will find the multiscale solution ( u H , p H , (cid:98) p H ) ∈ V H × Q H × (cid:98) Q H such that a DG ( u H , v ) + b DG ( v, p H , (cid:98) p H ) = ( f, v ) + (cid:90) Γ D (cid:16) γh g D · v − (( ∇ v ) n ) · g D (cid:17) + (cid:90) Γ N g N · v,b DG ( u H , q, (cid:98) q ) = (cid:90) Γ D ( g D · n ) (cid:98) q, (5)for all v ∈ V H , q ∈ Q H , (cid:98) q ∈ (cid:98) Q H . The derivation of the above scheme follows the standard DG derivationprocedures [4, 26, 35, 36]. We notice that the role of the variable (cid:98) p H is to enforce mass conservation oncoarse elements. In particular, taking q = 0 in (5), we have (cid:90) E (cid:98) q [ u H ] · n = 0 , ∀ E ∈ E Hint , ∀ (cid:98) q ∈ (cid:98) Q H . (cid:90) K q div u H = 0 , ∀ K ∈ T H , ∀ q ∈ Q H . The above is the key to the mass conservation, and we will discuss more in the numerical results section.We will show the accuracy of our method by comparing the multiscale solution to a reference solution,which is computed on the fine mesh. To find the reference solution ( u h , p h , (cid:98) p h ), we will solve the followingsystem a DG ( u h , v ) + b DG ( v, p h , (cid:98) p h ) = ( f, v ) + (cid:90) Γ D (cid:16) γh g D · v − (( ∇ v ) n ) · g D (cid:17) + (cid:90) Γ N g N · v,b DG ( u h , q, (cid:98) q ) = (cid:90) Γ D ( g D · n ) (cid:98) q, (6)for all v ∈ V DG h , q ∈ Q H , (cid:98) q ∈ (cid:98) Q H . We note that the reference velocity u h belongs to the fine scale velocityspace V DG h = { v ∈ L (Ω (cid:15) ) | v | K ∈ C ( K ) for every K ∈ T H , v | K ∈ ( P ( T )) for every K ∈ T h } . Thespace V DG h contains functions which are piecewise linear in each fine-grid element K and are continuousalong the fine-grid edges, but are discontinuous across coarse grid edges. Moreover, the reference pressure p h and (cid:98) p h belongs to the coarse scale pressure space Q H and (cid:98) Q H respectively. Notice that the pressure p h is determined up to a constant, we will achieve the uniqueness by requiring the averaging value of pressureover whole domain is zero. We remark that this reference solution ( u h , p h , (cid:98) p h ) is obtained using the coarsescale pressure spaces Q H and (cid:98) Q H since we only consider multiscale solutions and reduced spaces for thevelocity. The true fine scale solution ( u fine , p fine , (cid:98) p fine ) can be defined by a DG ( u fine , v ) + b DG ( v, p fine , (cid:98) p fine ) = ( f, v ) + (cid:90) Γ D (cid:16) γh g D · v − (( ∇ v ) n ) · g D (cid:17) + (cid:90) Γ N g N · v,b DG ( u fine , q, (cid:98) q ) = (cid:90) Γ D ( g D · n ) (cid:98) q, for all v ∈ V DG h , q ∈ Q h , (cid:98) q ∈ (cid:98) Q h , where Q h and (cid:98) Q h are suitable fine scale spaces. One can see that ( u fine , p fine )will converge to the exact solution ( u, p ) in the energy norm as the fine mesh size h →
0. Moreover, one canshow that (cid:107) u fine − u h (cid:107) A ≤ C inf q ∈ Q H , (cid:98) q ∈ (cid:98) Q H (cid:107) ( p fine − q, (cid:98) p fine − (cid:98) q ) (cid:107) Q where the norms are defined in (10) and (11). Thus, the reference solution defined in (6) can be consideredas the exact solution up to a coarse scale approximation error. In this section, we will present the construction of the multiscale space V H for the coarse scale approximationof velocity. To construct the coarse scale velocity space, we will follow the general idea of GMsFEM [23, 24],which contains two stages: (1) the construction of snapshot space, and (2) the construction of offline space.In the first stage, we will obtain the snapshot space, which contains a rich set of functions containing possiblefeatures in the solution. These snapshot functions are solutions of some local problems subject to all possibleboundary conditions up to the fine grid resolution. Notice that for the generalized multiscale DG schemeproposed in [14, 15], one solves the local problems in each coarse block. Thus the resulting system is muchsmaller compared with that of the CG approach [20, 12, 19], where the local problems are solved in eachoverlapping coarse neighborhood. Next, in order to reduce the dimension of the solution space, we will use aspace reduction technique to choose the dominated modes in the snapshot space. This procedure is achievedby defining proper local spectral problems. The resulting reduced order space is called the offline space andwill be used for coarse scale velocity approximation. Note that for approximating pressure on the coarse grid,we will use piecewise constant functions as defined before. In Section 3.1, we will present the constructionof the snapshot space, and in Section 3.2, we will present the construction of the offline space.5 .1 Snapshot space We will construct local snapshot basis in each coarse block K i , ( i = 1 , · · · , N ), where N is the number ofcoarse blocks in Ω (cid:15) . The local snapshot space consists of functions which are solutions u ∈ V h ( K i ) of − ∆ u + ∇ p = 0 , in K i div u = c, in K i (7)with u = δ ki on ∂K i , ( k = 1 , · · · , M i ), where M i is the number of fine grid nodes on the boundary of K i ,and δ ki is the discrete delta function defined on ∂K i . The above problem (7) is solved on the fine meshusing some appropriate approximation spaces. For instances, we take the space V h ( K i ) to be the standardconforming piecewise linear finite element space with respect to the fine grid on K i . Note that the constant c in (7) is chosen by the compatibility condition, that is, c = | K i | (cid:82) ∂K i δ ki · nds .Take these M i velocity solutions of (7) and denote them by ψ i, snap k ( k = 1 , · · · , M i ), we get the localsnapshot space V i snap = span { ψ i, snap1 , · · · , ψ i, snap M i } . Combining all the local snapshots, we can form the global snapshot space, that is V snap = span { ψ i, snap k , , ≤ k ≤ M i , ≤ i ≤ N } . In the above construction, the local problems are solved for every fine grid node on ∂K i . One can alsoapply the oversampling strategy [5, 24] in order to reduce the boundary effects. Applying this strategy, onecan solve the local problem for each fine node on the boundary of the oversampled domain. An illustrationof the original local domain K and the oversampled local domain K + are shown in Figure 2. Notice that inFigure 2, we present the triangular coarse grid in perforated domain with small inclusions on the left, andrectangular coarse grid perforated domain with multiple sizes of inclusions on the right. We will solve thelocal problem in an enlarged domain K + i of K i , − ∆ u + ∇ p = 0 , in K + i div u = c, in K + i with u = δ ki on ∂K + i , where k = 1 , · · · , M + i , where M + i is the number of fine nodes on the boundary of K + i .After removing linear dependence among these basis by POD, we denote the linearly independent functionsby ψ + ,ik , ( i = 1 , · · · , ˜ M i ). Note that the velocity solutions of these local problems are supported in thelarger domain K + i . There are usually several following choices for identification of basis. One of the straightforward way is that, we can restrict the basis ψ + ,ik on K i to form the snapshot basis, i.e. ψ i, snap k = ψ + ,ik | K i .Then the span of these basis function ψ i, snap k will form our new snapshot space. In this case, the localreduction will be performed in K i . Another choice is that, one can keep the snapshot basis ψ + ,ik withoutrestricting on K i . But in this case, one needs to solve the offline basis also in the oversampled domain K + i and finally restrict the offline basis on the original local domain K i . It is known that these oversamplingmethods can improve the accuracy of our multiscale methods ([24]).We remark that one can also use the idea of randomized snapshots (as in [9]) and reduce the computationalcost substantially. In randomized snapshots approach, instead of solving the local problem for each fine nodeon the boundary of oversampled local domain, one only computes a few snapshots in each oversampleddomain K + i with several random boundary conditions. These random boundary functions are constructedby independent identically distributed (i.i.d.) standard Gaussian random vectors defined on the fine degreesof freedoms on the boundary. The randomized snapshot requires much fewer calculations to achieve a goodaccuracy compared with the standard snapshot space. In this section, we will perform local model reduction on the snapshot space by solving some local spectralproblems. The reduced space consists of the important modes in the snapshot space, and is called the offline6igure 2: Illustration of oversampling domain. Left: Oversampling of a triangular coarse block for perforateddomain with small inclusions. Right: Oversampling of a rectangular coarse block for perforated domain withmultiple sizes of inclusions.space. The coarse scale approximation of velocity solution will be obtained in this space. We have multiplechoices of local spectral problems given the various constructions of snapshot space presented in the previoussection.First of all, if the snapshot basis obtained in the previous section is supported in each coarse element K i ,we will solve for ( λ, Φ) from the generalized eigenvalue problem in the snapshot space A Φ = λS Φ (8)where A is the matrix representation of the bilinear form a i ( u, v ) and S is the matrix representation of thebilinear form s i ( u, v ). The choices for a i and s i are based on the analysis. In particular, we take a i ( u, v ) = (cid:90) K i ∇ u : ∇ v,s i ( u, v ) = λH (cid:90) ∂K i u · v, where we remark that the integral in s i ( u, v ) is defined on the boundary of the coarse block. In this case,the number of the spectral problem equals the number of coarse blocks.We arrange the eigenvalues of (8) in increasing order. We will choose the first few eigenvectors corre-sponding to the first few small eigenvalues. Using these eigenvectors as the coefficients, we can form ouroffline basis. More precisely, assume we arrange the eigenvalues in increasing order λ ( i )1 < λ ( i )2 < · · · < λ ( i ) M i . The corresponding eigenvectors are denoted by Φ ( i ) k = (Φ ( i ) kj ) M i j =1 , where Φ ( i ) kj is the j -th component of theeigenvector. We will take the first L i ≤ M i eigenvectors to form the offline space, that is, the offline basisfunctions can be constructed as φ i, off k = M i (cid:88) j =1 Φ ( i ) kj ψ i, snap k , k = 1 , · · · , L i . On the other hand, one can use the snapshot basis ψ + ,ik (using oversampling strategy) without restrictingon K i in the space reduction process. To be more specific, since the snapshot basis are supported in the7versampled domain K + i , we will need another set of spectral problems, namely A + Φ + = λS + Φ + (9)where A + and S + are the matrix representations of the bilinear forms a + ,i ( u, v ) and s + ,i ( u, v ) respectively.Similar as before, we can choose a + ,i , s + ,i as follows a + ,i ( u, v ) = (cid:90) K + i ∇ u : ∇ v,s + ,i ( u, v ) = λH (cid:90) ∂K + i u · v. We then arrange the eigenvalues in increasing order λ ( i )1 < λ ( i )2 < · · · < λ ( i ) M + i . The corresponding eigenvectors are denoted by Φ + , ( i ) k . We will take the first L i ≤ M + i eigenvectors to forma basis supported in K + i φ + ,ik = M + i (cid:88) j =1 Φ + , ( i ) kj ψ + ,ik , k = 1 , · · · , L i . Then we will obtain our offline basis by restricting φ + ,ik on K i , namely φ i, off k = φ + ,ik | K i . Now we can finally form the local offline space, which is the span of these basis functions V i off = span { φ i, off1 , · · · , φ i, off L i } . The global offline space V off is the combination of the local ones, i.e. V off = span { φ i, off k , , ≤ k ≤ L i , ≤ i ≤ N } . This space will be used as the coarse scale approximation space for velocity V H := V off . In this section we will present numerical results of our method for various types of perforations, boundaryconditions and sources. We will illustrate the performance of our method using two kinds of perforateddomains: (1) perforated domain with small inclusions and (2) perforated domain with big inclusions as wellas some extremely small inclusions, see Figure 3. We will also illustrate the performance of the oversamplingstrategy.We set Ω = [0 , × [0 , H = . For the fine scale discretization, the size of thesystem is 69146 for domain with small inclusions (Figure 3, left) and 91588 for domain with multiple size ofinclusions (Figure 3, right).We will consider two different boundary conditions and force terms: • Example 1 : Source term f = (0 , u = (1 ,
0) on ∂ Ω and u = (0 ,
0) on ∂ B (cid:15) . • Example 2 : Source term f = (1 , ∂u∂n − pn = (0 ,
0) on ∂ Ω and u = (0 ,
0) on ∂ B (cid:15) .8igure 3: Illustration of the perforated domain with fine and coarse mesh. Left: perforated domain withsmall inclusions. Right: perforated domain with multiple sizes of inclusions.The errors will be measured in relative L , H and DG norms for velocity, and L norm for pressure || e u || L = (cid:107) u h − u H (cid:107) L (Ω (cid:15) ) (cid:107) u h (cid:107) L (Ω (cid:15) ) , || e u || H = (cid:107) u h − u H (cid:107) H (Ω (cid:15) ) (cid:107) u h (cid:107) H (Ω (cid:15) ) , || e u || DG = (cid:112) a DG ( u h − u H , u h − u H ) (cid:112) a DG ( u h , u h ) , || e p || L = (cid:107) ¯ p h − p H (cid:107) L (Ω (cid:15) ) (cid:107) ¯ p h (cid:107) L (Ω (cid:15) ) . where ¯ p h is the cell average of the fine scale pressure, that is, ¯ p h = | K i | (cid:82) K i p h for all K i ∈ T H . In this section, we show the numerical results for the Stokes problem in perforated domain with smallinclusions (left of Figure 3), see Table 1 for Example 1 and Table 2 for Example 2. Remark that the finescale system has size 69146, while our coarse scale systems only have size 1280 − u = (1 ,
0) on the global boundary, and u = (0 ,
0) on the boundary ofinclusions. The force term f = (0 , L velocity error reduce from 33 .
2% to 6 .
5% when the number of basis increase from 4 to 8. Moreover,the energy error for velocity is 28% and the L error for pressure is 12% as we take 32 offline basis for non-oversampling case. To get a faster convergence, we employ oversampling strategy when calculating the basis,that is, we solve the local problems in an oversampled coarse domain and then restrict the local velocitysolution to the original coarse block to form our snapshot basis. In our numerical example, the oversampleddomain is the original coarse block plus four fine cells layers neighboring the original domain. We can seethat, the oversampling case gives us better accuracy with respect to velocity energy error and pressure error.For instance, the velocity energy error reduces from 25% to 18% and the pressure error decreased from 12%to 2% when the number of offline basis is 32 comparing the non-oversampling with oversampling case.For the second example in perforated domain with small inclusions, we take Neumann boundary condition ∂u∂n − pn = (0 ,
0) on the global boundary and Dirichlet condition u = (0 ,
0) on the boundary of inclusions.9he convergence history is shown in Table 2. From this table, we find that the velocity L error reduce from39 .
9% to 7 . .
8% to 5 .
0% when the basis number increase from 4 to8 for the non-oversampling case. Moreover, the velocity L error reduces to 4 .
9% when we take 32 basis. Wealso observe that the oversampling strategy works efficiently to speed up the convergence rate for both the L error and the energy error for velocity. For example, the velocity L error is 7 .
6% when we take 8 basis innon-oversampling case, however, it is only 2 .
6% when we take the same number of basis in oversampling case.The velocity H error reduce from 30 . . (cid:82) ∂K i u · n ds in Table 3. From the table, we see that the maximum of the values (cid:82) ∂K i u · n ds isalmost zero for all cases. This shows that we have exact mass conservation in the coarse grid level. Weremark that we also have fine grid mass conservation by the construction of the basis functions.Figure 4 and Figure 5 shows the corresponding solution plots for Example 1 and Example 2 in perforateddomain with small inclusions, where we compare the fine scale velocity solution with different coarse scalevelocity solution. In Figure 4, we take 8 and 16 basis functions per coarse element for coarse scale com-putations. We observe that some fine scale features are lost in the solution when we take 8 basis, and theframe of the coarse edges can be seen in the figure. However, when we take 16 basis, we can observe a muchsmoother solution which capture the fine features well. Similar behavior can be found in Figure 5, where weobserve higher contrast between 4 basis per element and 16 basis per element coarse scale solutions. M off DOF || e u || L || e u || DG || e u || H || e p || L Non-oversampling4 1280 33.2 96.8 76.8 –8 2080 6.5 48.8 43.7 38.116 3680 2.6 31.9 28.9 1232 6880 1.9 28.3 25.3 12Oversampling, K + = K + 44 1280 32.6 85.7 69.9 –8 2080 6.6 39.6 36.7 23.416 3680 1.9 21.7 19.4 2.732 6880 1.8 20.3 18.5 2.7Table 1: Stokes problem in perforated domain with small inclusions. Numerical results for Example 1 .Non-oversampling and oversampling with 4 fine layers. M off DOF || e u || L || e u || DG || e u || H || e p || L Non-oversampling4 1280 39.9 87.6 71.2 69.88 2080 7.6 49.4 39.5 5.016 3680 6.7 36.7 31.8 2.632 6880 4.9 35.9 30.5 2.9Oversampling, K + = K + 44 1280 31.7 69.6 52.6 –8 2080 2.6 36.7 27.8 16.816 3680 1.8 25.5 20.7 3.632 6880 1.5 20.3 17.4 3.5Table 2: Stokes problem in perforated domain with small inclusions. Numerical results for Example 2 .Non-oversampling and oversampling with 4 fine layers.10xample 1 M off DOF Non-oversampling Oversampling4 1280 2.9e-20 -4.4e-228 2080 6.6e-18 -4.2e-1816 3680 5.7e-19 -9.5e-1832 6880 -4.0e-18 1.2e-1532 6880 -4.0e-18 1.2e-15Example 2 M off DOF Non-oversampling Oversampling4 1280 -8.4e-22 4.1e-228 2080 -1.3e-19 -1.9e-2016 3680 1.9e-19 -5.8e-2232 6880 9.7e-20 -5.1e-18Table 3: Stokes problem in perforated domain with small inclusions. Verification of local mass conservationon coarse edges by computing the maximum of (cid:82) ∂K i u · n ds over all coarse blocks. Top: Example 1. Bottom:Example 2 In this section, we show the numerical results for the Stokes problem in perforated domain with various sizeof inclusions (right of Figure 3), see Table 4 for Example 1 and Table 5 for Example 2. The fine degrees offreedoms for this domain is 91588, and the coarse degrees of freedoms range only from 680 for 4 basis perelement to 3480 for 32 basis per coarse element. Note that, in this domain we use the coarse mesh whereeach block is a rectangle, thus the coarse degrees of freedom is less than that in the previous section wherewe used triangular blocks for coarse mesh. From the tables, we can see that for Example 1, the velocity L errors can be less than 10% when we take more than 8 basis. Moreover, for Example 2, the velocity L errorsare already 6 .
1% (or 3 . H error become 12 .
9% in the oversampling case, which is muchsmaller than 20 .
1% in the non-oversampling case. The oversampling strategy works even better to improvethe velocity results for Example 2. Table 5 shows that the velocity L , H and DG errors are almost reducedby half when we take 8, 16 or 32 basis applying the oversampling strategy. The local mass conservation isalso verified by the data presented in Table 6. Figure 6 and Figure 7 demonstrate the velocity solution plotsfor Example 1 and Example 2 respectively. In Figure 6, we compare the fine scale velocity solution with 8basis coarse scale solution and 16 basis coarse scale solution. It is clear to see that when we take 8 basis, thehigher value regions in the solution shrinks, and some properties of the solution between two inclusions arenot captured well. These drawbacks are recovered better when we take 16 basis, and the solution is morecomparable with fine scale solution. The solution is reported in Figure 7 for Example 2, where we compare4 basis and 16 basis coarse scale solution with fine scale solution. The behavior is similar as before.In addition, in Figure 8, we present the comparison the solutions for Example 2 in perforated domain withsmall inclusions (left of Figure 3) in oversampling and non-oversampling case respectively. The x-componentof velocity is shown on the top, and the y-component is on the bottom, the results for non-oversamplingare on the left ( L error 6 . H error 31 . L error1 . H error 20 . off DOF || e u || L || e u || DG || e u || H || e p || L Non-oversampling4 680 46.6 93.4 79.7 –8 1080 11.5 55.0 52.1 39.616 1880 2.9 27.9 25.9 9.132 3480 1.9 22.3 20.1 5.6Oversampling, K + = K + 44 680 50.8 83.3 76.3 –8 1080 10.8 48.1 45.3 31.616 1880 4.5 23.4 21.6 2.532 3480 1.6 14.5 12.9 2.1Table 4: Stokes problem in perforated domain with additional small inclusions. Numerical results for Example1 . Non-oversampling and oversampling with 4 fine layers. M off DOF || e u || L || e u || DG || e u || H || e p || L Non-oversampling4 680 63.1 96.6 82.1 33.68 1080 6.1 47.7 36.5 3.716 1880 3.8 28.4 24.2 1.532 3480 2.9 26.6 22.3 1.4Oversampling, K + = K + 44 680 41.6 65.6 54.3 –8 1080 3.5 29.3 23.1 11.816 1880 1.7 15.5 13.0 4.332 3480 1.3 12.9 11.0 2.8Table 5: Stokes problem in perforated domain with additional small inclusions. Numerical results for Example2 . Non-oversampling and oversampling with 4 fine layers.Example 1 M off DOF Non-oversampling Oversampling4 680 2.3e-20 2.6e-208 1080 1.8e-20 -5.5e-2016 1880 -8.2e-18 5.5e-1832 2480 3.9e-20 3.5e-17Example 2 M off DOF Non-oversampling Oversampling4 680 1.8e-23 1.0e-228 1080 -5.1e-22 -1.8e-2216 1880 4.7e-19 1.2e-1932 2480 1.4e-20 -5.2e-21Table 6: Stokes problem in perforated domain with small inclusions. Verification of local mass conservationon coarse edges by computing the maximum value of (cid:82) ∂K i u · n ds over all coarse blocks. Top: Example 1.Bottom: Example 2 12igure 4: Stokes problem for perforated domain with small inclusions. Numerical solution for Example 1.Top: x-component of velocity. Bottom: y-component of velocity. Left: Fine-scale solution. Middle: Coarse-scale solution with 8 basis, non-oversampling. Right: Coarse-scale solution with 16 basis, non-oversampling. In this section, we will present the analysis of our multiscale method (5). First, we will prove the existenceand uniqueness of the problem (5) by showing the coercivity and continuity of a DG , the continuity of b DG and the discrete inf-sup condition for b DG . Next, we will derive a convergence result for our method. Forour analysis, we define the energy norm (cid:107) u (cid:107) A = (cid:90) Ω |∇ u | + 1 h (cid:88) E ∈E H (cid:90) E | [ u ] | . (10)Moreover, we define the following L norm (cid:107) ( q, (cid:98) q ) (cid:107) Q = (cid:107) q (cid:107) L (Ω (cid:15) ) + (cid:88) E ∈E H h (cid:107) (cid:98) q (cid:107) L ( E ) . (11)The notation α (cid:46) β means that α ≤ Cβ for a constant C independent of the mesh size. We notice that the Q -norm in (11) is a weaker norm compared with the more usual choice (cid:107) q (cid:107) L (Ω (cid:15) ) + (cid:80) E ∈E H H (cid:107) (cid:98) q (cid:107) L ( E ) .First, we consider the continuity and coercivity of the bilinear form a DG , as well as the continuity of thebilinear form b DG . These properties are summarized in the following lemma. Lemma 5.1.
Assume that γ = O (1) is large enough. The bilinear form a DG is continuous and coercive, that is | a DG ( u, v ) | ≤ a (cid:107) u (cid:107) A (cid:107) v (cid:107) A (12) a DG ( u, u ) ≥ a (cid:107) u (cid:107) A (13) and the bilinear form b DG is also continuous: | b DG ( v, q, (cid:98) q ) | ≤ b (cid:107) v (cid:107) A (cid:107) ( q, (cid:98) q ) (cid:107) Q . (14) Proof.
The proof for continuity and coercivity of a DG is classical [35, 14, 17], and will be omitted here. Forthe continuity of b DG , it follows from the Cauchy-Schwarz inequality. In this section, we will prove an inf-sup condition for the bilinear form b DG ( v, q, (cid:98) q ). We will assume thecontinuous inf-sup condition holds for b ( v, q ). That is, for any q ∈ L (Ω (cid:15) ), we havesup u ∈ H (Ω (cid:15) ) b ( u, q ) (cid:107) u (cid:107) H (Ω (cid:15) ) ≥ β (cid:107) q (cid:107) L (Ω (cid:15) ) . (15)We will also assume the following independence condition for the multiscale basis. For every coarse block K i ∈ T H , there are at least 4 basis functions, denoted by φ i, off j , j = 1 , , ,
4, in the local offline space V i off d jk such that (cid:90) E l (cid:16) (cid:88) j =1 d jk φ i, off j (cid:17) · n = δ kl , k, l = 1 , , , , (16)for all coarse edges E l on the boundary of K i . We remark that the above independence condition says thatwe can construct a function in V i off with normal component having mean value one on one coarse edge andmean value zero on the other coarse edges. In particular, for each coarse element K i , and for every edge E j ∈ ∂K i , there is a basis function Ψ j such that (cid:82) E j Ψ j · n = 1 and (cid:82) E k Ψ j · n = 0 for other coarse edges E k ∈ ∂K i .The next lemma is the main result of this section. Lemma 5.2.
For all q ∈ Q H and (cid:98) q ∈ (cid:98) Q H , we have (cid:107) ( q, (cid:98) q ) (cid:107) Q ≤ C infsup sup v ∈ V H b DG ( v, q, (cid:98) q ) (cid:107) v (cid:107) A (17) where C infsup > is a constant independent of the mesh size, provided the fine mesh size h is small enough.Proof. Let q ∈ Q H and (cid:98) q ∈ (cid:98) Q H be arbitrary. By the continuous inf-sup condition (15), there is u ∈ H (Ω (cid:15) ) such that div u = q and (cid:107) u (cid:107) H (Ω (cid:15) ) ≤ β − (cid:107) q (cid:107) L (Ω (cid:15) ) . By the assumption (16), for each coarse element K i , andfor every edge E j ∈ ∂K i , there is a basis function Ψ j such that (cid:82) E j Ψ j · n = 1, and (cid:82) E k Ψ j · n = 0 for other15igure 7: Stokes problem for perforated domain with large inclusions. Numerical solution for Example 2.Top: x-component of velocity. Bottom: y-component of velocity. Left: Fine-scale solution. Middle: Coarse-scale solution with 4 basis, non-oversampling. Right: Coarse-scale solution with 16 basis, non-oversampling.coarse edges E k ∈ ∂K i . Note that we suppress the dependence of Ψ j on i to simplify the notations. Thenwe define v ∈ V off by v = (cid:88) K i ∈T H (cid:88) E j ∈ ∂K i c j,i Ψ j , with c j,i = (cid:90) E j u · n. (18)It is clear that (cid:90) E j v · n = (cid:90) E j u · n, and (cid:90) E k v · n = 0 . In addition, we define v so that (cid:82) E v · n = 0 for all boundary edges E ∈ ∂ Ω (cid:15) . We also choose the normalvectors in (18) so that the average jumps of v · n across all interior coarse edges are zero. This conditioncan be achieved by choosing a fixed normal direction for each coarse edge in the definition (18). By thedefinition of b DG , integration by parts and using the definition of v , we have b DG ( v , q, (cid:98) q ) = b ( u, q ) = (cid:107) q (cid:107) L (Ω (cid:15) ) . Next, we will show that (cid:107) v (cid:107) A ≤ α (cid:107) q (cid:107) L (Ω (cid:15) ) for some positive constant α . We define the energy D j of thebasis function Ψ j by D j := (cid:90) K i |∇ Ψ j | + 1 h (cid:90) ∂K i | Ψ j | . So, by the definition of (cid:107) · (cid:107) A , the trace inequality and the continuous inf-sup condition, (cid:107) v (cid:107) A ≤ (cid:88) K i ∈T H (cid:88) E j ∈ ∂K i c j,i D j (cid:46) α (cid:107) q (cid:107) L (Ω (cid:15) ) , α = max K i ∈T H max E j ∈ ∂E i D j . On the other hand, we can choose v ∈ V off such that (cid:90) E j v · n = 12 ( hH ) (cid:98) q, and (cid:90) E k v · n = 0if E j is an interior edge, or (cid:90) E j v · n = ( hH ) (cid:98) q, and (cid:90) E k v · n = 0if E j is a boundary edge, where n is the outward normal vector on the boundary of K i . This can be achievedby defining v = (cid:88) K i ∈T H (cid:88) E j ∈ ∂K i d j,i Ψ j , with d j,i = σ ( hH ) (cid:98) q (19)17here σ = 1 or σ = 1 / E j . Thus, we have (cid:90) E j [ v ] · n = ( hH ) (cid:98) q on all interior coarse edges. By the definition of b DG , b DG ( v , q, (cid:98) q ) = − (cid:88) K ∈T H (cid:90) K q div v + h (cid:88) E ∈E H (cid:107) (cid:98) q (cid:107) L ( E ) . We can show that (cid:107) v (cid:107) A ≤ C α ( hH ) h (cid:80) E ∈E H (cid:107) (cid:98) q (cid:107) L ( E ) using arguments similar as above, where the constant C is independent of the mesh size.Finally, we let v = α v + v ∈ V off . Then b DG ( v, q, (cid:98) q ) = α (cid:107) q (cid:107) L (Ω (cid:15) ) − (cid:88) K ∈T H (cid:90) K q div v + h (cid:88) E ∈E H (cid:107) (cid:98) q (cid:107) L ( E ) . Using the Young’s inequality, we have b DG ( v, q, (cid:98) q ) ≥ α (cid:107) q (cid:107) L (Ω (cid:15) ) − C α ( hH ) (cid:88) K ∈T H (cid:90) K div v − C α ( hH )2 (cid:88) K ∈T H (cid:90) K q + h (cid:88) E ∈E H (cid:107) (cid:98) q (cid:107) L ( E ) which implies b DG ( v, q, (cid:98) q ) ≥ ( α − C α ( hH )2 ) (cid:107) q (cid:107) L (Ω (cid:15) ) + 12 h (cid:88) E ∈E H (cid:107) (cid:98) q (cid:107) L ( E ) . Taking α = C α ( hH ) and assuming that the fine mesh size h is small enough so that C α ( hH ) = O (1), weobtain b DG ( v, q, (cid:98) q ) ≥ C (cid:107) ( q, (cid:98) q ) (cid:107) Q where C is a constant independent of the mesh size. Moreover, (cid:107) v (cid:107) A (cid:46) α (cid:107) v (cid:107) A + (cid:107) v (cid:107) A (cid:46) α α (cid:107) q (cid:107) L (Ω (cid:15) ) + α h (cid:88) E ∈E H (cid:107) (cid:98) q (cid:107) L ( E ) . Thus, choosing h small enough, we have (cid:107) v (cid:107) A (cid:46) (cid:107) ( q, (cid:98) q ) (cid:107) Q . In this section, we will derive an error estimate between the fine scale solution u h and coarse scale solution u H . First, we construct a projection of the fine grid velocity in the snapshot space, and estimate the errorfor this projection. Second, we will estimate the difference between this projection and coarse scale velocity.Combine these two errors, we obtain the results as desired. Theorem 5.3.
Let u h be the fine scale velocity solution in (6) , and u H be the coarse scale velocity solutionof (5) . The following estimate holds (cid:107) u h − u H (cid:107) A (cid:46) N (cid:88) i =1 Hλ ( i ) L i +1 (1 + Hhλ ( i ) L i +1 ) (cid:90) ∂K i | ( ∇ u snap ) n | + H (cid:107) f (cid:107) L (Ω (cid:15) ) , where u snap is the snapshot solution defined in (20) . roof. Let ( u h , p h ) ∈ V DG h × Q H be the fine scale solution satisfying (6). We will next define a projection,denoted u snap , of u h in the snapshot space V snap . For each coarse element K , the restriction of u snap on K is defined by solving − ∆ u snap + ∇ p snap = 0 , in K div u snap = c, in Ku snap = u h , on ∂K (20)where p snap is a constant, and c is chosen by the compatibility condition, c = | K | (cid:82) ∂K u h · n ds . We remarkthat u snap is obtained on the fine grid, and we therefore have u snap ∈ V snap . We define u off as the projectionof u snap in the offline space V H . Using [14], we obtain (cid:107) u snap − u off (cid:107) A ≤ N (cid:88) i =1 Hλ ( i ) L i +1 (1 + Hhλ ( i ) L i +1 ) (cid:90) ∂K i | ( ∇ u snap ) n | . (21)Next, by comparing (5) and (6), we have a DG ( u h − u H , v ) + b DG ( v, p h − p H , (cid:98) p h − (cid:98) p H ) = 0 ,b DG ( u h − u H , q, (cid:98) q ) = 0 , (22)for all v ∈ V H , q ∈ Q H , (cid:98) q ∈ (cid:98) Q H . Then, using the inf-sup condition (17) and standard arguments, we have (cid:107) u h − u H (cid:107) A (cid:46) (cid:107) u h − u off (cid:107) A . (23)Finally, we define u h = u snap + u , where u = u h − u snap . Then (22) and (21) imply that (cid:107) u h − u H (cid:107) A (cid:46) N (cid:88) i =1 Hλ ( i ) L i +1 (1 + Hhλ ( i ) L i +1 ) (cid:90) ∂K i | ( ∇ u snap ) n | + (cid:107) u (cid:107) A . (24)By (6), we have a DG ( u , v ) = − a DG ( u snap , v ) + ( f, v ) + (cid:90) Γ D (cid:16) γh g D · v − (( ∇ v ) n ) · g D (cid:17) − b DG ( v, p h , (cid:98) p h ) (25)for all v ∈ V DG h . By the definition of u , we see that u = 0 on ∂K for all coarse element K ∈ T H . Thus,using (13) and taking v = u in (25), we have (cid:107)∇ u (cid:107) A (cid:46) − a DG ( u snap , u ) + ( f, u ) . (26)Notice that( f, u ) = (cid:88) K ∈T H (cid:90) K f u ≤ (cid:88) K ∈T H (cid:107) f (cid:107) L ( K ) (cid:107) u (cid:107) L ( K ) (cid:46) H (cid:88) K ∈T H (cid:107) f (cid:107) L ( K ) (cid:107)∇ u (cid:107) L ( K ) (27)where the last inequality follows from the Poincare inequality. So, we obtain( f, u ) (cid:46) H (cid:107) f (cid:107) L (Ω (cid:15) ) (cid:107) u (cid:107) A . (28)By the definition of a DG and u , we have a DG ( u snap , u ) = (cid:90) Ω (cid:15) ∇ u snap : ∇ u − (cid:88) E ∈E H (cid:90) E { ( ∇ u ) n } · [ u snap ] . u snap ] = [ u h ] for all E . Thus, by the results in [14], we obtain (cid:88) E ∈E H (cid:90) E { ( ∇ u ) n } · [ u snap ] (cid:46) (cid:107) u (cid:107) A (cid:16) h (cid:88) E ∈E H (cid:90) E | [ u h ] | (cid:17) . (29)By the variational form of (20), we have, for all coarse elements K (cid:90) K ∇ u snap : ∇ u = (cid:90) K p snap div u = 0since p snap is a constant and u = 0 on ∂K . Combining the above results, we have (cid:107) u (cid:107) A (cid:46) N (cid:88) i =1 Hλ ( i ) L i +1 (1 + Hhλ ( i ) L i +1 ) (cid:90) ∂K i | ( ∇ u snap ) n | + H (cid:107) f (cid:107) L (Ω (cid:15) ) . (30)This completes the proof. In this paper, we develop a new GMsFEM for Stokes problems in perforated domains. The method is basedon a discontinuous Galerkin formulation, and constructs local basis functions for each coarse region. Theconstruction of basis follows the general framework of GMsFEM by using local snapshots and local spectralproblems. In addition, we use a hybridized technique in order to achieve mass conservation. Our numericalresults show that only a few basis functions per coarse region are needed in order to obtain a good accuracy.We also show numerically that the multiscale solution satisfies the mass conservation property. Furthermore,we prove the stability and the convergence of the scheme. In the future, we plan to develop adaptivity ideas[18, 21] for this method.
EC’s research is partially supported by Hong Kong RGC General Research Fund (Project: 400813) andCUHK Faculty of Science Research Incentive Fund 2015-16. MV’s work is partially supported by RussianScience Foundation Grant RS 15-11-10024 and RFBR 15-31-20856.
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