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A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media

  • Received: 01 January 2016 Revised: 01 October 2016
  • Primary: 65N99

  • In this paper, we present a multiscale model reduction framework within Generalized Multiscale Finite Element Method (GMsFEM) for nonlinear elliptic problems. We consider an exemplary problem, which consists of nonlinear p-Laplacian with heterogeneous coefficients. The main challenging feature of this problem is that local subgrid models are nonlinear involving the gradient of the solution (e.g., in the case of scale separation, when using homogenization). Our main objective is to develop snapshots and local spectral problems, which are the main ingredients of GMsFEM, for these problems. Our contributions can be summarized as follows. (1) We re-cast the multiscale model reduction problem onto the boundaries of coarse cells. This is important and allows capturing separable scales as discussed. (2) We introduce nonlinear eigenvalue problems in the snapshot space for these nonlinear "harmonic" functions. (3) We present convergence analysis and numerical results, which show that our approaches can recover the fine-scale solution with a few degrees of freedom. The proposed methods can, in general, be used for more general nonlinear problems, where one needs nonlinear local spectral decomposition.

    Citation: Eric Chung, Yalchin Efendiev, Ke Shi, Shuai Ye. A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media[J]. Networks and Heterogeneous Media, 2017, 12(4): 619-642. doi: 10.3934/nhm.2017025

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  • In this paper, we present a multiscale model reduction framework within Generalized Multiscale Finite Element Method (GMsFEM) for nonlinear elliptic problems. We consider an exemplary problem, which consists of nonlinear p-Laplacian with heterogeneous coefficients. The main challenging feature of this problem is that local subgrid models are nonlinear involving the gradient of the solution (e.g., in the case of scale separation, when using homogenization). Our main objective is to develop snapshots and local spectral problems, which are the main ingredients of GMsFEM, for these problems. Our contributions can be summarized as follows. (1) We re-cast the multiscale model reduction problem onto the boundaries of coarse cells. This is important and allows capturing separable scales as discussed. (2) We introduce nonlinear eigenvalue problems in the snapshot space for these nonlinear "harmonic" functions. (3) We present convergence analysis and numerical results, which show that our approaches can recover the fine-scale solution with a few degrees of freedom. The proposed methods can, in general, be used for more general nonlinear problems, where one needs nonlinear local spectral decomposition.



    Many processes in nature have multiscale nature and nonlinearities. The interaction between nonlinearities and multiple scales can be complex and non separable. This occurs in many applications. A specific feature for this non-separability is that nonlinearities change the multiscale nature of the solution. To discuss some main concepts, we consider an example

    div(a(x,u,u))=f. (1)

    We assume a(x,,) is highly heterogeneous with respect to x. From the point of view of the interaction between nonlinearities and the multiple scales, one can distinguish several classes. In some nonlinear problems, the nonlinearities within coarse regions (a computational grid), that induce the change in the heterogeneities, can be parametrized with a low dimensional parameter, e.g., a(x,u,u)=a0(x,u)u (assuming smoothness and boundedness of a). Within each coarse region, one can approximate the solution u by a constant and thus, can handle these nonlinearities via a low dimensional parametrization (see, e.g., [30, 46] for homogenization and numerical homogenization discussions). If the nonlinearities and heterogeneities are separable in this case, i.e., a(x,u,u)=a(x)b(u)u, then, in fact, one can use a linear theory of multiscale methods (cf. [30, 46]). The situation is very different when a(x,u,u)=a1(x,u)u. Because u is highly heterogeneous, one can not use any low dimensional approximation and linear theories (see e.g., [30, 46, 37, 36, 13, 3, 42, 44] for homogenization and numerical homogenization). This is true even for a separable case a(x,u,u)=a0(x)b(|u|)u. These problems require nonlinear cell problems ([30, 46, 37, 36, 13, 3, 42, 44]). In the paper, we focus on a(x,u,u)=a(x,u) and discuss these nonlinear cell problems for GMsFEM.

    Many previous research on multiscale methods have considered nonlinear problems. The approaches including homogenization [47, 41], numerical homogenization [2, 49, 21], heterogeneous multiscale methods [22, 1, 37, 43], multiscale network approximations [8, 9, 7], multiscale finite element methods [21, 49, 4, 27, 24, 28, 34, 33, 23, 25, 29, 14], variational multiscale methods [39, 38, 6, 40], polyharmonic homogenization [45, 10], generalized multiscale finite element methods [24, 26, 11, 18, 16, 17] have been developed and applied. These approaches approximate the solution of nonlinear PDEs on a coarse grid (see Figure 1 for illustration of coarse and fine grids) by using subgrid models. Some common ingredients in these methods for linear problems are that local solutions are calculated and used to form equations on a coarse grid. GMsFEM approaches propose a systematic enrichment, which calculates multiscale basis functions via local spectral decomposition in each coarse cell. The extensions of these methods to nonlinear problems (as (1)) use nonlinear local problems. For example, in numerical homogenization methods, one can use as a local problem in each coarse cell, for the case a(x,u,u)=a(x,u),

    Figure 1.  Illustration of a multiscale discretization..
    div(a(x,ϕξ))=0

    with the boundary conditions ϕ=ξx. The homogenized fluxes are computed by averaging the flux a(ξ)=a(x,ϕξ). These approaches follow homogenization theory ([46, 37, 36, 13, 3, 42, 44], see also [1, 37, 43] and, the references therein, for numerical homogenization), which is well developed. For GMsFEM, a systematic local enrichment via appropriate local nonlinear spectral problems is needed.

    The main idea of GMsFEM for linear problems is to form snapshot spaces and perform local spectral decomposition in the snapshot space. In this paper, we will follow the same general concept and introduce nonlinear eigenvalue problems. Previous approach [31] develops local nonlinear eigenvalue problems in each coarse cell. Our main contribution is the development of a systematic model reduction using nonlinear harmonic functions. The latter is important as it allows capturing the effects of separable scales. Without using nonlinear harmonic functions, one can not, in general, capture the effects of small separable scales. This is in contrast to linear problems, where one can construct one linear basis function per every coarse node that contains the effects of small scales. Using local solutions allows compressing the effects of small scales within a coarse block and we work with a system reduced to the boundaries of coarse cells. In this case, we can also guarantee that our approaches recover homogenization results when there is a scale separation (note that previous approaches [31] can not guarantee it). The proposed method is in a spirit of hybridization techniques [20, 32, 15] and necessary to eliminate the scale interactions in each block.

    In the paper, we present a local enrichment procedure for the degrees of freedom defined on the boundaries. Our computations are performed in a nonlinear space, where any function defined on the boundaries of coarse cells is extended harmonically to the interior by using local nonlinear PDEs. Our snapshot space can be thought as a nonlinear map from the boundaries to the interior. We discuss the use of nonlinear eigenvalue problems, which are motivated by the analysis. Our analysis allows removing some of major assumptions that are used when not using local nonlinear-harmonic functions (see [31]).

    The numerical results are presented for several examples. We consider a high-contrast permeability field in p-Laplacian example with heterogeneous coefficients. The high-contrast permeability field contains several channels and inclusions with a high permeability. In our numerical results, we increase the number of local multiscale basis functions and compute the errors. The results show that the error decreases rapidly as we increase the number of basis functions and we can approximate the global solution accurately with a very few degrees of freedom.

    The paper is organized as follows. In Section 2, we present some preliminaries and also present a motivation. The description of the GMsFEM for nonlinear problem is presented in Section 3. The convergence analysis of the method is given in Section 4. We present numerical results in Section 5. The conclusion is drawn in Section 6.

    Let D be a bounded open set in R2 with Lipschitz boundary D. We consider the following heterogeneous p-Laplacian equation

    div(a(x,u))=f(x)inD,u=gonD, (2)

    where a(x,u)=κ(x)|u|p2u, p2, κ(x)κ0>0 is a high-contrast coefficient (i.e., κmax/κmin is large), fW1,q(D) (1/p+1/q=1) is an external forcing term, and gW1/q,p(D) is the Dirichlet boundary data.

    The corresponding weak formulation is: (P) Find uW1,pg(D){vW1,p(D):v=g on D} such that

    Da(x,u)v=Dfv,vW1,p0(D).

    The well-posedness of (P) is well established, and one can refer to, for example, Glowinski and Marrocco [35] or the account in Ciarlet [19]. Throughout the paper, we define the energy norm of uW1,p(D) as

    u1,p(D)=(Dκ(x)|u|pdx)1/p.

    Next, we describe the finite element approximation of the solution. We let Th be a fine triangulation, and denote by Vh=Vh(D) the usual finite element space containing continuous piecewise linear functions with respect to Th. We also let Vh0(D) be the subset of Vh(D) containing functions that vanish on D. Similar notations, Vh(Ω),Vh0(Ω), are used for ΩD.

    The discrete fine-scale problem is defined in the following: (Ph) Find uhVh(D) such that

    Da(x,uh)v=Dfv,vVh0(D).

    Additionally, we introduce a coarse discretization TH in which each coarse element is comprised of a localized fine mesh. See Figure 1 for an illustration of a multiscale discretization containing both fine and coarse elements. We use {yi}Nvi=1 to denote the vertices of the coarse mesh, and define a coarse neighborhood of yi by

    ωi={KjTH;  yi¯Kj}, (3)

    where Kj denotes a coarse element in the domain and Nv is the number of coarse vertices. See Figure 2 for an illustration of a coarse neighborhood and elements. Inside each coarse neighborhood ωi (i=1,,Nv), we call the collection of the coarse edges with yi being a common vertex the cross of yi.

    Figure 2.  Illustration of a coarse neighborhood and elements..

    First, we introduce the concept of pharmonicextension.

    Definition 2.1. Let uW1,p(K) (p2) be a given function. Let ˜uW1,p(K) be defined so that ˜uuW1,p0(K), and that ˜u satisfies:

    div(a(x,˜u))=0inK,

    where a(x,˜u)=κ(x)|˜u|p2˜u. Then ˜u is called the pharmonicextension of u. We denote ˜u:=Hp(u).

    Remark 1. The p-harmonic extension minimizes the energy norm, i.e.

    Kκ(x)|˜u|pdx=minvW1,pu(K)Kκ(x)|v|pdx,

    where W1,pu(K)={vW1,p(K)|v=uon K}.

    Remark 2. In this context, all p-harmonic extensions are accomplished coarse-element by coarse-element. Though, we might use the notation Hp directly on a larger domain such as a coarse neighborhoods ωi or the whole domain D, it means that the p-harmonic extension is performed on each coarse element contained in ωi or D.

    Our main idea is solving for the Generalized Multiscale Finite Element solution of Equation (2) on the crosses of the coarse mesh and then the solution in the whole domain can be approximated by p-harmonically extending the obtained cross values into the domain. This idea is motivated by the technique of Numerical Homogenization (NH), which is described in the following. Our goal is to show that our proposed Generalized Multiscale Finite Element Method recovers NH.

    First, we describe a well known numerical homogenization technique. This method can be regarded as using a limited number of degrees of freedom per coarse element. Our objective is to show that the numerical homogenization is a finite element approximation on a coarse grid using p-harmonic extension with only one degree of freedom per edge. We consider

    div(a(x,u))=finD,

    with u=0 on D. We consider a coarse-grid block K and our goal for each coarse-grid block is to compute the effective property. This is done by solving local problem

    div(a(x,Nξ))=0inK,

    with boundary condition Nξ=ξx on K. According to the previous definition, we can write Nξ=Hp(ξx). Then a() is defined as

    a(ξ)=1|K|Ka(y,Nξ)dy.

    The coarse-grid equation is given by

    div(a(x,u))=finD,

    with u=0 on D. Suppose u=ckϕk, where {ϕk} is a linear basis, then

    FNH(c)=Da(x,ckϕk)ϕjdx=KDKa(ckϕk)ϕjdx,

    At this step, we denote ckϕk=ξ=constant, then Nξ=Hp(ckϕkx) and

    FNH(c)=KDKa(ξ)ϕjdx=KDK(1|K|Ka(x,Nξ)dx)ϕjdx=D1|K|(Ka(x,Hp(ckϕkx))dx)ϕjdx=D1|K|(Ka(x,Hp(ckϕk))dx)ϕjdx=Dfϕjdx.

    Compared with the numerical homogenization, Generalized Multiscale Finite Element method seeks the approximation of the solution in the form i,kχicωikϕωik, which solves

    F(c)=Da(x,Hp(i,kχicωikϕωik)ϕωijdx=Dfϕωijdx,

    where {ϕωik}Lik=1 (Li is the number of basis chosen in ωi) are generalized multiscale basis constructed in each ωi (i=1,,Nv), {χi}Nvi=1 is the set of partition of unity functions. Our main approach is to construct multiscale basis functions in a systematic way and provide a priori error. We see from the above discussion that Generalized Multiscale Finite Element Method can be thought as an extension of Numerical Homogenization, where we need to identify appropriate procedures for finding multiscale basis functions. In the following section, we will describe the details of constructing multiscale basis as well as partition of unity functions.

    The goal of our proposed generalized multiscale finite element method is to find a numerical approximation of the solution as well as employing the degree of freedoms only on the crosses in order to exhibit model reduction. Suppose the generalized multiscale finite element solution we are seeking for is ums=Hp(iLik=1χicωikϕωik), where {ϕωik}Lik=1 are multiscale basis constructed in each coarse neighborhood ωi, {χi}Nvi=1 is the set of partition of unity functions, then the generalized multiscale finite element formulation for Equation (2) is the following: Find c={cωik}i,k such that

    Da(x,Hp(iLik=1χicωikϕωik)ϕωijdx=Dfϕωijdxfor any j, (4)

    where a(x,u)=κ(x)|u|p2u, as defined in Section 2.1.

    Remark 3. The GMsFEM numerical solution ums in (4) is uniquely defined. We suppose there are two solutions u1=Hp(iLik=1χicωik,1ϕωik) and u2=Hp(iLik=1χicωik,2ϕωik) to (4). Then, we have

    D(a(x,u1)a(x,u2))vdx=0,

    for any test function. Since u1 and u2 are harmonic in each coarse block, we have

    Dκ(x)|u1u2|pdxD(a(x,u1)a(x,u2))(u1u2)dx=0,

    which guarantees u1=u2. So the solution to (4) is uniquely defined.

    To propose our method, we first need to construct a set of partition of unity functions {χi}Nvi=1. These functions are supported in coarse neighborhoods, and summed to one. Specifically, the support of χi is ωi, and Nvi=1χ1=1. In addition, χi has value 1 at the vertex yi. There are two commonly used sets of partition of unity functions, which are presented below.

    A bilinear partition of unity: χi is defined as usual bilinear basis functions χ0i on ωi, which is equal to 1 at node yi and equal to 0 on ωi.

    A multiscale partition of unity (with linear boundary conditions): χi is defined by

    div(a(x,χi)=0in Kωi,χi=χ0ion K,for all Kωi.

    We remark that using the second choice of partition of unity functions can, in general, provide better numerical performance.

    Let ωi be a given coarse neighborhood. The construction of the multiscale basis functions on ωi starts with a snapshot space Vωisnap. The snapshot space Vωisnap is a set of functions defined on ωi and contains all or most necessary components of the fine-scale solution restricted to ωi. A spectral problem is then solved in the snapshot space to extract the dominant modes in the snapshot space. These dominant modes are the offline basis functions and the resulting reduced space is called the offline space. There are two choices of Vωisnap that are commonly used.

    The first choice is to use all possible fine-grid functions in ωi. This snapshot space provides accurate approximation for the solution space; however, this snapshot space can be very large. The second choice for the snapshot space consists of harmonic extensions. In particular, we denote by Mh(ωi) the set of all nodes of the fine mesh Th which lie on ωi. For each fine-grid node xjMh(ωi), we construct a discrete delta function δhj(x) defined on Mh(ωi) by

    δhj(xk)={1 for k=j0 for kj,xkMh(ωi).

    Then the jth snapshot basis function ψωij is defined as the solution of

    div(κ(x)ψωij)=0in ωi,ψωij=δhjon ωi. (5)

    The dimension of Vωisnap is equal to the size of Mh(ωi). We note that one can use randomized snapshots in conjunction with oversampling to reduce the computational cost associated with the snapshot calculations. We refer to [12] for more details.

    With these snapshots, we follow the procedure in the following subsection to generate offline basis functions by using an auxiliary spectral decomposition.

    The construction of generalized multiscale basis for solving p-Laplacian equation in the fashion of p-harmonic extension is based on the design of a proper nonlinear spectral problem which will be solved in the snapshot space. In each coarse neighborhood ωi, we define the following nonlinear eigenvalue problem which can be characterized by the Rayleigh-Ritz method (RRM).

    {ϕωi1=cωi,λωi1=0,ϕωik=argminvVωisnapGωi(v)Gωiχ(vPk1(v)),λωik=Gωi(ϕωik)Gωiχ(ϕωikPk1(ϕωik)), for k2, (6)

    where cωiVωisnap is a constant function in ωi, the functionals are given by Gωi(v)=ωiκ(x)|Hp(v)|pdx and Gωiχ(v)=ωiκ(x)|Hp(χiv)|pdx, the projector Pk(u)=argminvVωik1Gωi(uv), Vωik1=span{ϕωi1,,ϕωik1}. This nonlinear eigenvalue problem is a standard orthogonal subspace minimization method and is well-defined (see e.g., [50]).

    The eigenfunctions {ϕωik}k in each coarse neighborhood ωi will contribute as offline basis (or we call them generalized multiscale basis or eigenbasis) after being multiplied by the associated partition of unity function χi. We choose the first Li eigenfunctions on each ωi and denote the offline space as

    Vc=span{χiϕωik:k=1,,Li;i=1,,Nv}W1,p(D).

    Recall that our solution assumes the form of ums=Hp(iLik=1cωikχiϕωik), which means ums is obtained by p-harmonically extending iLik=1cωikχiϕωik in each coarse block K, thus only the values of iLik=1cωikχiϕωik on each coarse edge matter in this sense. If we consider one coarse neighborhood ωi, for example the coarse neighborhood of an interior coarse vertex (see yi and ωi in Figure 2), it is the restriction of Lik=1cωikχiϕωik on the 12 coarse edges that will matter in the process of p-harmonic extension. Notice that the partition of unity function χi vanishes on and beyond the boundary of ωi, thus merely the restriction of Lik=1cωikχiϕωik on the cross (that is, the inside 4 coarse edges) makes an influence. Therefore, we can restrict χiϕωik (k=1,,Li) on the cross of ωi and denote the restricted basis (which we call cross basis in this context) by ˆϕωik. Then we can write ums=Hp(iLik=1cωikˆϕωik). We denote

    ˆVc=span{ˆϕωik:k=1,,Li;i=1,,Nv}.

    In this way, we can focus on the degree of freedoms on the crosses and perform spectral decomposition on these crosses.

    Remark 4. In the computation, we use a simpler eigenvalue problem

    {ϕωi1=cωi,λωi1=0,ϕωik=argminvXωikGωi(v)Gωiχ(v),λωik=Gωi(ϕωik)Gωiχ(ϕωik),fork2, (7)

    where Xωik is a subspace of Vωisnap and defined as Xωik=(span{ϕωi1,,ϕωik1}) where the orthogonality is defined with respect to the H1 norm in Vωisnap.

    To analyze the convergence of our proposed method, we first prove several lemmas. The first two lemmas are the direct applications of Lemmas 5.1 and 5.3 in Glowinski and Marrocco [35] and prove the monotonicity and continuity of p-Laplacian operator a(x,u)=κ(x)|u|p2u, respectively. In the following proof, we introduce the notation FG to represent FCG with a constant C independent of the mesh, contrast and the functions involved.

    Lemma 4.1. u,vW1,p(K), p2, the following inequality holds:

    κ(x)|uv|p(a(x,u)a(x,v))(uv). (8)

    Proof. We use Lemma 5.1 in Glowinski and Marrocco [35], which proves the following inequality: p2, α>0 such that y,zR2,

    (|z|p2z|y|p2y,zy)R2α|zy|p.

    If we take z=(κ(x))1/pu,y=(κ(x))1/pv, then (8) is proved.

    Lemma 4.2. u,vW1,p(K), p2, the following inequality holds:

    |a(x,u)a(x,v))|M(x,u,v)|uv|, (9)

    where M(x,u,v)=κ(x)(|u|+|v|)p2.

    Proof. According to Lemma 5.3 in Glowinski and Marrocco [35], there holds the following inequality: p2, β>0 such that y,zR2,

    ||z|p2z|y|p2y|β|zy|(|z|+|y|)p2.

    If we take z=(κ(x))1/(p1)u,y=(κ(x))1/(p1)v, then (9) is proved.

    The next two lemmas deal with the properties of p-harmonic extension operator.

    Lemma 4.3. Suppose ˜u=Hp(u0),˜v=Hp(v0),˜w=Hp(u0v0), where u0,v0W1,p(K), p2. Then we have

    Kκ(x)|(˜u˜v)|pdx(Kκ(x)|˜w|pdx)qp(Kκ(x)|˜u|pdx+Kκ(x)|˜v|pdx)p2p1, (10)

    where 1/p+1/q=1.

    Proof. Using Lemma 4.1 and integrating by parts, we immediately obtain the following inequality:

    Kκ(x)|(˜u˜v)|pdxK(a(x,˜u)a(x,˜v))(˜u˜v)=K(a(x,˜u)a(x,˜v))n(˜u˜v)dsKdiv(a(x,˜u)a(x,˜v))(˜u˜v)dx=K(a(x,˜u)a(x,˜v))n(˜u˜v)ds0=K(a(x,˜u)a(x,˜v))n˜wdsKdiv(a(x,˜u)a(x,˜v))˜wdx=K(a(x,˜u)a(x,˜v))˜wdxKM(x,˜u,˜v)|˜u˜v||˜w|dx, (11)

    where on the last line we have used the continuity property of a(x,u) proved in Lemma 4.2. Applying Hölder's inequality to (11), we have

    Kκ(x)|(˜u˜v)|pdx(Kκ(x)|˜u˜v|pdx)1p(Kκ(x)|˜w|pdx)1p(K(κ(x)2p|M(x,˜u,˜v)|)pp2dx)p2p.

    Dividing both sides by (Kκ(x)|˜u˜v|pdx)1p gives

    Kκ(x)|(˜u˜v)|pdx(Kκ(x)|˜w|pdx)qp(K(κ(x)2p|M(x,˜u,˜v)|)pp2dx)p2p1. (12)

    Recall from Lemma 4.2 that M(x,˜u,˜v)=κ(x)(|˜u|+|˜v|)p2, thus

    (K(κ(x)2p|M(x,˜u,˜v)|)pp2dx)p2p1=(Kκ(x)(|˜u|+|˜v|)pdx)p2p1(Kκ(x)|˜u|pdx+Kκ(x)|˜v|pdx)p2p1. (13)

    We substitute (13) into (12), and we then see that (10) is proved.

    Lemma 4.4. For any u0,v0W1,p(K), p2, we have

    Kκ(x)|Hp(u0+v0)|pdxKκ(x)|(Hp(u0)+Hp(v0))|pdx. (14)

    Proof. Recall Remark 1, we have

    Kκ(x)|Hp(u)|pdx=minvW1,pu(K)Kκ(x)|v|pdx.

    Therefore,

    Kκ(x)|Hp(u0+v0)|pdx=minwW1,pu0+v0(K)Kκ(x)|w|pdx=minw1+w2W1,pu0+v0(K)Kκ(x)|(w1+w2)|pdxminw1W1,pu0(K),w2W1,pv0(K)Kκ(x)|(w1+w2)|pdx.

    Taking w1=Hp(u0),w2=Hp(v0), we obtain

    Kκ(x)|Hp(u0+v0)|pdxKκ(x)|(Hp(u0)+Hp(v0))|pdx.

    We will show the main convergence result in this section. First, we will prove Lemma 4.5 which approximates the error of GMsFEM solution by using functions from the offline space Vc.

    Lemma 4.5. Suppose u is the exact solution of Equation (2), ums is the GMsFEM solution from Equation (4), then for any p2, we have

    uums1,p(D)uHp(vH)qp1,p(D)up2p11,p(D)for any vHVc,

    where 1/p+1/q=1, u1,p(D)=(Dκ(x)|u|pdx)1/p is the energy norm defined in Section 2.1.

    Proof. Using Lemma 4.1, we immediately obtain the following inequality:

    Dκ(x)|(uums)|pdxD(a(x,u)a(x,ums))(uums)=D(a(x,u)a(x,ums))(uHp(vH))DM(x,u,ums)|uums||uHp(vH)|dx,

    for any vHVc, where on the last line we have used the continuity property of a(x,u). Applying Hölder's inequality, we have

    Dκ(x)|(uums)|pdx(Dκ(x)|(uums)|pdx)1p(Dκ(x)|(uHp(vH))|pdx)1p(D(κ(x)2p|M(x,u,ums)|)pp2dx)p2p.

    Dividing both sides by (Dκ(x)|(uums)|pdx)1p gives

    Dκ(x)|(uums)|pdx(Dκ(x)|(uHp(vH))|pdx)qp×(D(κ(x)2p|M(x,u,ums)|)pp2dx)p2p1=(Dκ(x)|(uHp(vH))|pdx)qp×(D(κ(x)2pκ(x)(|u|+|ums|)p2)pp2dx)p2p1=(Dκ(x)|(uHp(vH))|pdx)qp(Dκ(x)(|u|+|ums|)pdx)p2p1(Dκ(x)|(uHp(vH))|pdx)qp(Dκ(x)(|u|p+|ums|p)dx)p2p1(Dκ(x)|(uHp(vH))|pdx)qp(Dκ(x)|u|pdx+Dκ(x)|ums|pdx)p2p1(Dκ(x)|(uHp(vH))|pdx)qp(Dκ(x)|u|pdx)p2p1.

    It follows immediately

    uums1,p(D)uHp(vH)qp1,p(D)up2p11,p(D).

    Lemma 4.6. Suppose u is the exact solution of Equation (2), K is any coarse block of size H, p2, then we have

    Kκ(x)|(uHp(u))|pdxHqK|f|qdx, (15)

    where 1/p+1/q=1.

    Proof. It's clear that

    div(a(x,u)a(x,Hp(u)))=f.

    Thus,

    K(a(x,u)a(x,Hp(u)))(uHp(u))dx=K(uHp(u))fdx.

    By Lemma 4.1,

    Kκ(x)|(uHp(u))|pdxK|uHp(u)||f|dx(K|uHp(u)|pdx)1p(K|f|qdx)1q.

    Using Poincaré's inequality, we get

    Kκ(x)|(uHp(u))|pdxH(K|(uHp(u))|pdx)1p(K|f|qdx)1q=κ1p0H(Kκ0|(uHp(u))|pdx)1p(K|f|qdx)1qH(Kκ(x)|(uHp(u))|pdx)1p(K|f|qdx)1q.

    Dividing both sides by (Kκ(x)|(uHp(u))|pdx)1p, we get

    Kκ(x)|(uHp(u))|pdxHqK|f|qdx.

    Remark 5. This local error estimate proved in Lemma 4.6 immediately deduces the global error estimate:

    uHp(u)p1,p(D)HqfqLq(D). (16)

    Now, we come to the main convergence theorem.

    Theorem 4.7. Suppose u is the exact solution of Equation (2), ums is the GMsFEM solution from Equation (4), then for any p2, we have

    uums1,p(D)up2p11,p(D){H1(p1)2f1(p1)2Lq(D)+(1Λ)1p(p1)2u1p11,p(D)}, (17)

    where Λ=minωiλωiLi+1, {λωij} are the eigenvalues defined by (6) in Section 3.2.2, Li is the number of eigenbasis chosen in each coarse neighborhood ωi.

    Proof. We first define the interpolation of u onto the offline space Vc as

    I0u=argminvVc{uHp(v)1,p(D)}.

    Since I0uVc, we denote I0u=iχiuωi0, where uωi0=Lik=1cωikϕωik. By Lemma 4.5 and (16), it follows that

    uums1,p(D)up2p11,p(D)uHp(I0u)qp1,p(D)up2p11,p(D)(uHp(u)1,p(D)+Hp(u)Hp(I0u)1,p(D))qpup2p11,p(D)(uHp(u)qp1,p(D)+Hp(u)Hp(I0u)qp1,p(D))up2p11,p(D)(Hq2p2fq2p2Lq(D)+Hp(u)Hp(I0u)qp1,p(D))=up2p11,p(D)(H1(p1)2f1(p1)2Lq(D)+Hp(u)Hp(I0u)1p11,p(D)). (18)

    In the following, we will estimate Hp(u)Hp(I0u)1p11,p(D).

    Using Lemma 4.3, we have

    Hp(u)Hp(I0u)p1,p(D)(Dκ(x)|Hp(uI0u)|pdx)qp× (19)
    (Dκ(x)|Hp(u)|pdx+Dκ(x)|Hp(I0u)|pdx)p2p1(Dκ(x)|Hp(ωiχi(uuωi0))|pdx)qp(Dκ(x)|u|pdx)p2p1 (20)

    Applying the property of Hp() proved in Lemma 4.4 to (20), we achieve

    Hp(u)Hp(I0u)p1,p(D)(Dωiκ(x)|Hp(χi(uuωi0))|pdx)qp×(Dκ(x)|u|pdx)p2p1(ωiωiκ(x)|Hp(χi(uuωi0))|pdx)qp(Dκ(x)|u|pdx)p2p1. (21)

    Recall that uωi0=Lik=1cωikϕωik with {ϕωik} being eigenfunctions defined by (6) in Section 3.2.2. We have the following inequality

    ωiκ(x)|Hp(χi(uuωi0))|pdx1λLi+1ωiκ(x)|Hp(uuωi0)|pdx. (22)

    Define Λ=minωiλωiLi+1, then through (21) and (22) we get

    |Hp(u)Hp(I0u)p1,p(D)(ωi1λLi+1ωiκ(x)|Hp(uuωi0)|pdx)qp× (23)
    (Dκ(x)|u|pdx)p2p1(1Λωiωiκ(x)|Hp(uuωi0)|pdx)qp(Dκ(x)|u|pdx)p2p1=(1Λ)qp(ωiωiκ(x)|Hp(uuωi0)|pdx)qp(Dκ(x)|u|pdx)p2p1(1Λ)qp(Dκ(x)|Hp(u)|pdx)qp(Dκ(x)|u|pdx)p2p1. (24)

    Using the energy minimization property of Hp() claimed in Remark 1, we obtain from (24) that

    Hp(u)Hp(I0u)p1,p(D)(1Λ)qp(Dκ(x)|u|pdx)qp(Dκ(x)|u|pdx)p2p1=(1Λ)qp(Dκ(x)|u|pdx)qp+p2p1=(1Λ)1p1Dκ(x)|u|pdx.

    This gives

    Hp(u)Hp(I0u)1p11,p(D)(1Λ)1p(p1)2u1p11,p(D). (25)

    Substituting (25) into (18), we obtain

    uums1,p(D)up2p11,p(D){H1(p1)2f1(p1)2Lq(D)+(1Λ)1p(p1)2u1p11,p(D)}.

    Remark 6. We notice that Λ will increase to infinity. Considering a function u with highly oscillating boundary conditions, the value of Gωi(u)Gωiχ(u) will be very large. More specifically, for u having highly oscillating boundary conditions, the value of Gωi(u) is large. But χiu will have less oscillation on the cross since u solves the harmonic problem. Therefore, Gωiχ(u) will be small and the ratio of Gωi(u) over Gωiχ(u) will be large.

    Besides, we can improve the offline convergence rate by using multiple oversampled spectral problems. To be simple, we start with two eigenvalue problems. We denote Iωi0(u)=uωi0, ~χi=χiχ+i, ˜u=χ+i(uIωi0(u)), where χ+i is the partition of unity function on the oversampled domain ω+i. Following inequality (21),

    ωiωiκ(x)|Hp(χi(uIωi0(u)))|pdx=ωiωiκ(x)|Hp(~χi(˜uIωi0(˜u)))|pdxωi1λωiLi+1ωiκ(x)|Hp(˜uIωi0(˜u))|pdx1Λω+iω+iκ(x)|Hp(χ+i(uIωi0(u)))|pdx1Λω+i1λω+iLi+1ω+iκ(x)|Hp(uIωi0(u))|pdx1Λ1Λ+ω+iω+iκ(x)|Hp(uIωi0(u))|pdx1Λ1Λ+Dκ(x)|Hp(u)|pdx1Λ1Λ+Dκ(x)|u|pdx,

    where Λ=minωiλωiLi+1 and Λ+=minω+iλω+iLi+1. This result can be easily extended to multiple oversampled eigenvalue problems (instead of two eigenvalue problems), and the result would be

    ωiωiκ(x)|Hp(χi(uIωi0(u)))|pdx1Λ1Λ+1Λ+NDκ(x)|u|pdx,

    where Λ+N=minω+iλω+NiLi+1, ω+Ni is a N layers oversampled domain (ω+i is a 1 layer oversampled domain). We note that if we choose all these 1Λ and 1Λ+k's to be less than some δ (0<δ<1), then 1Λ1Λ+1Λ+N<δN+1 and the offline error would be exponential decay as the number of oversampled layers increases.

    In this part, we exhibit the process of numerically implementing the proposed method for p-Laplacian equation. From Glowinski and Marrocco [35], or Ciarlet [19], (P) is equivalent to the following minimization problem: (Q) Find uW1,pg(D){vW1,p(D):v=g on D} such that

    JD(u)=infvW1,pg(D)JD(v), (26)

    where JD(u)=1pDκ(x)|u|pdxDfudx.

    It is easily established that JD(u) is strictly convex and continuous on W1,pg(D). Besides, JD(u) is Gateaux differentiable with

    JD(u)(w)=Dκ(x)|u|p2uwdxDfwdxwW1,p0(D).

    Hence, there exists a unique solution to (Q), and (Q) is equivalent to its Euler equation (P). The corresponding discrete problem of (Q) is: (Qh) Find uhVh(D) such that

    JD(uh)=minvhVh0(D)JD(vh). (27)

    The well-posedness of (Qh) = (Ph) follows in an analogous way to that of (Q) and (P), see Glowinski and Marrocco [35] or Ciarlet [19].

    Recall the discussion in Section 3.2.2, we can represent the GMsFEM solution by uh=Hp(iLik=1cωikˆϕωik). For notational brevity we use a single-index notation to write uh=Hp(Nj=1cjˆϕj). Then we apply Broyden's method (which is a Quasi-Newton's method) to solve the minimization problem (Qh), see Algorithm 1.

    Algorithm 1 A Quasi-Newton algorithm
    1: Initialization: An initial guess c(0)=(c(0)j)Nj=1 and B(0)RN×N
    2:  (1) Compute the gradient vector g(0)(c(0))=JD(Hp(Nj=1c(0)j^ϕj)).
    3:  (2) Compute the stepsize τ(0).
    4:  (3) Set: c(1)=c(0)τ(0)B(0)g(0).
    5:  (4) If c(1)c(0)<δ, where is a suitable norm, return.
    6: for k=1 to N: do
    7:  (1) Compute the gradient vector g(k)(c(k))=JD(Hp(Nj=1c(k)j^ϕj)).
    8:  (2) Compute the approximation of the inverse of Hessian matrix:
    B(k)=B(k1)+[(c(k)c(k1))B(k1)(g(k)g(k1))](c(k)c(k1))TB(k1)(c(k)c(k1))TB(k1)(g(k)g(k1)).
    9:  (3) Compute the stepsize τ(k).
    10:  (4) Set: c(k+1)=c(k)τ(k)B(k)g(k).
    11:  (5) If c(k+1)c(k)<δ, return.
    12: end For

     | Show Table
    DownLoad: CSV

    In this section, we offer a number of representative numerical results to verify the proposed methods in the previous sections. In particular, we solve Equation (2) using the proposed GMsFEM to validate the effectiveness of the respective approaches. To obtain benchmark fine-grid solutions we solve (2) on the unit square D=[0,1]×[0,1] using a uniform fine grid of 100×100 square finite elements which is divided into 10×10 square coarse elements uniformly. We also use a forcing term f=1 and impose a linear Dirichlet boundary condition u(x,y)=x+y. The high-contrast permeability field κ1(x) used in our experiments is shown in Figure 3, with high-contrast ratio κmax/κmin being 105. We note that the fine-grid discretization leads to a system of size Nf=10201. As such, we aim to construct a reduced-order system that can accurately approximate the benchmark solutions from the original fine-scale system.

    Figure 3.  Illustration of the high-contrast permeability field κ1(x)..

    We employ both fine-grid (FEM) and coarse-grid (GMsFEM) methods to solve the model equation (2). In comparing the respective approaches, we introduce relative Lp errors and relative energy errors, which are defined as

    Lperror=uumsLp(D)uLp(D)×100%,Energy error=uums1,p(D)u1,p(D)×100%, (28)

    where we recall that u denotes the FEM solution and ums denotes the GMsFEM solution.

    For the first set of experiments, we take p=3,4,5,6 separately and use different numbers of cross basis (Li for each ωi) for each fixed value of p. Then we check the relative errors of the GMsFEM solutions. Numerical results are shown in Table 1 and Figure 4. Note that in the first column of each sub-table, we show the numbers of basis functions used for each coarse neighborhood ωi, and the degrees of freedom (DOF) of offline space which are the numbers in parentheses. To visually observe the accuracy of GMsFEM, we plot the solutions obtained by both FEM and GMsFEM in the case p=3 using 4 cross basis functions in each coarse neighborhood, see Figure 5.

    Table 1.  Relative errors for p=3,4,5,6 using different numbers of cross basis..
    Li (DOF)p=3
    Lp errorEnergy error
    1(81)9.52 %41.03 %
    2(162)6.45 %34.38 %
    3(243)5.76 %27.76 %
    4(324)0.52 %6.55 %
    5(405)0.45 %5.15 %
    Li (DOF)p=4
    Lp errorEnergy error
    1(81)10.88 %42.35 %
    2(162)6.47 %32.93 %
    3(243)5.12 %24.13 %
    4(324)0.92 %8.57 %
    5(405)0.82 %6.65 %
    Li (DOF)p=5
    Lp errorEnergy error
    1(81)10.12 %40.46 %
    2(162)7.71 %34.05 %
    3(243)5.17 %27.88 %
    4(324)0.94 %9.94 %
    5(405)0.81 %7.92 %
    Li (DOF)p=3
    Lp errorEnergy error
    1(81)8.95 %39.68 %
    2(162)6.94 %30.92 %
    3(243)4.37 %23.85 %
    4(324)1.07 %8.70 %
    5(405)0.91 %7.08 %

     | Show Table
    DownLoad: CSV
    Figure 4.  Relative error vs Li for p=3, 4, 5, 6..
    Figure 5.  FEM v.s. GMsFEM node-wise solutions, p=3, DOF=324..

    By observing the columns in Table 1 (or the curves in Figure 4), we can clearly see that for each p, the relative error decays as we use more cross basis functions. We note that as Li increases, the the value of (Li+1)'s eigenvalue increases, and the error bound 1/Λ will correspondingly decrease. In other words, the analysis suggests (and the results validate) that keeping more basis functions for the coarse space construction will indeed yield a decreasing global error. Through a more careful examination, we notice that for each p, when 4 or more than 4 cross basis are chosen in each coarse neighborhood (i.e. Li4 for each ωi), the errors are much smaller. This might suggest that if we use 4 or more than 4 cross basis in each coarse neighborhood, we would get a better convergence. We will explore this in more details in the following subsection.

    Aside from the accuracy of our proposed method, we are interested in determining how many cross basis (or DOFs) should be used. As we mentioned earlier, there is a "jump" in the relative energy errors when we take 4 cross basis in each coarse neighborhood (i.e. Li=4 for each ωi, see Table 1). Thus, Li=4 might be a good choice. According to our analysis in Section 4, that is probably due to a sudden decrease in the quantity of 1/Λ, where Λ=minωiλωiLi+1, {λωij} are the eigenvalues defined in (7) in Section 3.2.2. To verify this theory, we calculate the corresponding 1/Λ for each Li in the case of p=3. The results are shown in Table 2. In this table, we see the jump in Λ and 1/Λ at Li=4, which explains our earlier inference. Hence, we conclude that the proper number of cross basis is chosen at the spot where there is a sudden increase in the values of Λ (or a sudden decrease in the values of 1/Λ). We would like to remark that an adaptive method can be employed to determine a best choice of Li for each coarse neighborhood ωi. Moreover, to see a more quantitative relationship between the relative errors and the values of Λ as well as being inspired by the result in Theorem 4.7, we calculate the cross-correlation coefficient between the relative energy errors and the corresponding values of (1Λ)1p(p1)2 for the case p=3. We recall that the quantity (1Λ)1p(p1)2 comes from (17) in Theorem 4.7. The evaluated cross-correlation coefficient is 0.99. This indicates a linear relationship between the relative energy error and the corresponding (1Λ)1p(p1)2, which verifies our result in (17).

    Table 2.  Values of Λ and 1/Λ when p=3..
    LiΛ1/Λ
    18.86e-41.13e3
    22.59e-33.86e2
    34.46e-32.24e2
    41.55e26.44e-3
    54.01e22.50e-3

     | Show Table
    DownLoad: CSV

    Remark 7. We note that the choice of Li is highly related to the number of "channels" and "inclusions" in each coarse neighborhood. In more details, if there are m inclusions and channels in a coarse neighborhood ωi, then one can observe m small, asymptotically vanishing, eigenvalues. In the example presented in this section, we can see that there are at most 4 inclusions and channels in each coarse neighborhood. This suggests the choice of Li=4, and we verify this choice by observing the values of Λi.

    Since Li is the number of eigen-pairs solved from the nonlinear eigenvalue problem, it's a finite number and can not grow to infinity. We can only guarantee that as Li grows (not necessary to be a large number), the error will decay, which is observed by our numerical results.

    To verify that our proposed method is applicable to more situations, we examine other choices of permeability field κ(x). First, we would like to check that the GMsFEM solution errors do not depend on the high-contrast ratio κmax/κmin. To see this, we increase the high-contrast ratio of κ1(x), which is used in the previous subsections, from 105 to 107. We denote the new permeability field by κ2(x). Then we solve Equation (2) using both FEM and proposed GMsFEM, and calculate the relative errors and the error bound quantity 1/Λ. Numerical results for p=3 are shown in Table 3. Comparing these results with the top left sub-table in Table 1 and Table 2, we can observe similar trend inside the columns as well as a slight increase in the values of both relative energy errors and 1/Λ's. The jump at Li=4 still occurs. The cross-correlation coefficient between the relative energy errors and (1Λ)1p(p1)2 is calculated to be 0.98.

    Table 3.  Relative energy errors and values of 1/Λ using κ2(x), p=3..
    LiEnergy error1/Λ
    144.15 %1.42e3
    236.44 %4.04e2
    327.99 %2.35e2
    46.77 %6.49e-3
    55.30 %2.50e-3

     | Show Table
    DownLoad: CSV

    We also consider a different high-contrast permeability field κ3(x), see Figure 6. We solve Equation (2) for p=3 and the results are presented in Table 4. The cross-correlation coefficient between the relative energy errors and (1Λ)1p(p1)2 is calculated to be 0.94. Similar conclusions as made in Section 5.1.1 and 5.1.2 can be drawn for this new choice of permeability field. We can see that our proposed method works well for this permeability field.

    Figure 6.  Illustration of the high-contrast permeability field κ3(x)..
    Table 4.  Relative energy errors and values of 1/Λ using κ3(x), p=3..
    LiEnergy error1/Λ
    147.08 %1.85e1
    227.68 %4.64e0
    320.81 %2.68e0
    44.33 %2.26e-3
    52.69 %1.01e-3

     | Show Table
    DownLoad: CSV

    The online cost is independent of fine mesh parameters, while it will grow as the spectral basis parameters increase. We note that the online cost is proportional to that of solving homogeneous p-Laplacian equation with polynomial basis. In practise, we usually only use a few spectral basis, so the online cost is close to that of solving homogeneous p-Laplacian equation with low order polynomial basis. We note that solving the nonlinear eigenvalue problem in each coarse neighborhood is one source of the computational cost. However, this is an offline step, which means when dealing with different forcing terms and boundary conditions we only need to solve this nonlinear eigenvalue problem for a single time. Thus, the computation of this eigenvalue problem will not affect the online cost of our method. We also note that in Algorithm 1, a nonlinear function of the form N(u)=κ(x)|u|p2 requires a fine-grid update at each iterative step. In particular, at each iterative step, we must use the fine-scale solution values to construct a gradient and its norm, and we must subsequently multiply the resulting expressions with the original coefficient κ(x) at all fine grid points in order to update the nonlinear permeability coefficient. This is a fine-grid dependent process that adds an increasing computational cost depending on the size of the fine grid. To decrease this cost, we introduce the discrete empirical interpolation method (DEIM), which allows us to approximate the nonlinear function on the fine grid while only evaluating at a few carefully selected points. In itself, DEIM is a snapshot-based preprocessing procedure in which the dominant spectral behavior of the global nonlinear function is extracted. We refer to [5] [48] for more detailed discussions on the use of DEIM. Moreover, by comparing the degrees of freedoms listed in Table 1 with the size of the fine-scale finite element system Nf=10201, we see that we obtain a reduced sized system by applying generalized multiscale finite element method, which will reduce the computational cost.

    Remark 8. Compared with the online cost, the offline cost depends on the fine mesh parameter, considering that each local snapshot problem is solved on the local coarse neighborhood consisting of fine grids. We note that the cost of numerical homogenization is high because the local problem div(a(x,Nξi))=0inK with boundary condition Nξi=ξix on K (K is a coarse block) is solved for all ξ1,ξ2,,ξN. Similarly, the local problem in offline stage is solved for all possible boundary conditions, which is consistent to numerical homogenization. So the offline cost is high. However, as mentioned in Section 3.2.1, one can use randomized snapshots in conjunction with oversampling to reduce the offline computational cost associated with the snapshot calculations. Also, one can select and compute eigenbasis adaptively in each coarse neighborhood which can eliminate the use of non-dominated modes and reduce offline cost. Besides, as mentioned above, DEIM is introduced to reduce offline computational cost when it comes to the evaluation of nonlinear functions.

    Remark 9. To illustrate how the error of DEIM affects the global error estimate of Theorem 4.8, we adopt the notation aDEIM(x,u)=N1(u)u, where N1(u)(κ(x)|u|p2) is evaluated using DEIM. We denote ˜ums the DEIM solution which is obtained by solving

    DaDEIM(x,˜ums)v=Dfv,vVh0(D).

    Then we have

    ums˜umsp1,p(D)=Da(x,(ums˜ums))(ums˜ums)Da(x,ums)(ums˜ums)Da(x,˜ums)(ums˜ums)=D(ums˜ums)fDa(x,˜ums)(ums˜ums)=D(aDEIM(x,˜ums)a(x,˜ums))(ums˜ums).

    We note that aDEIM(x,˜ums)a(x,˜ums) is the DEIM error, which can be assumed to be a small quantity (ref. [5]). Then from the above inequality, it follows

    ums˜umsp1,p(D)δp1ums˜ums1,p(D), (29)

    for some small δ(0<δ<1).

    Combining (29) with Theorem 4.8, we obtain

    u˜ums1,p(D)uums1,p(D)+ums˜ums1,p(D)δ+up2p11,p(D){H1(p1)2f1(p1)2Lq(D)+(1Λ)1p(p1)2u1p11,p(D)}.

    In this paper, our objective is to develop a multiscale model reduction using the framework of GMsFEM. We re-cast the problem and use the degrees of freedom defined on the boundaries of coarse elements (cf. hybridization techniques [20, 32, 15]). Our motivation stems from homogenization and the analysis of multiscale methods. Homogenization and numerical homogenization methods rely on nonlinear harmonic extensions of boundary values in order to capture the effects of scales within the domain. Via these local solutions, we can capture the effects of small separable scales. In the linear case, one can use a single basis per coarse element to capture these effects; however, for nonlinear problems, this is not possible because of non-additivity. Moreover, the use of degrees of freedom on the boundaries of coarse elements is important for achieving low dimensional approximate models. If nonlinear harmonic extensions are not used, one can not estimate the residuals (see [31]). In our framework, we propose a local nonlinear spectral decomposition, which select dominant modes in these nonlinear snapshot spaces. We present a convergence analysis and numerical results.

    The authors would like to thank Wing Tat Leung for a number of insightful discussions regarding this work. YE would like to thank the partial support from NSF 1620318, the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-FG02-13ER26165 and National Priorities Research Program grant NPRP grant 7-1482-1278 from the Qatar National Research Fund.

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