1 Introduction

Many materials in nature are highly heterogeneous and their properties can vary at different scales. Direct numerical simulations in such multiscale media are prohibitively expensive and some type of model reduction is needed. Multiscale approaches such as homogenization and numerical homogenization (Cao 2005; Abdulle 2006; Schröder 2014; Buck et al. 2013; Francfort and Murat 1986; Oleinik et al. 2009; Vinh and Tung 2011; Liu et al. 2009) have been routinely used to model macroscopic properties and macroscopic behavior of elastic materials. These approaches compute the effective material properties based on representative volume simulations. These properties are further used to solve macroscale equations. In this paper, our goal is to design multiscale method for elasticity equations in the media when the media properties do not have scale separation and classical homogenization and numerical homogenization techniques do not work. We are motivated by seismic wave applications when elastic wave propagation in heterogeneous subsurface formation is studied where the subsurface properties can contain vugs, fractures, and cavities of different sizes. In this paper, we develop multiscale methods for static problems and present their analysis.

In this paper, we design a multiscale model reduction techniques using GMsFEM for steady state elasticity equation in heterogeneous media

$$\begin{aligned} \frac{\partial }{\partial x_i} (c_{ijkl}(x) e_{kl}(u))=f_j(x), \end{aligned}$$
(1)

where \(e_{kl}(u)=\frac{1}{2}(\frac{\partial u_k}{\partial x_l}+\frac{\partial u_l}{\partial x_k})\) and \(c_{ijkl}(x)\) is a multiscale field with a high contrast. GMsFEM has been studied for a various applications related to flow problems (see Efendiev et al. 2013a, 2014a; Chung et al. 2014; Efendiev et al. 2013c, 2014b). In GMsFEM, we solve Eq. (1) on a coarse grid each coarse grid consists of a union of fine-grid blocks. In particular, we design (1) a snapshot space (2) an offline space for each coarse patch. The offline space consists of multiscale basis functions that are coupled in a global formulation. In this paper, we consider several choices for snapshot spaces, offline spaces, and global coupling. The main idea of the snapshot space in each coarse patch is to provide an exhaustive space where an appropriate spectral decomposition is performed. This space contains local functions that can mimic the global solution behavior in the coarse patch for all right hand sides or boundary conditions. We consider two choices for the snapshot space. The first one consists of all fine-grid functions in each coarse patch and the second one consists of harmonic extensions. Next, we propose a local spectral decomposition in the snapshot space which allows selecting multiscale basis functions. This local spectral decomposition is based on the analysis and depends on the global coupling mechanisms. We consider several choices for the local spectral decomposition including oversampling approach where larger domains are used in the eigenvalue problem. The oversampling technique uses larger domains to compute snapshot vectors that are more consistent with local solution space and thus can have much lower dimension.

To couple multiscale basis functions constructed in the offline space, we consider two methods, conforming Galerkin (CG) approach and discontinuous Galerkin (DG) approach based on symmetric interior penalty method for (1). These approaches are studied for linear elliptic equations in Efendiev et al. (2013a, b). Both approaches provide a global coupling for multiscale basis functions where the solution is sought in the space spanned by these multiscale basis functions. This representation allows approximating the solution with a reduced number of degrees of freedom. The constructions of the basis functions are different for continuous Galerkin and discontinuous Galerkin methods as the local spectral decomposition relies on the analysis. In particular, for continuous Galerkin approach, we use partition of unity functions and discuss several choices for partition of unity functions. We provide an analysis of both approaches. The offline space construction is based on the analysis.

We present numerical results where we study the convergence of continuous and discontinuous Galerkin methods using various snapshot spaces as well as with and without the use of oversampling. We consider highly heterogeneous coefficients that contain high contrast. Our numerical results show that the proposed approaches allow approximating the solution accurately with fewer degrees of freedom. In particular, when using the snapshot space consisting of harmonic extension functions, we obtain better convergence results. In addition, oversampling methods and the use of snapshot spaces constructed in the oversampled domains can substantially improve the convergence.

The paper is organized as follows. In Sect. 2, we state the problem and the notations for coarse and fine grids. In Sect. 3, we give the construction of multiscale basis functions, snapshot spaces and offline spaces, as well as global coupling via CG and DG. In Sect. 4, we present numerical results. Sections 5–6 are devoted to the analysis of the methods.

2 Preliminaries

In this section, we will present the general framework of GMsFEM for linear elasticity in high-contrast media. Let \(D\subset {\mathbb {R}}^{2}\) (or \({\mathbb {R}}^{3}\)) be a bounded domain representing the elastic body of interest, and let \( {u} = (u_1,u_2)\) be the displacement field. The strain tensor \( {\epsilon }( {u}) = (\epsilon _{ij}( {u}))_{1\le i,j \le 2}\) is defined by

$$\begin{aligned} {\epsilon }({u}) = \frac{1}{2} (\nabla {u} + \nabla {u}^T), \end{aligned}$$

where \(\nabla {u} = (\frac{\partial u_i}{\partial x_j})_{1\le i,j \le 2}\). In the component form, we have

$$\begin{aligned} \epsilon _{ij}( {u}) = \frac{1}{2} \Big ( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \Big ), \quad 1\le i,j \le 2. \end{aligned}$$

In this paper, we assume the medium is isotropic. Thus, the stress tensor \( {\sigma }( {u}) = (\sigma _{ij}( {u}))_{1\le i,j \le 2}\) is related to the strain tensor \( {\epsilon }( {u})\) in the following way

$$\begin{aligned} {\sigma } = 2\mu {\epsilon } + \lambda \nabla \cdot {u} \, {I}, \end{aligned}$$

where \(\lambda >0\) and \(\mu >0\) are the Lamé coefficients. We assume that \(\lambda \) and \(\mu \) have highly heterogeneous spatial variations with high contrasts. Given a forcing term \( {f} = (f_1,f_2)\), the displacement field \( {u}\) satisfies the following

$$\begin{aligned} -\nabla \cdot {\sigma }={f},\quad \text {in }D \end{aligned}$$
(2)

or in component form

$$\begin{aligned} - \Big (\frac{\partial \sigma _{i1}}{\partial x_1} + \frac{\partial \sigma _{i2}}{\partial x_2} \Big ) = f_i,\quad \text {in }D,\ i=1,2. \end{aligned}$$
(3)

For simplicity, we will consider the homogeneous Dirichlet boundary condition \({u}={0}\) on \(\partial D\).

Let \({\mathcal T}^H\) be a standard triangulation of the domain \(D\) where \(H>0\) is the mesh size. We call \({\mathcal T}^H\) the coarse grid and \(H\) the coarse mesh size. Elements of \({\mathcal T}^H\) are called coarse grid blocks. The set of all coarse grid edges is denoted by \({\mathcal E}^H\) and the set of all coarse grid nodes is denoted by \({\mathcal S}^H\). We also use \(N_S\) to denote the number of coarse grid nodes, \(N\) to denote the number of coarse grid blocks. In addition, we let \({\mathcal T}^h\) be a conforming refinement of the triangulation \({\mathcal T}^H\). We call \({\mathcal T}^h\) the fine grid and \(h>0\) is the fine mesh size. We remark that the use of the conforming refinement is only to simplify the discussion of the methodology and is not a restriction of the method.

Let \(V^h\) be a finite element space defined on the fine grid. The fine-grid solution \({u}_h\) can be obtained as

$$\begin{aligned} a({u}_h,{v})=({f}, {v}), \quad \forall {v}\in V^h, \end{aligned}$$
(4)

where

$$\begin{aligned} a( {u}, {v}) = \int _D \Big ( 2\mu {\epsilon }( {u}) : {\epsilon }( {v}) + \lambda \nabla \cdot {u}\,\nabla \cdot {v}\Big ) \; d {x}, \quad ( {f}, {v}) = \int _D {f} \cdot {v} \; d {x} \end{aligned}$$
(5)

and

$$\begin{aligned} {\epsilon }( {u}) : {\epsilon }( {v}) = \sum _{i,j=1}^2 \epsilon _{ij}( {u}) \epsilon _{ij}( {v}), \quad {f} \cdot {v} = \sum _{i=1}^2 f_i v_i. \end{aligned}$$
(6)

Now, we present GMsFEM. The discussion consists of two main steps, namely, the construction of local basis functions and the global coupling. In this paper, we will develop and analyze two types of global coupling, namely, the continuous Galerkin coupling and the discontinuous Galerkin coupling. These two couplings will require two types of local basis functions. In essence, the CG coupling will need vertex-based local basis functions and the DG coupling will need element-based local basis functions.

For each vertex \( {x}_i \in {\mathcal S}^H\) in the coarse grid, we define the coarse neighborhood \(\omega _i\) by

$$\begin{aligned} \omega _i = \bigcup \{ K_j \; : \; K_j \subset {\mathcal T}^H, \; {x}_i \in K_j \}. \end{aligned}$$

That is, \(\omega _i\) is the union of all coarse grid blocks \(K_j\) having the vertex \( {x}_i\) (see Fig. 1). A snapshot space \(V^{i,\text {snap}}\) is constructed for each coarse neighborhood \(\omega _i\). The snapshot space contains a large set that represents the local solution space. A spectral problem is then constructed to get a reduced dimensional space. Specifically, the spectral problem is solved in the snapshot space and eigenfunctions corresponding to dominant modes are used as the final basis functions. To obtain conforming basis functions, each of these selected modes will be multiplied by a partition of unity function. The resulting space is denoted by \(V^{i,\text {off}}\), which is called the offline space for the \(i\)-th coarse neighborhood \(\omega _i\). The global offline space \(V^{\text {off}}\) is then defined as the linear span of all these \(V^{i,\text {off}}\), for \(i=1,2,\ldots , N_S\). The CG coupling can be formulated as to find \({u}_H^{\text {CG}} \in V^{\text {off}}\) such that

$$\begin{aligned} a( {u}_H^{\text {CG}}, {v}) = ( {f}, {v}), \quad \forall {v}\in V^{\text {off}}. \end{aligned}$$
(7)

The DG coupling can be constructed in a similar fashion. A snapshot space \(V^{i,\text {snap}}\) is constructed for each coarse grid block \(K_i\). A spectral problem is then solved in the snapshot space and eigenfunctions corresponding to dominant modes are used as the final basis functions. This space is called the offline space \(V^{i,\text {off}}\) for the \(i\)-th coarse grid block. The global offline space \(V^{\text {off}}\) is then defined as the linear span of all these \(V^{i,\text {off}}\), for \(i=1,2,\ldots ,N\). The DG coupling can be formulated as: find \( {u}_H^{\text {DG}} \in V^{\text {off}}\) such that

$$\begin{aligned} a_{\text {DG}}( {u}_H^{\text {DG}},{v})=({f},{v}),\quad \forall {v}\in V^{\text {off}}, \end{aligned}$$
(8)

where the bilinear form \(a_{\text {DG}}\) is defined as

$$\begin{aligned} a_{\text {DG}}( {u}, {v})&= a_H( {u}, {v}) - \sum _{E\in {\mathcal E}^H} \int _E \Big ( \{{ {\sigma }( {u}) \, {n}_E}\} \cdot [{ {v}}] + \{{ {\sigma }( {v}) \, {n}_E}\} \cdot [{ {u}}] \Big )~ds \nonumber \\&+ \sum _{E\in {\mathcal E}^H} \frac{\gamma }{h} \int _E \{{\lambda +2\mu }\} [u] \cdot [v]~ds \end{aligned}$$
(9)

with

$$\begin{aligned} a_H( {u}, {v}) = \sum _{K\in {\mathcal T}_{H}} a_H^K(u,v), \quad a_H^K(u,v) = \int _{K} \Big ( 2\mu {\epsilon }({u}): {\epsilon }({v}) + \lambda \nabla \cdot {u} \nabla \cdot {v} \Big )~d {x},\nonumber \\ \end{aligned}$$
(10)

where \(\gamma > 0\) is a penalty parameter, \( {n}_E\) is a fixed unit normal vector defined on the coarse edge \(E\) and \( {\sigma }( {u}) \, {n}_E\) is a matrix-vector product. Note that, in (9), the average and the jump operators are defined in the classical way. Specifically, consider an interior coarse edge \(E\in {\mathcal E}^H\) and let \(K^{+}\) and \(K^{-}\) be the two coarse grid blocks sharing the edge \(E\). For a piecewise smooth function \(G\), we define

$$\begin{aligned} \{{G}\} = \frac{1}{2}(G^{+} + G^{-}), \quad [{G}] = G^{+} - G^{-},\qquad \text {on }E, \end{aligned}$$

where \(G^{+} = G|_{K^{+}}\) and \(G^{-} = G|_{K^{-}}\) and we assume that the normal vector \( {n}_E\) is pointing from \(K^{+}\) to \(K^{-}\). For a coarse edge \(E\) lying on the boundary \(\partial D\), we define

$$\begin{aligned} \{{G}\} = [{G}] = G, \quad \text {on }E, \end{aligned}$$

where we always assume that \( {n}_E\) is pointing outside of \(D\). For vector-valued functions, the above average and jump operators are defined component-wise. We note that the DG coupling (8) is the classical interior penalty discontinuous Galerkin (IPDG) method with our multiscale basis functions.

Fig. 1
figure 1

Illustration of a coarse neighborhood, oversampled coarse neighborhood, coarse block and oversampled coarse block

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Finally, we remark that, we use the same notations \(V^{i,\text {snap}}, V^{i,\text {off}}\) and \(V^{\text {off}}\) to denote the local snapshot, local offline and global offline spaces for both the CG coupling and the DG coupling to simplify notations.

3 Construction of multiscale basis functions

This section is devoted to the construction of multiscale basis functions.

3.1 Basis functions for CG coupling

We begin by the construction of local snapshot spaces. Let \(\omega _i\) be a coarse neighborhood, \(i=1,2,\ldots ,N_S\). We will define two types of local snapshot spaces. The first type of local snapshot space is

$$\begin{aligned} V_1^{i,\text {snap}} = V^h(\omega _i), \end{aligned}$$

where \(V^h(\omega _i)\) is the restriction of the conforming space to \(\omega _i\). Therefore, \(V_1^{i,\text {snap}}\) contains all possible fine scale functions defined on \(\omega _i\). The second type of local snapshot space contains all possible harmonic extensions. Next, let \(V^h(\partial \omega _i)\) be the restriction of the conforming space to \(\partial \omega _i\). Then we define the fine-grid delta function \(\delta _k \in V^h(\partial \omega _i)\) on \(\partial \omega _i\) by

$$\begin{aligned} \delta _k( {x}_l) = {\left\{ \begin{array}{ll} 1, &{} l = k \\ 0, &{} l \ne k, \end{array}\right. } \end{aligned}$$

where \(\{{x}_l\}\) are all fine grid nodes on \(\partial \omega _i\). Given \(\delta _k\), we find \( {u}_{k1}\) and \( {u}_{k2}\) by

$$\begin{aligned} \begin{aligned} -\nabla \cdot {\sigma }({u}_{k1})&= {0}, \quad \text {in }\omega _i \\ {u}_{k1}&= (\delta _k, 0)^T, \quad \text {on }\partial \omega _i \end{aligned} \end{aligned}$$
(11)

and

$$\begin{aligned} \begin{aligned} - \nabla \cdot {\sigma }( {u}_{k2})&= {0}, \quad \text {in }\omega _i \\ {u}_{k2}&= (0,\delta _k)^T, \quad \text {on }\partial \omega _i. \end{aligned} \end{aligned}$$
(12)

The linear span of the above harmonic extensions is our second type of local snapshot space \(V^{i,\text {snap}}_2\). To simplify the notations, we will use \(V^{i,\text {snap}}\) to denote \(V^{i,\text {snap}}_1\) or \(V^{i,\text {snap}}_2\) when there is no need to distinguish the two type of spaces. Moreover, we write

$$\begin{aligned} V^{i,\text {snap}} = \text {span}\{ {\psi }^{i,\text {snap}}_k, \quad k=1,2,\ldots , M^{i,\text {snap}}\}, \end{aligned}$$

where \(M^{i,\text {snap}}\) is the number of basis functions in \(V^{i,\text {snap}}\).

We will perform a dimension reduction on the above snapshot spaces by the use of a spectral problem. First, we will need a partition of unity function \(\chi _i\) for the coarse neighborhood \(\omega _i\). One choice of a partition of unity function is the coarse grid hat functions \(\Phi _i\), that is, the piecewise bi-linear function on the coarse grid having value \(1\) at the coarse vertex \( {x}_i\) and value \(0\) at all other coarse vertices. The other choice is the multiscale partition of unity function, which is defined in the following way. Let \(K_j\) be a coarse grid block having the vertex \( {x}_i\). Then we consider

$$\begin{aligned} \begin{aligned} - \nabla \cdot {\sigma }( {\zeta }_i)&= {0}, \quad \text {in }K_j \\ {\zeta }_{i}&= (\Phi _i,0)^T, \quad \text {on }\partial K_j. \end{aligned} \end{aligned}$$
(13)

Then we define the multiscale partition of unity as \(\widetilde{\Phi }_i = ( {\zeta }_i)_1\). The values of \(\widetilde{\Phi }_i \) on the other coarse grid blocks are defined similarly.

Based on our analysis to be presented in the next sections, we define the spectral problem as

$$\begin{aligned} \int _{\omega _i} \Big ( 2\mu {\epsilon }( {u}) : {\epsilon }( {v}) + \lambda \nabla \cdot {u} \, \nabla \cdot {v} \Big )~d {x} = \xi \int _{\omega _i} \tilde{\kappa } {u} \cdot {v}~d {x}, \end{aligned}$$
(14)

where \(\xi \) denotes the eigenvalue and

$$\begin{aligned} \tilde{\kappa }= \sum _{i=1}^{N_S}(\lambda +2\mu ) | \nabla \chi _i |^2. \end{aligned}$$
(15)

The above spectral problem (14) is solved in the snapshot space. We let \((\phi _k, \xi _k)\) be the eigenfunctions and the corresponding eigenvalues. Assume that

$$\begin{aligned} \xi _1 \le \xi _2 \le \cdots \le \xi _{M^{i,\text {snap}}}. \end{aligned}$$

Then the first \(L_i\) eigenfunctions will be used to construct the local offline space. We define

$$\begin{aligned} {\psi }^{i,\text {off}}_l = \sum _{k=1}^{M^{i,\text {snap}}} \phi _{lk} {\psi }^{i,\text {snap}}_k,\quad l=1,2,\ldots ,L_i, \end{aligned}$$
(16)

where \(\phi _{lk}\) is the \(k\)-th component of \(\phi _l\). The local offline space is then defined as

$$\begin{aligned} V^{i,\text {off}} = \text {span} \{ \chi _i {\psi }^{i,\text {off}}_l, \quad l=1,2,\ldots ,L_i\}. \end{aligned}$$

Next, we define the global continuous Galerkin offline space as

$$\begin{aligned} V^{\text {off}} = \text {span} \{ V^{i,\text {off}}, \quad i=1,2,\ldots ,N_S \}. \end{aligned}$$

3.2 Basis functions for DG coupling

We will construct the local basis functions required for the DG coupling. We also provide two types of snapshot spaces as in CG case. The first type of local snapshot space is all possible fine grid bi-linear functions defined on \(K_i\). The second type of local snapshot space \(V^{i,\mathrm{{snap}}}\) for the coarse grid block \(K_i\) is defined as the linear span of all harmonic extensions. Specifically, given \(\delta _k\), we find \( {u}_{k1}\) and \( {u}_{k2}\) by

$$\begin{aligned} \begin{aligned} - \nabla \cdot {\sigma }( {u}_{k1})&= {0}, \quad \text {in }K_i \\ {u}_{k1}&= (\delta _k, 0)^T, \quad \text {on }\partial K_i \end{aligned} \end{aligned}$$
(17)

and

$$\begin{aligned} \begin{aligned} - \nabla \cdot {\sigma }({u}_{k2})&={0},\quad \text {in }K_i \\ {u}_{k2}&=(0,\delta _k)^T,\quad \text {on }\partial K_i. \end{aligned} \end{aligned}$$
(18)

The linear span of the above harmonic extensions is the local snapshot space \(V^{i,\text {snap}}\). We also write

$$\begin{aligned} V^{i,\mathrm{{snap}}} = \text {span} \{ {\psi }^{i,\text {snap}}_k, \quad k=1,2,\ldots , M^{i,\text {snap}} \}, \end{aligned}$$

where \(M^{i,\text {snap}}\) is the number of basis functions in \(V^{i,\text {snap}}\).

We will perform a dimension reduction on the above snapshot spaces by the use of a spectral problem. Based on our analysis to be presented in the next sections, we define the spectral problem as

$$\begin{aligned} \int _{K_i} \big (2\mu {\epsilon }( {u}): {\epsilon }( {v}) + \lambda \nabla \cdot {u} \nabla \cdot {v}\big )~d {x} = \frac{\xi }{H} \int _{\partial K_i}\left\langle \lambda +2\mu \right\rangle {u}\cdot {v}~ds, \end{aligned}$$
(19)

where \(\xi \) denotes the eigenvalues and \(\left\langle \lambda +2\mu \right\rangle \) is the maximum value of \(\{{ \lambda +2\mu }\}\) on \(\partial K_i\). The above spectral problem (19) is again solved in the snapshot space \(V^{i,\text {snap}}\). We let \((\phi _k, \xi _k)\), for \(k=1,2,\ldots ,M^{i,\text {snap}}\) be the eigenfunctions and the corresponding eigenvalues. Assume that

$$\begin{aligned} \xi _1\le \xi _2 \le \cdots \le \xi _{M^{i,\text {snap}}}. \end{aligned}$$

Then the first \(L_i\) eigenfunctions will be used to construct the local offline space. Indeed, we define

$$\begin{aligned} {\psi }^{i,\text {off}}_l = \sum _{k=1}^{M^{i,\text {snap}}} \phi _{lk} {\psi }^{i,\text {snap}}_k,\quad l=1,2,\ldots ,L_i, \end{aligned}$$
(20)

where \(\phi _{lk}\) is the \(k\)-th component of \(\phi _l\). The local offline space is then defined as

$$\begin{aligned} V^{i,\text {off}} = \text {span} \{ {\psi }^{i,\text {off}}_l, \quad l=1,2,\ldots , L_i\}. \end{aligned}$$

The global offline space is also defined as

$$\begin{aligned} V^{\text {off}} = \text {span} \{ V^{i,\text {off}}, \quad i=1,2,\ldots , N \}. \end{aligned}$$

3.3 Oversampling technique

In this section,we present an oversampling technique for generating multiscale basis functions. The main idea of oversampling is to solve local spectral problem in a larger domain. This allows obtaining a snapshot space that has a smaller dimension since snapshot vectors contain solutions oscillating near the boundaries. In our previous approaches, we assume that the snapshot vectors can have an arbitrary value on the boundary of coarse blocks which yield to large dimensional coarse spaces.

For the harmonic extension snapshot case,we solve equation (11) and (12) in \(\omega _i^+\) (see Fig. 1) instead of \(\omega _i\) for CG case, and solve the Eqs. (17) and (18) in \(K_i^+\) instead of \(K_i\) for DG case. We denote the solutions as \(\psi _i^{+,\text {snap}}\), and their restrictions on \(\omega _i\) or \( K_i \) as \(\psi _i^{\text {snap}}\). We reorder these functions according to eigenvalue behavior and write

$$\begin{aligned} R_{\text {snap}}^+ = \left[ \psi _{1}^{+,\text {snap}}, \ldots , \psi _{M_{\text {snap}}}^{+,\text {snap}} \right] \quad \text {and} \quad R_{\text {snap}} = \left[ \psi _{1}^{\text {snap}}, \ldots , \psi _{M_{\text {snap}}}^{\text {snap}} \right] . \end{aligned}$$

where \({M_\mathrm{{snap}}}\) denotes the total number of functions kept in the snapshot space.

For CG case we define the following spectral problems in the space of snapshot:

$$\begin{aligned} R_{\text {snap}}^TAR_{\text {snap}}\Psi _k=\zeta (R_{\text {snap}}^{+})^TM^+R_{\text {snap}}^+\Psi _k, \end{aligned}$$
(21)

or

$$\begin{aligned} (R_{\text {snap}}^{+})^TA^+R_{\text {snap}}^+\Psi _k=\zeta (R_{\text {snap}}^{+})^TM^+R_{\text {snap}}^+\Psi _k, \end{aligned}$$
(22)

where

$$\begin{aligned} A&= [a_{kl}] =\int _{\omega _i} \Big ( 2\mu {\epsilon }( {\psi _k^{\text {snap}}}) : {\epsilon }( {\psi _l^{\text {snap}}}) + \lambda \nabla \cdot {\psi _k^{\text {snap}}} \, \nabla \cdot {\psi _l^{\text {snap}}} \Big )~d {x},\\ A^+&= [a_{kl}^+] =\int _{\omega _i^+} \Big ( 2\mu {\epsilon } ({\psi _k^{+,\text {snap}}}) : {\epsilon }( {\psi _l^{+,\text {snap}}}) + \lambda \nabla \cdot {\psi _k^{+,\text {snap}}} \, \nabla \cdot {\psi _l^{+,\text {snap}}} \Big )~d {x},\\ M^+&= [m_{kl}^+] =\int _{\omega _i^+} \tilde{\kappa } {\psi _k^{+,\text {snap}}} \cdot {\psi _l^{+,\text {snap}}}~d {x}, \end{aligned}$$

where \(\tilde{\kappa }\) is defined through (15).

The local spectral problem for DG coupling is defined as

$$\begin{aligned} (R_{\text {snap}}^{+})^TA^+R_{\text {snap}}^+\Psi _k=\zeta (R_{\text {snap}}^{+})^TM_1^+R_{\text {snap}}^+\Psi _k \end{aligned}$$
(23)

or

$$\begin{aligned} (R_{\text {snap}}^{+})^TA^+R_{\text {snap}}^+\Psi _k=\zeta (R_{\text {snap}}^{+})^TM_2^+R_{\text {snap}}^+\Psi _k \end{aligned}$$
(24)

in the snapshot space, where

$$\begin{aligned}&A^+=[a_{kl}^+] =\int _{K_i^+} \Big ( 2\mu {\epsilon }( {\psi _k^{+,\text {snap}}}) : {\epsilon }( {\psi _l^{+,\text {snap}}}) + \lambda \nabla \cdot {\psi _k^{+,\text {snap}}} \, \nabla \cdot {\psi _l^{+,\text {snap}}} \Big )~d {x},\\&M_1^+=[m_{1,kl}^+] =\frac{1}{H}\int _{K_i^+} \{{\lambda +2\mu }\} {\psi _k^{+,\text {snap}}} \cdot {\psi _l^{+,\text {snap}}}~d {x}, \\&M_2^+=[m_{2,kl}^+] =\frac{1}{H}\int _{\partial K_i^+} \{{\lambda +2\mu }\} {\psi _k^{+,\text {snap}}} \cdot {\psi _l^{+,\text {snap}}}~d {s}. \end{aligned}$$

After solving above local spectral problems, we form the offline space as in the no oversampling case, see Sect. 3.1 for CG coupling and Sect. 3.2 for DG coupling.

4 Numerical result

In this section, we present numerical results for CG-GMsFEM and DG-GMsFEM with two models. We consider different choices of snapshot spaces such as local-fine grid functions and harmonic functions and use different local spectral problems such as no-oversampling and oversampling described in the previous section. For the first model, we consider the medium that has no-scale separation and features such as high conductivity channels and isolated inclusions. The Young’s modulus \(E(x)\) is depicted in Fig. 2, \(\lambda (x)=\frac{\nu }{(1+\nu )(1-2\nu )}E(x)\), \(\mu (x)=\frac{1}{2(1+\nu )}E(x)\), the Poisson ratio \(\nu \) is taken to be \(0.22\). For the second example, we use the model that is used in Gao et al. (2014) for the simulation of subsurface elastic waves (see Fig. 3). In all numerical tests, we use constant force and homogeneous Dirichlet boundary condition. In all tables below, \(\Lambda _*\) represent the minimum discarded eigenvalue of the corresponding spectral problem. We note that the first three eigenbasis (that correspond to the first three smallest eigenvalues) are constant and linear functions, therefore we present our numerical results starting from fourth eigenbasis in all cases. In the below, dimension of a solution represents the total number of basis used for the finite element space.

Fig. 2
figure 2

Young’s modulus (Model 1)

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Fig. 3
figure 3

Left \(\lambda \), Right \(\mu \) (Model 2)

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  • We observe a fast decay in the error as more basis functions are added in both CG-GMsFEM and DG-GMsFEM

  • We observe the use of multiscale partition of unity improves the accuracy of CG-GMsFEM compared to the use of piecewise bi-linear functions

  • We observe an improvement in the accuracy (a slight improvement in CG case and a large improvement in DG case) when using oversampling for the examples we considered and the decrease in the snapshot space dimension.

4.1 Numerical results for Model 1 with conforming GMsFEM (CG-GMsFEM)

For the first model, we divide the domain \(D=[0,1]\times [0,1]\) into \(10\times 10\) coarse grid blocks, inside each coarse block we use \(10\times 10\) fine scale square blocks, which results in a \(100\times 100\) fine grid blocks. The number of basis functions used to get the reference solution is 20,402. We will show the performance of CG-GMsFEM with the use of local fine-scale snapshots and harmonic extension snapshots. Both bi-linear and multiscale partition of unity functions (see Sect. 3.1) will be considered. For each case, we will provide the comparison using oversampling and no-oversampling. For the error measure, we use relative weighted \(L^2\) norm error and weighted \(H^1\) norm error to compare the accuracy of CG-GMsFEM, which is defined as

$$\begin{aligned} e_{L^2}=\frac{\Vert (\lambda +2\mu )( u_{H}- u_h)\Vert _{L^{2}(D)}}{\Vert (\lambda +2\mu ) u_h\Vert _{L^{2}(D)}},\quad e_{H^{1}}=\sqrt{\frac{a( {u_{H}-u_h}, {u_{H}-u_h})}{a( {u_h}, {u_h})}} \end{aligned}$$

where \( {u_H}\) and \( {u_h}\) are CG-GMsFEM defined in (7) and fine-scale CG-FEM solution defined in (4) respectively.

Tables 1 and 2 show the numerical results of using local fine-scale snapshots with piecewise bi-linear function and multiscale functions as partition of unity respectively. As we observe, when using more multiscale basis, the errors decay rapidly, especially for multiscale partition of unity. For example, we can see that the weighted \(L^2\) error drops from 24.9 to 1.1 % in the case of using bi-linear function as partition of unity with no oversampling, while the dimension increases from 728 to 2,672. If we use multiscale partition of unity, the corresponding weighted \(L^2\) error drops from 8.4 to 0.6 %, which demonstrates a great advantage of multiscale partition of unity. Oversampling can help improve the accuracy as our results indicate. The local eigenvalue problem used for oversampling is Eq. (22).

Table 1 Relative errors between CG-MsFEM solution and the fine-scale CG-FEM solution, piecewise bi-linear partition of unity functions are used
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Table 2 Relative errors between CG-MsFEM solution and the fine-scale CG-FEM solution, multiscale partition of unity functions are used
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Next, we present the numerical results when harmonic extensions are used as snapshots in Tables 3 and 4. We can observe similar trends as in the local fine-scale snapshot case. The errors decrease as the number of basis functions increase. The \(L^2\) error is less than \(1\) % when about \(13\) % of degrees of freedom is used. Similarly, the oversampling method helps to improve the accuracy. In this case, the local eigenvalue problem used for oversampling is Eq. (21).

Table 3 Relative errors between CG-MsFEM solution and the fine-scale CG-FEM solution, piecewise bi-linear partition of unity functions are used
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Table 4 Relative errors between CG-MsFEM solution and the fine-scale CG-FEM solution, multiscale partition of unity functions are used
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4.2 Numerical results for Model 1 with DG-GMsFEM

In this section, we consider numerical results for DG-GMsFEM discussed in Sect. 3.2. To show the performance of DG-GMsFEM, we use the same model (see Fig. 2) and the coarse and fine grid settings as in the CG case. We will also present the result of using both harmonic extension and eigenbasis (local fine-scale) as snapshot space. To measure the error, we define broken weighted \(L^2\) norm error and \(H^1\) norm error

$$\begin{aligned} e_{H^1} =\sqrt{\frac{\sum _{K\in {\mathcal T}_{H}}\int _{K} {\sigma }( u_{H}- u_h)): {\varepsilon }( u_{H}- u_h))~d {x}}{\sum _{K\in {\mathcal T}_{H}}\int _{K} {\sigma }( u_h): {\varepsilon }(u_h)~d {x}}} \end{aligned}$$

where \({u_H}\) and \({u_h}\) are DG-GMsFEM defined in (8) and fine-scale DG-FEM solution defined in (50) respectively. We note that the dimension of the reference solution \({u_h}\) here is 24,200.

In Table 5, the numerical results of DG-MsFEM with local fine-scale functions as the snapshot space is shown. We observe that DG-MsFEM shows a better approximation compared to CG-MsFEM if oversampling is used. The error decreases more rapidly as we add basis. More specifically, the relative broken \(L^2\) error and \(H^1\) error decrease from \(14.1\), \(52.5\) to \(0.2\) and \(5.8\) % respectively, while the degrees of freedom of the coarse system increase from 728 to 2,696, where the latter is only \(13.2\) % of the reference solution. The local eigenvalue problem used for oversampling is Eq. (23).

Table 5 Relative errors between DG-MsFEM solution and the fine-scale DG-FEM solution
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Table 6 shows the corresponding results when harmonic functions are used to construct the snapshot space. We observe similar errors decay trend as local fine-scale snapshots are used. Oversampling can help improve the results significantly. Although the error is very large when the dimension of coarse system is \(728\) (\(4\) multiscale basis is used), the error becomes very small when the dimension reaches 1,728 (\(9\) multiscale basis is used). The local eigenvalue problem used for oversampling here is Eq. (24). We remark that oversampling can not only help decrease the error, but also decrease the dimension of the snapshot space greatly in periodic case.

Table 6 Relative errors between DG-MsFEM solution and the fine-scale DG-FEM solution
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4.3 Numerical results for Model 2

The purpose of this example is to test a method for an earth model that is used in Gao et al. (2014). The domain for the second model is \(D=(0,6{,}000)^2\) (in meters) which is divided into \(900=30\times 30\) square coarse grid blocks, inside each coarse block we generate \(20\times 20\) fine scale square blocks. The reference solution is computed through standard CG-FEM on the resulting \(600\times 600\) fine grid. We note that the dimension of the reference solution is 722,402. The numerical results for CG-MsFEM and DG-MsFEM are presented in Tables 7 and 8 respectively. We observe the relatively low errors compared to the high contrast case and the error decrease with the dimension increase of the offline space. Both coupling methods (CG and DG) show very good approximation ability.

Table 7 Relative errors between CG-MsFEM solution and the fine-scale CG-FEM solution, piecewise bi-linear partition of unity functions are used
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Table 8 Relative errors between DG-MsFEM solution and the fine-scale CG-FEM solution
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5 Error estimate for CG coupling

In this section, we present error analysis for both no oversampling and oversampling cases. In the discussions below, \(a\preceq b\) means \(a\le Cb\), where \(C\) is a constant independent of the mesh size and the contrast of the coefficient.

5.1 No oversampling case

Lemma 1

Let \(\omega _n\) coarse neighborhood. For any \(\psi \in H^1(\omega _n),\) we define \(r=-\mathrm{{div}}(\sigma (\psi ))\). Then we have

$$\begin{aligned}&\int _{\omega _n}2\mu \chi _n^2 \epsilon (\psi ):\epsilon (\psi )~d {x}+\int _{\omega _n}\lambda \chi _n^2 (\nabla \cdot {\psi })^{2}~d{x}\nonumber \\&\quad \preceq \left| \int _{\omega _n} \chi _n^2{r}\cdot {{\psi }}~d {x}\right| +\int _{\omega _n} (\lambda + 2\mu )|\nabla \chi _n|^2 {\psi }^2~d{x}, \end{aligned}$$
(25)

where \(\chi _n\) is a scalar partition of unity subordinated to the coarse neighborhood \(\omega _n\).

Proof

Multiplying both sides of \(-\mathrm{{div}}(\sigma (\psi ))=r\) by \(\chi _n^2 \psi \), we have

$$\begin{aligned} \int _{\omega _n} \chi _n^2{r}\cdot {\psi }~d{x}&= \int _{\omega _n} 2 \mu \epsilon (\psi ):\epsilon (\chi _n^2\psi )~d {x}+\int _{\omega _n}\lambda \nabla \cdot \psi \nabla \cdot (\chi _n^2{\psi })~d {x}\nonumber \nonumber \\&= \int _{\omega _n} 2\mu \chi _n^2 \epsilon (\psi ):\epsilon (\psi )~d {x} + \int _{\omega _n}2 \mu \chi _n \epsilon _{ij}({\psi }) \left( \psi _i \frac{\partial \chi _n}{\partial x_j} + \psi _j \frac{\partial \chi _n}{\partial x_i}\right) ~d {x}\nonumber \\&+\int _{\omega _n} \lambda \chi _n^2 (\nabla \cdot {\psi })^2~d {x} + \int _{\omega _n}2 \lambda \nabla \cdot \psi \chi _n \psi \cdot \nabla \chi _n~d {x}\nonumber \\&= \int _{\omega _n} 2\mu \chi _n^2 \epsilon (\psi ):\epsilon (\psi )~d {x} +\int _{\omega _n} 2 \big ( \sqrt{2\mu } \chi _n \epsilon _{ij}({\psi })\big )\nonumber \\&\times \left( \sqrt{\mu /2} \left( \psi _i {\partial \chi _n\over \partial x_j}~d{x}+ \psi _j \frac{\partial \chi _n}{\partial x_i}\right) \right) ~d {x}\nonumber \\&+\int _{\omega _n}\lambda \chi _n^2 (\nabla \cdot \psi )^2~d {x} +\int _{\omega _n} 2 \big ( \sqrt{\lambda } \chi _n \nabla \cdot \psi \big ) \big (\sqrt{\lambda }\psi \cdot \nabla \chi _n\big )~d {x}. \end{aligned}$$
(26)

Therefore,

$$\begin{aligned}&\int _{\omega _n}2\mu \chi _n^2 \epsilon (\psi ):\epsilon (\psi )~d {x} +\int _{\omega _n}\lambda \chi _n^2 (\nabla \cdot \psi )^2~d {x}\nonumber \\&\quad \le \left| \int _{\omega _n} \chi _n^2{r}\cdot {\psi }~d {x}\right| +\left| \int _{\omega _n} 2 \big ( \sqrt{2\mu } \chi _n \epsilon _{ij} ({\psi })\big ) \left( \sqrt{\mu /2} \left( \psi _i \frac{\partial \chi _n}{\partial x_j} +\psi _j \frac{\partial \chi _n}{\partial x_i}\right) \right) ~d {x}\right. \nonumber \\&\quad \quad \left. +\int _{\omega _n} 2 (\sqrt{\lambda } \chi _n \nabla \cdot \psi ) (\sqrt{\lambda }\psi \cdot \nabla \chi _n)~d {x}\right| \nonumber \\&\quad \preceq \left| \int _{\omega _n} \chi _n^2{r}\cdot {{\psi }}~d {x}\right| +\int _{\omega _n} (2\lambda \!+\! 4\mu )|\nabla \chi _n|^2 {\psi }^2~d {x}\nonumber \\&\quad \preceq \left| \int _{\omega _n} \chi _n^2{r}\cdot {{\psi }}~d {x}\right| \!+\!\int _{\omega _n} (\lambda \!+\! 2\mu )|\nabla \chi _n|^2 {\psi }^2~d {x}. \end{aligned}$$
(27)

In the last step, we have used \(2ab\le \epsilon a^2+\frac{1}{\epsilon }b^2\), and \((ab+cd)^2\le (a^2+c^2)(b^2+d^2)\). \(\square \)

Next, we will show the convergence of the CG-GMsFEM solution defined in (7) without oversampling. We take \(I^{\omega _n}{u_h}\) to be the first \(L_n\) terms of spectral expansion of \(u\) in terms of eigenfunctions of the problem \(-\text {div}(\sigma (\phi _{n}))=\xi \tilde{\kappa }\phi _{n}\) solved in \(V^h{(\omega _n)}\). Applying Cea’s lemma, Lemma 1 and using the fact that \(\chi _n\preceq 1\), we can get

$$\begin{aligned}&\int _D \left( 2\mu \epsilon ({u_h}-{u_H}):\epsilon ({u_h}-{u_H})+ \lambda (\nabla \cdot ({u_h}-{u_H}))^2\right) ~d {x}\nonumber \\&\quad \preceq \sum _{n=1}^{N_s} \int _{\omega _n}\left( 2\mu \epsilon (\chi _n({u_h}-I^{\omega _n}{u_h})):\epsilon (\chi _n ({u_h}-I^{\omega _n}{u_h})) \right. \nonumber \\&\qquad +\lambda (\nabla \cdot (\chi _n({u_h} -I^{\omega _n}{u_h})))^2\big )~d {x}\nonumber \\&\quad \preceq \sum _{n=1}^{N_s}\int _{\omega _n}2\mu \chi _n^2 \epsilon ({u_h}-I^{\omega _n}{u_h}):\epsilon ({u_h}-I^{\omega _n}{u_h})~d {x} \nonumber \\&\qquad +\sum _{n=1}^{N_s}\int _{\omega _n}\lambda \chi _n^2 (\nabla \cdot ({u_h} -I^{\omega _n}{u_h}))^2~d {x}\nonumber \\&\qquad +\sum _{n=1}^{N_s} \int _{\omega _n}(\lambda +2\mu ) |\nabla \chi _n|^2 ({u_h}-I^{\omega _n}{u_h})^2~d {x}\nonumber \\&\quad \preceq \sum _{n=1}^{N_s} \int _{\omega _n}(\lambda +2\mu ) |\nabla \chi _n|^2 ({u_h}-I^{\omega _n}{u_h})^2~d {x}+ \sum _{n=1}^{N_s}\left| \int _{\omega _n} \chi _n^2g\cdot ({u_h}-I^{\omega _n}{u_h})\right| ~d {x}\nonumber \\&\quad \preceq \sum _{n=1}^{N_s} \int _{\omega _n}(\lambda \!+\!2\mu )|\nabla \chi _n|^2 ({u_h}\!-\!I^{\omega _n}{u_h})^2~d {x} \!+\!\sum _{n=1}^{N_s}\int _{\omega _n}((\lambda \!+\!2\mu ) |\nabla \chi _n|^2)^{-1} {g}^2~d {x},\nonumber \\ \end{aligned}$$
(28)

where \(g=f_h+\text {div}(\sigma (I^{\omega _n}{u_h}))\), \(f_h\) is the \(L^2\) projection of \(f\) in \(V^h\), \(f\) is the right hand side of (2).

Using the properties of the eigenfunctions, we obtain

$$\begin{aligned}&\int _{\omega _n}(\lambda +2\mu )\sum _{s=1}^{N_s}|\nabla \chi _s|^2 ({u_h}-I^{\omega _n}{u_h})^2~dx\nonumber \\&\quad \preceq \frac{1}{\xi _{L_n+1}^{\omega _n}} \int _{\omega _n}\left( 2\mu \epsilon ({u_h}-I^{\omega _n}{u_h}): \epsilon ({u_h}-I^{\omega _n}{u_h})+ \lambda (\nabla \cdot ({u_h}-I^{\omega _n}{u_h}))^2\right) ~d {x}.\nonumber \\ \end{aligned}$$
(29)

Then, the first term in the right hand side of (28) can be estimated as follows

$$\begin{aligned}&\sum _{n=1}^{N_s} \int _{\omega _n}(\lambda +2\mu )|\nabla \chi _n|^2 ({u_h}-I^{\omega _n}{u_h})^2~d{x}\nonumber \\&\quad \preceq \sum _{n=1}^{N_s}\int _{\omega _n}(\lambda +2\mu ) \sum _{s=1}^{N_s}|\nabla \chi _s|^2 |({u_h}-I^{\omega _n}{u_h})^2~d {x}\nonumber \\&\quad \preceq \sum _{n=1}^{N_s}\frac{1}{\xi _{L_n+1}^{\omega _n}} \int _{\omega _n}\left( 2\mu \epsilon ({u_h}-I^{\omega _n}{u_h}): \epsilon ({u_h}-I^{\omega _n}{u_h})\right. \nonumber \\&\qquad \left. + \lambda (\nabla \cdot ({u_h}-I^{\omega _n}{u_h}))^2\right) ~d {x}\nonumber \\&\quad \preceq \sum _{n=1}^{N_s}\frac{\alpha ^{\omega _n}_{L_n+1}}{\xi _{L_n+1}^{\omega _n}}\int _{\omega _n}\left( 2\mu \chi _n^2 \epsilon ({u_h}-I^{\omega _n}{u_h}):\epsilon ({u_h}-I^{\omega _n}{u_h})\right. \nonumber \\&\qquad \left. +\lambda \chi _n^2 (\nabla \cdot ({u_h}-I^{\omega _n}{u_h}))^2\right) ~d {x} \nonumber \\&\quad \preceq \sum _{n=1}^{N_s}\frac{\alpha ^{\omega _n}_{L_n+1}}{\xi _{L_n+1}^{\omega _n}}\int _{\omega _n} (\lambda + 2\mu )|\nabla \chi _n|^2 ({u_h}-I^{\omega _n}{u_h})^2~d {x}\nonumber \\&\qquad +\sum _{n=1}^{N_s}\frac{\alpha ^{\omega _n}_{L_n+1}}{\xi _{L_n+1}^{\omega _n}}\left| \int _{\omega _n} \chi _n^2{g} \cdot ({u_h}-I^{\omega _n}{u_h})~d {x}\right| \nonumber \\&\quad \preceq \frac{1}{\Lambda _*}\left( \sum _{n=1}^{N_s} \int _{\omega _n}(\lambda + 2\mu )|\nabla \chi _n|^2 ({u_h} -I^{\omega _n}{u_h})^2~d {x}\right. \nonumber \\&\qquad \left. +\sum _{n=1}^{N_s}\left| \int _{\omega _n} \chi _n^2{g} \cdot ({u_h}-I^{\omega _n}{u_h})~d {x}\right| \right) , \end{aligned}$$
(30)

where

$$\begin{aligned} \Lambda _*=\text {min}_{\omega _n}\frac{\xi _{L_n+1}^{\omega _n}}{\alpha ^{\omega _n}_{L_n+1}}, \end{aligned}$$

and

$$\begin{aligned} {\alpha ^{\omega _n}_{L_n+1}}\!=\!\frac{\int _{\omega _n} 2\mu \epsilon ({u_h}-I^{\omega _n}{u_h}):\epsilon ({u_h}\!-\!I^{\omega _n}{u_h})~ d {x}\!+\!\int _{\omega _n}\lambda (\nabla \cdot ({u_h}\!-\!I^{\omega _n}{u_h}))^2~d {x}}{\int _{\omega _n}2\mu \chi _n^2 \epsilon ({u_h}\!-\!I^{\omega _n}{u_h}):\epsilon ({u_h}\!-\!I^{\omega _n}{u_h})~ d {x}\!+\!\int _{\omega _n}\lambda \chi _n^2 (\nabla \cdot ({u_h}\!-\!I^{\omega _n}{u_h}))^2~d {x}}. \end{aligned}$$

Applying inequality (30) \(m\) times, we have

$$\begin{aligned}&\sum _{n=1}^{N_s} \int _{\omega _n}(\lambda +2\mu )|\nabla \chi _n|^2 ({u_h}-I^{\omega _n}{u_h})^2~d {x}\nonumber \\&\quad \preceq \left( \frac{1}{\Lambda _*}\right) ^m \sum _{n=1}^{N_s}\int _{\omega _n}(\lambda + 2\mu )|\nabla \chi _n|^2 ({u_h}-I^{\omega _n}{u_h})^2~d {x}\nonumber \\&\qquad +\sum _{l=1}^m \left( \frac{1}{\Lambda _*}\right) ^l\sum _{n=1}^{N_s}\left| \int _{\omega _n} \chi _n^2{g} \cdot ({u_h}-I^{\omega _n}{u_h})~d {x}\right| \nonumber \\&\quad \preceq \left( \frac{1}{\Lambda _*}\right) ^m \sum _{n=1}^{N_s}\int _{\omega _n}(\lambda + 2\mu )|\nabla \chi _n|^2 ({u_h}-I^{\omega _n}{u_h})^2~d {x}\nonumber \\&\qquad +(\Lambda _*)^m\left( \frac{1-\Lambda _*^{-m}}{\Lambda _*-1} \right) \sum _{n=1}^{N_s}\int _{\omega _n}((\lambda +2\mu )|\nabla \chi _n|^2)^{-1}{g}^2~d {x}, \end{aligned}$$
(31)

Taking into account that

$$\begin{aligned}&\sum _{n=1}^{N_s} \int _{\omega _n}(\lambda +2\mu )|\nabla \chi _n|^2 ({u_h}-I^{\omega _n}{u_h})^2~d {x}\nonumber \\&\quad \preceq \sum _{n=1}^{N_s}\int _{\omega _n}(\lambda +2\mu )\sum _{s=1}^{N_s} |\nabla \chi _s|^2 ({u_h}-I^{\omega _n}{u_h})^2~d {x}, \end{aligned}$$
(32)

and

$$\begin{aligned}&\sum _{n=1}^{N_s}\int _{\omega _n} \left( 2\mu \epsilon ({u_h} -I^{\omega _n}{u_h}):\epsilon ({u_h}-I^{\omega _n}{u_h})~d {x} +\lambda (\nabla \cdot ({u_h}-I^{\omega _n}{u_h}))^2\right) ~d {x}\nonumber \\&\quad \preceq \int _{D}\left( 2\mu \epsilon ({u_h}):\epsilon (u_h)+ \lambda (\nabla \cdot {u_h})^2\right) ~d {x}. \end{aligned}$$
(33)

inequality (28) becomes

$$\begin{aligned}&\int _D\left( 2\mu \epsilon ({u_h}-{u_H}): \epsilon ({u_h}-{u_H})+ \lambda (\nabla \cdot ({u_h}-{u_H}))^2\right) ~d {x}\nonumber \\&\quad \preceq \left( \frac{1}{\Lambda _*}\right) ^{m+1} \sum _{n=1}^{N_s}\int _{\omega _n}\left( 2\mu \epsilon ({u_h} -I^{\omega _n}{u_h}):\epsilon ({u_h}-I^{\omega _n}{u_h}) \right. \nonumber \\&\qquad \left. +\lambda (\nabla \cdot ({u_h}-I^{\omega _n}{u_h}))^2\right) ~d {x}\nonumber \\&\qquad +\left( \Lambda _*^m\left( \frac{1-\Lambda _*^{-m}}{\Lambda _*-1}\right) +1\right) \sum _{n=1}^{N_s}\int _{\omega _n}((\lambda +2\mu )|\nabla \chi _n|^2)^{-1}{g}^2~d {x}\nonumber \\&\quad \preceq \left( \frac{1}{\Lambda _*}\right) ^{m+1}\int _{D} \left( 2\mu \epsilon ({u_h}):\epsilon (u)+\lambda (\nabla \cdot {u_h})^2\right) ~d {x}\nonumber \\&\qquad + \left( (\Lambda _*)^m\left( \frac{1-(\Lambda _*)^{-m}}{\Lambda _*-1}\right) +1\right) R, \end{aligned}$$
(34)

where \(R=\sum _{n=1}^{N_s}\int _{\omega _n}((\lambda +2\mu )|\nabla \chi _n|^2)^{-1}{g}^2\, d {x}\). If \(|{g}|\preceq 1\), then \(\int _{\omega _n}((\lambda +2\mu )|\nabla \chi _n|^2)^{-1}{g}^2\, d {x}\preceq H^2\), from which we obtain

$$\begin{aligned}&\int _D\left( 2\mu \epsilon ({u_h}-{u_H}):\epsilon ({u_h}-{u_H})+ \lambda (\nabla \cdot ({u_h}-{u_H}))^2\right) ~d {x}\nonumber \\&\quad \preceq \left( \frac{1}{\Lambda _*}\right) ^{m+1} \int _{D}\left( 2\mu \epsilon ({u_h}):\epsilon (u) +\lambda (\nabla \cdot {u_h})^2\right) ~d {x}\nonumber \\&\qquad +\left( (\Lambda _*)^m\left( \frac{1-(\Lambda _*)^{-m}}{\Lambda _*-1}\right) +1\right) H^2. \end{aligned}$$
(35)

Combining results above, we have

Theorem 1

Let \(u_h\in V^h_{CG}\) be the fine-scale CG-FEM solution defined in (4) and \(u_H\) be the CG-GMsFEM solution defined in (7) without oversampling. If \(\Lambda _*>1\) and \(\int _D (\lambda +2\mu )^{-1}g^2\; d {x}\preceq 1,\) let \( n=-\frac{\mathrm {log}(H)}{\mathrm {log}\Lambda _*},\) then

$$\begin{aligned}&\int _D\left( 2\mu \epsilon ({u_h}-{u_H}):\epsilon ({u_h}-{u_H})+\lambda (\nabla \cdot ({u_h}-{u_H}))^2\right) ~d {x} \\&\quad \preceq \left( \frac{H}{\Lambda _*}\right) \left( \int _{D}\left( 2\mu \epsilon (u_h):\epsilon (u_h)+\lambda (\nabla \cdot {u_h})^2 \right) ~d {x}+1\right) . \end{aligned}$$

5.2 Oversampling case

In this subsection, we will analyze the convergence of CG-GMsFEM solution defined in (7) with oversampling. We define \(I^{\omega _n^{+}}{u_h}\) as an interpolation of \({u_h}\) in \(\omega _n^{+}\) using the first \(L_n\) modes for the eigenvalue problem (21). Let \(\chi _n^+\) be a partition of unity subordinated to the coarse neighborhood \(\omega _n^+\). We require \(\chi _n^+\) to be zero on \(\partial \omega _n^+\) and

$$\begin{aligned} |\nabla \chi _n|^2\preceq |\nabla \chi _n^+|^2. \end{aligned}$$

Using the same argument as Lemma 1, it is easy to deduce

$$\begin{aligned}&\int _{\omega _n^+}\left( 2\mu |\chi _n^+|^2\epsilon ({u_h}-I^{\omega _n^+}{u_h}): \epsilon ({u_h}-I^{\omega _n^+}{u_h})+\lambda |\chi _n^+|^2 (\nabla \cdot ({u_h}-I^{\omega _n^+}{u_h}))^2 \right) ~d {x}\nonumber \\&\quad \preceq \left| \int _{\omega _n^+} |\chi _n^+|^2{g}\cdot ({u_h}-I^{\omega _n^+}{u_h})~d {x}\right| + \int _{\omega _n^+} (\lambda + 2\mu )|\nabla \chi _n^+|^2 ({u_h}-I^{\omega _n^+}{u_h})^2~d {x},\nonumber \\ \end{aligned}$$
(36)

where \(g=f_h+\text {div}(\sigma (I^{\omega _n}{u_h}))\), \(I^{\omega _n}{u_h} =I^{\omega _n^{+}}{u_h}\) in \(\omega _n\).

Applying eigenvalue problem (21), we obtain

$$\begin{aligned}&\int _{\omega _n^+}(\lambda +2\mu ) |\nabla \chi _n^+|^2 ({u_h}-I^{\omega _n^+}{u_h})^2~d {x}\nonumber \\&\quad \preceq \frac{1}{\xi _{L_n+1}^{\omega _n}}\int _{\omega _n} \left( 2\mu \epsilon ({u_h}-I^{\omega _n}{u_h}):\epsilon ({u_h} -I^{\omega _n}{u_h}) +\lambda (\nabla \cdot ({u_h}-I^{\omega _n}{u_h}))^2\right) ~d {x}.\nonumber \\ \end{aligned}$$
(37)

Using the definition of interpolation \(I^{\omega _n^+}{u_h}\) and inequality (37), we have

$$\begin{aligned}&\sum _{n=1}^{N_s}\int _{\omega _n}(\lambda +2\mu ) |\nabla \chi _n|^2 ({u_h}-I^{\omega _n}{u_h})^2~d {x}\nonumber \\&\quad \preceq \sum _{n=1}^{N_s}\int _{\omega _n^+}(\lambda +2\mu ) |\nabla \chi _n^+|^2 ({u_h}-I^{\omega _n^+}{u_h})^2~d {x}\nonumber \\&\quad \preceq \sum _{n=1}^{N_s} \frac{1}{\xi _{L_n+1}^{\omega _n}} \int _{\omega _n}\left( 2\mu \epsilon ({u_h}-I^{\omega _n}{u_h}): \epsilon ({u_h}-I^{\omega _n}{u_h})\right. \nonumber \\&\qquad + \lambda (\nabla \cdot ({u_h}-I^{\omega _n}{u_h}))^2\big )~d {x}\nonumber \\&\quad \preceq \sum _{n=1}^{N_s} \frac{1}{\xi _{L_n+1}^{\omega _n}} \int _{\omega _n^+} \left( 2\mu |\nabla \chi _n^+|^2\epsilon ({u_h} -I^{\omega _n^+}{u_h}):\epsilon ({u_h}-I^{\omega _n^+}{u_h})~d {x}\right. \nonumber \\&\qquad +\left. \lambda |\nabla \chi _n^+|^2(\nabla \cdot ({u_h} -I^{\omega _n^+}{u_h}))^2\right) ~d {x}\nonumber \\&\quad \preceq \sum _{n=1}^{N_s}\frac{1}{\xi _{L_n+1}^{\omega _n}} \int _{\omega _n^+} (\lambda + 2\mu )|\nabla \chi _n^+|^2 ({u_h} -I^{\omega _n^+}{u_h})^2~d {x}\nonumber \\&\qquad +\sum _{n=1}^{N_s} \frac{1}{\xi _{L_n+1}^{\omega _n}}\left| \int _{\omega _n^+}|\chi _n^+|^2{g}\cdot ({u_h}-I^{\omega _n^+}{u_h})~d {x}\right| \nonumber \\&\quad \preceq \frac{1}{\Lambda _*^+}\sum _{n=1}^{N_s} \int _{\omega _n^+} (\lambda + 2\mu )|\nabla \chi _n^+|^2 ({u_h}-I^{\omega _n^+}{u_h})^2~d {x}\nonumber \\&\qquad +\frac{1}{\Lambda _*^+}\sum _{n=1}^{N_s} \left| \int _{\omega _n^+} |\chi _n^+|^2{g}\cdot ({u_h}-I^{\omega _n^+}{u_h})~d {x}\right| \nonumber \\&\quad \preceq \frac{1}{\Lambda _*^+}\sum _{n=1}^{N_s} \frac{1}{\xi _{L_n+1}^{\omega _n}}\int _{\omega _n} \big (2\mu \epsilon ({u_h}-I^{\omega _n}{u_h}):\epsilon ({u_h}-I^{\omega _n}{u_h})\nonumber \\&\qquad +\lambda (\nabla \cdot ({u_h}-I^{\omega _n}{u_h}))^2\big )~d {x}\nonumber \\&\qquad + \frac{1}{\Lambda _*^+}\sum _{n=1}^{N_s}\left| \int _{\omega _n^+} |\chi _n^+|^2g\cdot ({u_h}-I^{\omega _n^+}u)~d {x}\right| , \end{aligned}$$
(38)

where \({\Lambda _*^+}=\text {min}_{\omega _n}\xi _{L_n+1}^{\omega _n}\).

Applying the last inequality \(m\) times with (37), we get

$$\begin{aligned}&\sum _{n=1}^{N_s}\int _{\omega _n^+}(\lambda +2\mu ) |\nabla \chi _n^+|^2 ({u_h}-I^{\omega _n^+}{u_h})^2~d {x}\nonumber \\&\quad \preceq \left( \frac{1}{\Lambda _*^+}\right) ^m \frac{1}{\xi _{L_n+1}^{\omega _n}}\sum _{n=1}^{N_s} \int _{\omega _n} (2\mu \epsilon ({u_h}-I^{\omega _n}{u_h}): \epsilon ({u_h}-I^{\omega _n}{u_h})\nonumber \\&\qquad + \lambda (\nabla \cdot ({u_h} -I^{\omega _n}{u_h}))^2)~d {x}\nonumber \\&\qquad +\sum _{l=1}^{m}\left( \frac{1}{\Lambda _*^+}\right) ^l \sum _{n=1}^{N_s} \left| \int _{\omega _n^+}|\chi _n^+|^2 {g} \cdot ({u_h}-I^{\omega _n^+}{u_h})~d {x}\right| \nonumber \\&\quad \preceq \left( \frac{1}{\Lambda _*^+}\right) ^{m+1} \sum _{n=1}^{N_s}\int _{\omega _n} (2\mu \epsilon ({u_h} -I^{\omega _n}{u_h}):\epsilon ({u_h}-I^{\omega _n}{u_h})\nonumber \\&\qquad +\lambda (\nabla \cdot ({u_h}-I^{\omega _n}{u_h}))^2)~d {x}\nonumber \\&\qquad +(\Lambda _*^+)^m\left( \frac{1-(\Lambda _*^+)^{-m}}{\Lambda _*^+-1}\right) \sum _{n=1}^{N_s}\int _{\omega _n^+} ((\lambda +2\mu )|\nabla \chi _n^+|^2)^{-1}{g}^2~d {x}. \end{aligned}$$
(39)

Using Cea’s lemma and inequality (33) , we have

$$\begin{aligned}&\int _{D}\left( 2\mu \epsilon ({u_h}-{u_H}):\epsilon ({u_h} -{u_H})+\lambda (\nabla \cdot ({u_h}-{u_H}))^2\right) ~d {x}\nonumber \\&\quad \preceq \sum _{n=1}^{N_s} \int _{\omega _n}(\lambda +2\mu )| \nabla \chi _n|^2 ({u_h}-I^{\omega _n}{u_h})^2~d {x}\nonumber \\&\qquad +\sum _{n=1}^{N_s}\left| \int _{\omega _n} \chi _n^2g\cdot ({u_h}-I^{\omega _n}{u_h})~d {x}\right| \nonumber \\&\quad \preceq \left( \frac{1}{\Lambda _*^+}\right) ^{m+1}\int _{D} \left( 2\mu \epsilon (u_h):\epsilon (u_h)+\lambda (\nabla \cdot {u_h})^2\right) ~d {x}\nonumber \\&\qquad + \left( (\Lambda _*^+)^m\left( \frac{1-(\Lambda _*^+)^{-m}}{\Lambda _*^+-1} \right) +1\right) R. \end{aligned}$$
(40)

where \(R=\sum _{n=1}^{N_s}\int _{\omega _n^+}((\lambda +2\mu )|\nabla \chi _n^+|^2)^{-1}{g}^2~d {x}\).

Therefore, similar with the no oversampling case, we have

Theorem 2

Let \(u_h\in V^h_{CG}\) be the fine-scale CG-FEM solution defined in (4) and \(u_H\) be the CG-GMsFEM solution defined in (7) with oversampling. If \(\Lambda _*^+>1\) and \(\int _D (\lambda +2\mu )^{-1}g^2\; d {x}\preceq 1,\) let \( n=-\frac{\mathrm {log}(H)}{\mathrm {log}\Lambda _*^+},\) then

$$\begin{aligned}&\int _D\left( 2\mu \epsilon ({u_h}-{u_H}):\epsilon ({u_h}-{u_H})+ \lambda (\nabla \cdot ({u_h}-{u_H}))^2\right) ~d {x} \\&\quad \preceq \left( \frac{H}{\Lambda _*^+}\right) \left( \int _{D}\left( 2\mu \epsilon (u_h):\epsilon (u_h)+ \lambda (\nabla \cdot {u_h})^2 \right) ~d {x}+1\right) . \end{aligned}$$

6 Error estimate for DG coupling

In this section, we will analyze the DG coupling of the GMsFEM (8). For any \( {u}\), we define the DG-norm by

$$\begin{aligned} \Vert {u} \Vert _{\text {DG}}^2 = a_H( {u}, {u}) + \sum _{E\in {\mathcal E}_{H}}\frac{\gamma }{h}\int _{E} \{{\lambda +2\mu }\} [{{u}}]^2 ~d {s}. \end{aligned}$$

Let \(K\) be a coarse grid block and let \( {n}_{\partial K}\) be the unit outward normal vector on \(\partial K\). We denote \(V^h(\partial K)\) by the restriction of the conforming space \(V^h\) on \(\partial K\). The normal flux \( {\sigma }( {u}) \, {n}_{\partial K}\) is understood as an element in \(V^h(\partial K)\) and is defined by

$$\begin{aligned} \int _{\partial K} ( {\sigma }( {u}) \, {n}_{\partial K}) \cdot {v} ~d {s}= \int _{K} \Big ( 2\mu {\epsilon }( {u}): {\epsilon }(\widehat{ {v}}) + \lambda \nabla \cdot {u} \nabla \cdot \widehat{ {v}} \Big )~d {x}, \quad {v} \in V^h(\partial K), \nonumber \\ \end{aligned}$$
(41)

where \(\widehat{ {v}}\) is the harmonic extension of \( {v}\) in \(K\). By the Cauchy–Schwarz inequality,

$$\begin{aligned} \int _{\partial K} ( {\sigma }( {u}) \, {n}_{\partial K}) \cdot {v}~d {s} \le a_H^K(u,u)^{\frac{1}{2}} \, a_H^K(\widehat{v},\widehat{v})^{\frac{1}{2}}. \end{aligned}$$

By an inverse inequality and the fact that \(\widehat{v}\) is the harmonic extension of \(v\)

$$\begin{aligned} a_H^K(\widehat{v},\widehat{v}) \le \kappa _K C^2_{\text {inv}} h^{-1} \int _{\partial K} |v|^2~d {x}, \end{aligned}$$
(42)

where \(\kappa _K = \max _K \{ \lambda +2\mu \}\) and \(C_{\text {inv}}>0\) is the constant from inverse inequality. Thus,

$$\begin{aligned} \int _{\partial K} ( {\sigma }( {u}) \, {n}_{\partial K}) \cdot {v}~d {s} \le \kappa _K^{\frac{1}{2}} C_{\text {inv}} h^{-\frac{1}{2}} \Vert v\Vert _{L^2(\partial K)} \, a_H^K(u,u)^{\frac{1}{2}}. \end{aligned}$$

This shows that

$$\begin{aligned} \int _{\partial K} | {\sigma }( {u}) \, {n}_{\partial K}|^2~d {s} \le \kappa _K C^2_{\text {inv}} h^{-1} a_H^K(u,u). \end{aligned}$$
(43)

Our first step in the convergence analysis is to establish the continuity and the coercivity of the bilinear form (9) with respect to the DG-norm.

Lemma 2

Assume that the penalty parameter \(\gamma \) is chosen so that \(\gamma > 2 C_{\mathrm {inv}}^2\). The bilinear form \(a_{\text {DG}}\) defined in (9) is continuous and coercive\(,\) that is\(,\)

$$\begin{aligned} a_{\text {DG}}( {u}, {v})&\le \Vert {u} \Vert _{\text {DG}} \, \Vert {v} \Vert _{\text {DG}},\end{aligned}$$
(44)
$$\begin{aligned} a_{\text {DG}}( {u}, {u})&\ge a_0 \Vert {u} \Vert _{\text {DG}}^2, \end{aligned}$$
(45)

for all \({u},{v},\) where \(a_0 = 1 - \sqrt{2} C_{\mathrm {inv}} \gamma ^{-\frac{1}{2}} >0.\)

Proof

By the definition of \(a_{\text {DG}}\), we have

$$\begin{aligned} a_{\text {DG}}( {u}, {v})&= a_H( {u}, {v}) - \sum _{E\in {\mathcal E}^H} \int _E \Big ( \{{ {\sigma }( {u}) \, {n}_E}\} \cdot [v] + \{{ {\sigma }( {v}) \, {n}_E}\} \cdot [{ {u}}] \Big )~d {s} \\&+ \sum _{E\in {\mathcal E}^H}\frac{\gamma }{h} \int _E \{{\lambda +2\mu }\} [u]\cdot [{{v}}]~d {s}. \end{aligned}$$

Notice that

$$\begin{aligned} a_H( {u}, {v}) + \sum _{E\in {\mathcal E}^H} \frac{\gamma }{h} \int _E \{{\lambda +2\mu }\} [u] \cdot [v]~d {s} \le \Vert u\Vert _{\text {DG}} \, \Vert v\Vert _{\text {DG}}. \end{aligned}$$

For an interior coarse edge \(E \in {\mathcal E}^H\), we let \(K^{+}, K^{-}\in {\mathcal T}^H\) be the two coarse grid blocks having the edge \(E\). By the Cauchy–Schwarz inequality, we have

$$\begin{aligned} \int _E \{{ {\sigma }( {u}) \, {n}_E}\} \cdot [v]~d {s} \!\le \! \Big ( h\int _E \{{ {\sigma }( {u}) \, {n}_E}\}^2 \{{\lambda \!+\!2\mu }\}^{-1}~d {s} \Big )^{\frac{1}{2}} \Big ( \frac{1}{h} \int _E \{{\lambda \!+\!2\mu }\} [v]^2~d {s} \Big )^{\frac{1}{2}}. \nonumber \\ \end{aligned}$$
(46)

Notice that

$$\begin{aligned}&h\int _E \{{ {\sigma }( {u}) \, {n}_E}\}^2 \{{\lambda +2\mu }\}^{-1}~d {s} \\&\quad \le h\Big ( \int _E ( {\sigma }( {u}^{+}) \, {n}_E)^2 (\lambda ^{+} + 2\mu ^{+})^{-1}~d {s} + \int _E ( {\sigma }( {u}^{-}) \, {n}_E)^2 (\lambda ^{-} + 2\mu ^{-})^{-1}~d {s} \Big ), \end{aligned}$$

where \( {u}^{\pm } = {u}|_{K^{\pm }}\), \(\lambda ^{\pm } = \lambda |_{K^{\pm }}\) and \(\mu ^{\pm } = \mu |_{K^{\pm }}\). So, we have

$$\begin{aligned} h\int _E \{{ {\sigma }( {u}) \, {n}_E}\}^2 \{{\lambda +2\mu }\}^{-1}~d {s} \le C_{\text {inv}}^2 \Big ( a_H^{K^{+}}(u^{+},u^{+}) + a_H^{K^{-}}(u^{-},u^{-}) \Big ). \end{aligned}$$

Thus (46) becomes

$$\begin{aligned} \int _E \{{ {\sigma }( {u}) \, {n}_E}\} \cdot [v]~d {s} \!&\le \! C_{\text {inv}} \Big ( a_H^{K^{+}}(u^{+},u^{+}) \!+\! a_H^{K^{-}}(u^{-},u^{-}) \Big )^{\frac{1}{2}} \nonumber \\&\quad \times \Big ( \frac{1}{h} \int _E \{{\lambda +2\mu }\} [v]^2~d {s} \Big )^{\frac{1}{2}}. \end{aligned}$$
(47)

When \(E\) is a boundary edge, we have

$$\begin{aligned} \int _E \{{ {\sigma }( {u}) \, {n}_E}\} \cdot [v]~d {s} \le C_{\text {inv}} a_H^{K}(u,u)^{\frac{1}{2}} \Big ( \frac{1}{h} \int _E \{{\lambda +2\mu }\} [v]^2~d {s} \Big )^{\frac{1}{2}}, \end{aligned}$$
(48)

where \(K\) denotes the coarse grid block having the edge \(E\). Summing (47) and (48) for all edges \(E\in {\mathcal E}^H\), we have

$$\begin{aligned} \sum _{E\in {\mathcal E}^H} \int _E \{{ {\sigma }( {u}) \, {n}_E}\} \cdot [v]~d {s} \le \sqrt{2} C_{\text {inv}} a_H(u,u)^{\frac{1}{2}} \left( \sum _{E\in {\mathcal E}^H} \frac{1}{h} \int _E \{{\lambda +2\mu }\} [v]^2~d {s} \right) ^{\frac{1}{2}}. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \sum _{E\in {\mathcal E}^H} \int _E \{{ {\sigma }( {v}) \, {n}_E}\} \cdot [u]~d {s} \le \sqrt{2} C_{\text {inv}} a_H(v,v)^{\frac{1}{2}} \left( \sum _{E\in {\mathcal E}^H} \frac{1}{h} \int _E \{{\lambda +2\mu }\} [u]^2~d{s} \right) ^{\frac{1}{2}}. \end{aligned}$$

Hence

$$\begin{aligned} \sum _{E\in {\mathcal E}^H} \int _E \left( \{{ {\sigma }( {u}) \, {n}_E}\} \cdot [v] + \{{ {\sigma }( {v}) \, {n}_E}\} \cdot [u] \right) ~d {s} \le \sqrt{2} C_{\text {inv}} \gamma ^{-\frac{1}{2}} \Vert u\Vert _{\text {DG}} \, \Vert v\Vert _{\text {DG}}.\qquad \quad \end{aligned}$$
(49)

This proves the continuity.

For coercivity, we have

$$\begin{aligned} a_{\text {DG}}( {u}, {u}) = \Vert u\Vert _{\text {DG}}^2 - \sum _{E\in {\mathcal E}^H} \int _E \Big ( \{{ {\sigma }( {u}) \, {n}_E}\} \cdot [u] + \{{ {\sigma }( {u}) \, {n}_E}\} \cdot [{ {u}}] \Big )~d {s}. \end{aligned}$$

By (49), we have

$$\begin{aligned} a_{\text {DG}}( {u}, {u}) \ge (1 - \sqrt{2} C_{\text {inv}} \gamma ^{-\frac{1}{2}} ) \Vert u\Vert _{\text {DG}}^2, \end{aligned}$$

which gives the desired result. \(\square \)

We will now prove the convergence of the method (8). Let \(u_h \in V^h_{\text {DG}}\) be the fine grid solution which satisfies

$$\begin{aligned} a_{\text {DG}}(u_h, v) = (f,v), \quad \forall v\in V^h_{\text {DG}}. \end{aligned}$$
(50)

It is well-known that \(u_h\) converges to the exact solution \(u\) in the DG-norm as the fine mesh size \(h\rightarrow 0\). Next, we define a projection \(u_S\in V^{\text {snap}}\) of \(u_h\) in the snapshot space by the following construction. For each coarse grid block \(K\), the restriction of \(u_S\) on \(K\) is defined as the harmonic extension of \(u_h\), that is,

$$\begin{aligned} \begin{aligned} - \nabla \cdot {\sigma }( {u}_{S})&= {0}, \quad \text {in }K,\\ {u}_{S}&=u_h, \quad \text {on }\partial K. \end{aligned} \end{aligned}$$
(51)

Now, we prove the following estimate for the projection \(u_S\).

Lemma 3

Let \(u_h\in V^h_{\text {DG}}\) be the fine grid solution defined in (50) and \(u_S\in V^{\text {snap}}\) be the projection of \(u_h\) defined in (51). Then we have

$$\begin{aligned} \Vert u_h - u_S \Vert _{\text {DG}} \le CH \Big ( \max _{K\in {\mathcal T}^H} \eta _K \Big ) \Vert f \Vert _{L^2(\Omega )}, \end{aligned}$$

where \(\eta _K = \min _K \{\lambda +2\mu \}.\)

Proof

Let \(K\) be a given coarse grid block. Since \(u_S=u_h\) on \(\partial K\), the jump terms in the DG-norm vanish. Thus, the DG-norm can be written as

$$\begin{aligned} \Vert u_h - u_S \Vert _{\text {DG}}^2 = \sum _{K\in {\mathcal T}^H} a_H^K(u_h-u_S,u_h-u_S). \end{aligned}$$

Since \(u_S\) satisfies (51) and \(u_h-u_S=0\) on \(\partial K\), we have

$$\begin{aligned} a_H^K(u_S, u_h-u_S) = 0. \end{aligned}$$

So,

$$\begin{aligned} \Vert u_h - u_S \Vert _{\text {DG}}^2 = \sum _{K\in {\mathcal T}^H} a_H^K(u_h,u_h-u_S) = a_{\text {DG}}(u_h, u_h-u_S) = (f, u_h-u_S). \end{aligned}$$

By the Poincaré inequality, we have

$$\begin{aligned} \Vert u_h - u_S \Vert _{L^2(K)} \le C H^2 \eta ^2_K a_H^K(u_h-u_S,u_h-u_S), \end{aligned}$$

where \(\eta _K = \min _K \{ \lambda +2\mu \}\). Hence, we have

$$\begin{aligned} \Vert u_h - u_S \Vert _{\text {DG}} \le CH \Big ( \max _{K\in {\mathcal T}^H} \eta _K \Big ) \Vert f \Vert _{L^2(\Omega )}. \end{aligned}$$

\(\square \)

In the following theorem, we will state and prove the convergence of the GMsFEM (8).

Theorem 3

Let \(u_h\in V^h_{\text {DG}}\) be the fine grid solution defined in (50) and \(u_H\) be the GMsFEM solution defined in (8). Then we have

$$\begin{aligned} \Vert u_h - u_H \Vert _{\text {DG}}^2&\le C\left( \sum _{i=1}^{N_E} \frac{H}{\left\langle \lambda +2\mu \right\rangle \xi _{L_i+1}} \Big ( 1 + \frac{\gamma H}{h \xi _{L_i+1}} \Big ) \int _{\partial K_i} ( \sigma (u_S)\cdot n_{\partial K})^2~d {s}\right. \\&\left. + H^2 \Big ( \max _{K\in {\mathcal T}^H} \eta _K^2 \Big ) \Vert f \Vert ^2_{L^2(\Omega )} \right) , \end{aligned}$$

where \(u_S\) is defined in (51).

Proof

First, we will define a projection \(\widehat{u}_S\in V^{\text {off}}\) of \(u_S\) in the offline space. Notice that, on each \(K_i\), \(u_S\) can be represented by

$$\begin{aligned} u_S = \sum _{l=1}^{M_i} c_l \psi _l^{i,\text {off}}, \end{aligned}$$

where \(M_i = M^{i,\text {snap}}\) and we assume that the functions \(\psi _l^{i,\text {off}}\) are normalized so that

$$\begin{aligned} \int _{\partial K_i} \left\langle \lambda +2\mu \right\rangle (\psi _l^{i,\text {off}})^2~d {s}=1. \end{aligned}$$

Then the function \(\widehat{u}_S\) is defined by

$$\begin{aligned} \widehat{u}_S = \sum _{l=1}^{L_i} c_l \psi _l^{i,\text {off}}. \end{aligned}$$

We will find an estimate of \(\Vert u_S-\widehat{u}_S\Vert _{\text {DG}}\). Let \(K\) be a given coarse grid block. Recall that the spectral problem is

$$\begin{aligned} \int _{K} 2\mu {\epsilon }({u}): {\epsilon }({v})~d {x} + \int _K \lambda \nabla \cdot {u} \nabla \cdot {v}~d {x} = \frac{\xi }{H} \int _{\partial K} \left\langle \lambda +2\mu \right\rangle {u} {v}~ds. \end{aligned}$$

By the definition of the flux (41), the spectral problem can be represented as

$$\begin{aligned} \int _{\partial K} (\sigma (u)\cdot n_{\partial K}) v~ds= \frac{\xi }{H} \int _{\partial K}\left\langle \lambda +2\mu \right\rangle {u} {v}~ds. \end{aligned}$$

By the definition of the DG-norm, the error \(\Vert u_S - \widehat{u}_S\Vert _{\text {DG}}\) can be computed as

$$\begin{aligned}&\Vert \widehat{{u}}_S - {u}_S \Vert _{\text {DG}}^2 \le \sum _{K} \left( \int _K 2\mu { \epsilon }(\widehat{{u}}_S - {u}_S )^2~d {s} + \int _K \lambda (\nabla \cdot (\widehat{{u}}_S - {u}_S) )^2~d {s}\right. \\&\quad \left. + \frac{\gamma }{h} \int _{\partial K} \{{\lambda +2\mu }\} (\widehat{{u}}_S - {u}_S))^2~d {s}\right) . \end{aligned}$$

Note that

$$\begin{aligned}&\int _{K_i} 2\mu { \epsilon }(\widehat{{u}}_S - {u}_S )^2~d {s} \!+\! \int _{K_i} \lambda (\nabla \cdot (\widehat{{u}}_S - {u}_S) )^2~d {s} \le \frac{1}{h}\int _{\partial {K_i}}\left\langle \lambda +2\mu \right\rangle (\widehat{{u}}_S \!-\! {u}_S)^2~d {s} \\&\quad = \sum _{l={L_i}+1}^{M_i} \frac{\xi _l}{H} c_l^2 \le \frac{H}{\xi _{L_i+1}} \sum _{l=L_i+1}^{M_i} \left( \frac{\xi _l}{H}\right) ^2 c_l^2. \end{aligned}$$

Also,

$$\begin{aligned} \frac{1}{h} \int _{\partial K_i} \{{\lambda +2\mu }\} (\widehat{{u}}_S - {u}_S)^2~d {s} = \frac{1}{h} \sum _{l=L_i+1}^{M_i} c_l^2 \le \frac{H^2}{h \xi _{L_i+1}^2} \sum _{l=L_i+1}^{M_i} \left( \frac{\xi _l}{H}\right) ^2 c_l^2. \end{aligned}$$

Moreover,

$$\begin{aligned} \sum _{l=L_i+1}^{M_i} \left( \frac{\xi _l}{H}\right) ^2 c_l^2 \le \sum _{l=1}^{M_i} \left( \frac{\xi _l}{H}\right) ^2 c_l^2 \le \frac{1}{\left\langle \lambda +2\mu \right\rangle }\int _{\partial K_i} ( \sigma (u_S)\cdot n_{\partial K})^2~ds. \end{aligned}$$

Consequently, we obtain the following bound

$$\begin{aligned} \Vert u_S - \widehat{{u}}_S \Vert ^2_{\text {DG}} \le \sum _{i=1}^{N_E} \frac{H}{\left\langle \lambda +2\mu \right\rangle \xi _{L_i+1}} \left( 1 + \frac{\gamma H}{h \xi _{L_i+1}}\right) \int _{\partial K_i} ( \sigma (u_S)\cdot n_{\partial K})^2~ds. \end{aligned}$$

Next, we will prove the required error bound. By coercivity,

$$\begin{aligned} a_0 \Vert \widehat{u}_S - u_H \Vert _{\text {DG}}^2&= a_{\text {DG}}(\widehat{u}_S-u_H, \widehat{u}_S-u_H) \\&= a_{\text {DG}}(\widehat{u}_S-u_H, \widehat{u}_S-u_S) \\&+ a_{\text {DG}}(\widehat{u}_S-u_H, u_S-u_h) + a_{\text {DG}}(\widehat{u}_S-u_H, u_h-u_H). \end{aligned}$$

Note that \(a_{\text {DG}}(\widehat{u}_S-u_H, u_h-u_H) = 0\) since \(\widehat{u}-u_H \in V^{\text {off}}\). Using the above results,

$$\begin{aligned}&\Vert \widehat{u}_S - u_H \Vert _{\text {DG}}^2 \le C\left( \sum _{i=1}^{N_E}\frac{H}{\left\langle \lambda +2\mu \right\rangle \xi _{L_i+1}} \left( 1 +\frac{\gamma H}{h \xi _{L_i+1}}\right) \int _{\partial K_i} (\sigma (u_S)\cdot n_{\partial K})^2~ds \right. \nonumber \\&\left. \quad + H^2 \left( \max _{K\in {\mathcal T}^H} \eta _K^2 \right) \Vert f \Vert ^2_{L^2(\Omega )} \right) . \end{aligned}$$
(52)

Finally, the desired bound is obtained by the triangle inequality

$$\begin{aligned} \Vert u_h - u_H\Vert _{\text {DG}} \le \Vert u_h - u_S\Vert _{\text {DG}} + \Vert u_S - \widehat{u}_S\Vert _{\text {DG}} + \Vert \widehat{u}_S - u_H\Vert _{\text {DG}}. \end{aligned}$$

\(\square \)

Remark 1

It is worthwhile to note that (42) can be replaced by

$$\begin{aligned} a_H^K(\widehat{v},\widehat{v}) \le \Lambda _K^{\mathrm{{snap}}}\left\langle \lambda +2\mu \right\rangle \int _{\partial K} |v|^2~d {s}, \end{aligned}$$

where \(\Lambda _K^{\mathrm{{snap}}} \) is the largest eigenvalue for the spectral problem (19). Therefore, (43) becomes

$$\begin{aligned} \int _{\partial K} | {\sigma }( {u}) \, {n}_{\partial K} |^2~d {s} \le \Lambda _K^{\mathrm{{snap}}} \left\langle \lambda +2\mu \right\rangle a_H^K(u,u). \end{aligned}$$

Repeating above steps, one can choose \(\gamma \) in (52) that satisfies

$$\begin{aligned} \gamma >\tilde{C}h\max \limits _{K\subset {\mathcal T}^H}\Lambda _{K}^{\mathrm{{snap}}}, \end{aligned}$$

where the constant \(\tilde{C}\) is defined as

$$\begin{aligned} \tilde{C}=\max \limits _{K\subset {\mathcal T}^H}\frac{{\mathrm {max}}_{E\subset \partial K}\{\lambda +2\mu \}}{\mathrm {min}_{E\subset \partial K}\{\lambda +2\mu \}}. \end{aligned}$$

If we assume every coarse element includes a high contrast region, then \(\tilde{C}\) is \(O(1)\).

Table 9 shows \(\Lambda _K^{\mathrm{{snap}}}\) with and without oversampling for different \(h\). We can see that \(\Lambda _{K^+}^{\mathrm{{snap}}}\) is much smaller than \(\Lambda _{K}^{\mathrm{{snap}}}\). Besides, the numerical experiments show \(\Lambda _{K^+}^{\mathrm{{snap}}}\) is a very weak function of \(h\), while \(\Lambda _{K}^{\mathrm{{snap}}}\) is proportional to \(h^{-1}\).

We can get similar error analysis for the case of oversampling by just following steps shown in the above no oversampling case. But we can have better estimate in the oversampling case. If we let \(\gamma =\alpha \tilde{C}{\mathrm {max}}_{K\subset {\mathcal T}^H}\Lambda _{K^+}^{\mathrm{{snap}}}\), then the term \(C_1=1+\frac{\gamma H}{h \xi _{L_i+1}}\) in (52) becomes \( 1+\frac{ \alpha \tilde{C}{\mathrm {max}}_{K\subset {\mathcal T}^H}\Lambda _{K^+}^{\mathrm{{snap}}}H}{ \xi _{L_i+1}}\). We have numerically shown that \(\Lambda _{K^+}^{\mathrm{{snap}}}\) is almost independent of \(h\), which means \(C_1\) can be controlled. Therefore, the dominated error comes from \(\frac{H}{\left\langle \lambda +2\mu \right\rangle \xi _{L_i+1}}\). We emphasize that this remark is based on our numerical observations while the analytical studies are complicated and it will be the subject of our future research.

Table 9 Largest eigenvalue for no oversampling and oversampling
Full size table

7 Conclusions

In this paper, we design a multiscale model reduction method using GMsFEM for elasticity equations in heterogeneous media. We design a snapshot space and an offline space based on the analysis. We present two approaches that couple multiscale basis functions of the offline space. These are continuous Galerkin and discontinuous Galerkin methods. Both approaches are analyzed. We present oversampling studies where larger domains are used for calculating the snapshot space. Numerical results are presented.