# On heterogeneous coupling of multiscale methods for problems with and without scale separation

- Assyr Abdulle
^{1}Email authorView ORCID ID profile and - Orane Jecker
^{1}

**4**:28

https://doi.org/10.1186/s40687-017-0118-9

© The Author(s) 2017

**Received: **24 October 2016

**Accepted: **28 July 2017

**Published: **4 December 2017

## Abstract

In this paper, we discuss partial differential equations with multiple scales for which scale resolution is needed in some subregions, while a separation of scale and numerical homogenization is possible in the remaining part of the computational domain. Departing from the classical coupling approach that often relies on artificial boundary conditions computed from some coarse grain simulation, we propose a coupling procedure in which virtual boundary conditions are obtained from the minimization of a coarse grain and a fine-scale model in overlapping regions where both models are valid. We discuss this method with a focus on interface control and a numerical strategy based on non-matching meshes in the overlap. A fully discrete a priori error analysis of the heterogeneous coupled multiscale method is derived, and numerical experiments that illustrate the efficiency and flexibility of the proposed strategy are presented.

## Keywords

## Mathematics Subject Classification

## 1 Introduction

In this contribution, we address an intermediate situation between separated and non-separated scales in the following sense: we assume that in a subset \(\omega _2\) of the computational domain \(\varOmega \) the macro-/micro-upscaling strategy can be applied but that in an other part \(\omega \) of the domain one needs full resolution of the scales. Here we assume that this second domain is sufficiently small so that standard resolved finite element method (FEM) can be used. In the region \(\omega _2\), we chose to use the finite element heterogeneous multiscale method (FE-HMM) [2]. While our method easily generalizes to multiple regions with and without scale separation, we assume here for simplicity that \(\varOmega =\omega _2\cup \omega \). The main issue for such a coupling strategy is to set adequate boundary conditions at the interfaces of both computational domains. We note that such problems have numerous applications in the sciences; we mention, for example, heterogeneous structures with defects [10, 16] or steady flow problems with singularities [17]. Coupling strategies between fine-scale and upscaled models have already been studied in the literature, for example, in [27] where a precomputed global homogenized solution is used to provide the boundary conditions in the fine-scale subregions. More recently, a coupling strategy based on an \(L^2\) projection of the homogenized solution onto harmonic fine-scale functions has been discussed [8].

We briefly describe the main contribution of this paper. First, in [7], the theory and the numerics have been developed for the cost function \(\left\| \cdot \right\| _{\mathrm {L}^{2}(\omega _0)}\), called distributed observation in the classical terminology of optimal control. Here we consider the cost function \(\left\| \cdot \right\| _{\mathrm {L}^{2}(\varGamma _1 \cup \varGamma _2)}\), called boundary or interface control. Such controls can reduce the cost of the iterative method to solve the optimality system compared to the cost of solving the optimization problem with distributed observation [14]. Second, in [7] we used the same mesh in the overlap \(\omega _0\) with the consequence of having to use a mesh size that scales with the fine mesh of \(\omega \). Here we discuss the use of independent meshes in the overlap through appropriate interpolation techniques. As a result, we have again a significant reduction in the computational cost of the coupling as the macroscopic numerical method in \(\omega _2=\varOmega \setminus \omega \) does not need an increasing number of micro-solvers as the mesh in \(\omega _1\) is refined. Finally, numerical examples were carried out in [7] only for the situation where \(\omega \Subset \varOmega \) (Fig. 1, left); here we discuss also the scenario for which \(\partial \omega \cap \partial \varOmega \ne \emptyset \) (Fig. 1, right).

The paper is organized as follows. In Sect. 2, we describe the model problem, introduce the two minimization costs functions considered in this paper, and give an a priori error analysis between the coupled and the fine-scale solutions. In Sect. 3, we define the multiscale numerical discretization of the optimization problem and perform a fully discrete a priori error analysis. Finally, Sect. 4 contains several numerical experiments that illustrate the theoretical results and the performance of the new coupling strategy.

**Notations** In what follows, \(C>0\) is used to denote a generic constant independent of \(\varepsilon \). We consider the usual Sobolev space \(H^1(\varOmega )=\{u\in L^2(\varOmega )\mid D^ru\in L^2(\varOmega ), |r|\le 1 \}\),where \(r \in \mathbb N^d, |r|=r_1 + \cdots + r_d\) and \(D^r= \partial _1^{r_1}\ldots \partial ^{r_d}_d\). The notation \(|\cdot |\) stands for the standard Euclidean norm in \(\mathbb R^d\). Let *Y* denote the unit cube \((0,1)^d\) and define \(W^1_{\mathrm{per}}(Y):=\{v\in H^1_{\mathrm{per}}(Y)\mid \int _{Y}v \text {d}y=0\}\) where the set \(H^1_{\mathrm{per}}(Y)\) is the closure of \(\mathcal {C}^{\infty }_{\mathrm{per}}(Y)\) for the \(H^1\) norm.

## 2 Problem formulation

*H*-converges toward an homogenized tensor \(a^0_2\) [25]. Further, we set \(a_1=a^\varepsilon \mathbb {1}_{\omega _1}\), \(u_1=u^\varepsilon _1\), and \(u_2=u_2^0\). The heterogeneous control restricted to Dirichlet boundary controls is given by the following problem: find \(u_1^{\varepsilon }\in H^1(\omega _1)\) and \(u_2^0\in H^1(\omega _2)\), such that \(\frac{1}{2}\left\| u_1^{\varepsilon }-u_2^0\right\| _\mathcal {H}^2\) is minimized under the following constraints, for \(i=1,2,\)

*Case*1. Minimization in \(L^2(\omega _0)\), with

*Case*2. Minimization in \(L^2(\varGamma _1\cup \varGamma _2)\), with

For the homogenization theory (*H*-convergence), we consider a family of problems (1) indexed by \(\varepsilon \). In what follows, we will often assume \(\varepsilon \le \varepsilon _0\), where \(\varepsilon _0\) is a parameter used in a strong Cauchy–Schwarz inequality (see Lemma 6.3). We assume that \(\theta _i\in \mathcal {U}_i^D\) and hence \(u_i(\theta _i)\) is in \(H^1(\omega _i)\), for \(i=1, 2\).

### 2.1 Minimization over \(\varGamma _1\cup \varGamma _2\)

### Lemma 2.1

The bilinear form \(\pi \) is a scalar product over \(\mathcal {U}\).

### Proof

The symmetry and positivity are clear, and it remains to prove that the form is positive definite; \(\pi (\theta _1, \theta _2)=0\) if and only if \(\theta _1=0\) and \(\theta _2=0\). We use the short-hand notation \(\pi (\theta _1, \theta _2)\) to denote \(\pi ((\theta _1, \theta _2),(\theta _1, \theta _2))\).

Assuming that \(\theta _1\) and \(\theta _2\) are zero, the state variables \(v_1^{\varepsilon }\) and \(v_2^0\) are solutions of boundary value problems with zero data; thus, \(v_1^{\varepsilon }\) and \(v_2^0\) are zero over \(\omega _1\) and \(\omega _2\), respectively. This leads to \(\pi (\theta _1, \theta _2)=0\).

*H*-convergence on the tensor \(a^\varepsilon _1\), to obtain an homogenized tensor \(a_1^0\) in \(\omega _1\). It holds that \(v_1^{\varepsilon }\) converges weakly in \(H^1\) toward \(v_1^0\) the homogenized solution of

*H*-convergence, the difference \(v_1^0-v_2^0\) satisfies

### 2.2 A priori error analysis

*Y*-periodic in

*y*, where \(Y=(0,1)^d\), explicit equations are available to compute the homogenized tensor \(a^0_2\)

*I*is the \(d\times d\) identity matrix. The functions \(\chi ^j\in W_{per}^1(Y)\) are called the first-order correctors and, for \(j=1, \ldots ,d\), \(\chi ^j\) is solution of the cell problem

*Estimates for the fine solution*Let us define an operator \(P: \mathcal {U}\rightarrow H^1(\omega _1) \times H^1(\varOmega \setminus \omega _1)\) such that

*Q*is bounded in the operator norm, i.e.,

*Q*is bounded for the norm in \(\mathcal {U}\) induced by the scalar product (6) for the cost function of case 2.

### Theorem 2.2

*C*depends on the constant of the Caccioppoli inequality, the bound \(\left\| Q\right\| \), and the trace constants associated with the trace operators \(\gamma _1\) and \(\gamma _2\) on \(\varGamma _1\) and \(\varGamma _2\), respectively.

### Lemma 2.3

### Proof

*J*, the Euler–Lagrange formulation (5) holds and

The next lemma gives an upper bound to the norm in Lemma 2.3.

### Lemma 2.4

*C*is independent of \(\varepsilon \).

### Proof

\(\square \)

The proof of Theorem 2.2 follows from (11) and Lemmas 2.3 and 2.4.

*Estimates for the coarse solution* The a priori error estimates to the coarse-scale solver follow from [7, Theorem 3.6] using Lemma 2.4. We skip the details.

### Theorem 2.5

*C*is independent of \(\varepsilon \), but depends on \(\tau ,\) \(\tau ^+\), and the ellipticity constants of \(a^\varepsilon _2\).

## 3 Fully discrete coupling method

In this section, we describe the fully discrete overlapping coupling method and perform an a priori error analysis. The fine-scale solver requires a triangulation of size \(\tilde{h}\) sufficiently small to resolve the multiscale nature of the tensor. In contrast, the coarse-scale solver on \(\omega _2\) takes full advantage of the scale separation and allows for a mesh size larger than the fine scale. We use the FEM in \(\omega _1\) and the FE-HMM in \(\omega _2\). As the finite elements of the fine and coarse meshes in \(\omega _0\) are different, an interpolation between the two meshes should be considered. One can also chose to use the same finite elements in the overlap, leading to a discontinuity at \(\varGamma _1\) in the mesh over \(\omega _2\). In that latter situation, the discontinuous Galerkin FE-HMM [4] should be used instead of the FE-HMM.

In what follows, we consider for simplicity the problem (1) with homogeneous Dirichlet boundary conditions, i.e., we set \(g_D=0\) and \(\varGamma _N=\emptyset \). Further, we assume that \(\varepsilon \) is small enough so that we can use the strong Cauchy–Schwarz lemma (Lemma 6.3) and its discrete version (Lemma 6.5) hold.

*Numerical method for the fine-scale problem*

*K*. In addition, we suppose that the family of partitions \(\{\mathcal {T}_{\tilde{h}}\}\) is admissible and shape regular [11], i.e.,

- (T1)
**admissible**: \(\overline{\omega }_1=\cup _{K\in \mathcal {T}_h} K\) and the intersection of two elements is either empty, a vertex, or a common face; - (T2)
**shape regular**: there exists \(\sigma >0\) such that \(h_K/ \rho _K\le \sigma \), for all \(K\in \mathcal {T}_{\tilde{h}}\) and for all \(\mathcal {T}_{\tilde{h}}\in \{\mathcal {T}_{\tilde{h}}\}\), where \(\rho _K\) is the diameter of the largest circle contained in the element*K*.

*p*on

*K*if

*K*is a triangle, and the space \(\mathcal {Q}^p\) of polynomials of degree at most

*p*in each variable if

*K*is a rectangle. Further, \(V_{0}^p(\omega _1, \mathcal {T}_{\tilde{h}})\) denotes the space of functions in \(V^p_D(\omega _1, \mathcal {T}_{\tilde{h}})\) that vanish on \(\partial \omega _1\).

*Numerical method for the coarse-scale problem*

*Quadrature formula* A macroscopic quadrature formula is given by the pair \(\{x_{j,K}, \omega _{j,K}\}\) of quadrature nodes \(x_{j,K}\) and weights \(\omega _{j,K}\), for \(j=1, \ldots , J\). The sampling domain of size \(\delta \) around each quadrature point is denoted by \(K_{\delta _j}= x_{j,K} + \delta [-1/2, 1/2]^d\). We assume that the quadrature formula verifies the necessary assumptions to guarantee that the standard error estimates for a FEM hold [11].

*K*with mesh size \(h=\max _{K\in \mathcal {T}_h} h_K\) satisfying \(h<\varepsilon \). The micro-FE space is

*Numerical algorithm*

In this section, we state the discrete coupling and give the main convergence results. The well posedness and the proofs of the errors estimates are done in details in [7]. In what follows, we use \(\mathcal {O}\) to denote either \(\omega _0\) or \(\varGamma _1\cup \varGamma _2\).

*U*is given by \(U=(v_{1,\tilde{h}}, v_{2, H}, \lambda _{1,\tilde{h}}, \lambda _{2,H})^{\top }\), and

*Fully discrete error estimates*The coupling solution, denoted by \(\bar{u}_{{\tilde{h}H}}\), is defined as

*Y*-periodic in

*y*, and we restrict the FE spaces to piecewise FE spaces. Periodic coupling is then used with sampling domains \(K_{\varepsilon }\) of size \(\varepsilon \). The reconstructed solution \(u_{2,H}^{rec}(\theta _{2,H})\) is given by

### Theorem 3.1

*H*, \(\tilde{h}\), and

*h*, and where the HMM error \(e_{HMM,L^2}\) is given by \(e_{HMM,L^2}=\left\| u^0-u^H\right\| _{\mathrm {L}^{2}(\omega _2)}\).

### Proof

It follows the lines of [7, Theorem 4.3], using a continuous macro-FEM (FE-HMM) instead of a discontinuous Galerkin FEM (DG-FE-HMM). \(\square \)

*C*is independent of \(\varepsilon \), \(\tilde{h}\),

*H*, and

*h*.

*C*is independent of \(\varepsilon \), \(\tilde{h}\),

*H*, and

*h*(we recall that for the reconstruction we use periodic boundary conditions in the micro-problems (12) over sampling domains are of size \(\delta =\varepsilon \)). If we collocate (i.e., freeze) the slow variable

*x*to the quadrature point \(x_{K}\) in the tensor \(a^\varepsilon _2\), i.e., we consider \(a^\varepsilon _2(x_{K}, x/\varepsilon )\) in the macro- and micro-bilinear forms, we obtain an optimal modeling error

*C*is independent of \(\varepsilon \), \(\tilde{h}\),

*H*, and

*h*.

### Remark 3.2

*H*, and

*h*.

### Remark 3.3

Higher-order FE macro- and micro-spaces can also be considered, and we refer to [2, 3] for details.

Next, we state an error estimates in the coarse-scale region for the optimization-based numerical solution with correctors.

### Theorem 3.4

*Y*-periodic in

*y*and satisfies \(a_2(x,y)\in \mathcal {C}(\overline{\omega }_2; L^{\infty }_{per}(Y))\). Let \(\psi ^j_{K_{\varepsilon }}(x)\in W_{\mathrm{per}}^1(K_{\varepsilon })\), \(j=1, \ldots , d\). If in addition, \(u^\varepsilon \in H^2( \varOmega )\), \(u_2^0(\theta _2) \in H^2(\omega _2)\), \(u_1^{\varepsilon } \in H^{s+1}(\omega _1)\), with \(s\le 1\), and \(\psi ^j_{K_{\varepsilon }}(x)\in W^{1, \infty }(K_{\varepsilon })\), \(j=1, \ldots , d\). It holds,

### Proof

It follows the lines of [7, Theorem 4.4], where DG-FE-HMM is replaced by FE-HMM. \(\square \)

## 4 Numerical experiments

In this section, we give three numerical experiments that can be seen as a complement of the ones carried in [7], where we focused on a minimization in \(L^2(\omega _0)\), with interior subdomains and matching grids in the overlap \(\omega _0\). In the first experiment, we still consider the minimization over \(L^2(\omega _0)\) and compare matching and non-matching meshes. The second experiment illustrates the coupling with the cost function of case 2 over \(\varGamma _1 \cup \varGamma _2\) and comparisons with the cost function of case 1 over \(\omega _0\). In the last example, we combine non-matching grids and a minimization over the boundary. We observe several order of magnitude of saving in computational cost when compared to the method proposed in [7]. In the experiments, we will use a tensor \(a^\varepsilon \) which is highly heterogeneous non-periodic and oscillate at several non-separated scales in \(\omega \), and which has scale separation in \(\omega _2\), with a locally periodic structure.

*Comparison of matching and non-matching grids on the overlap*

### Experiment 1

*H*and \(\varepsilon \), and when \(\tilde{h}\) is refined, we expect a saturation, depending on

*H*and \(\varepsilon \), in the convergence. Let \(\varepsilon =1/20\) and initialize the fine mesh to \(\tilde{h}=1/64\). We set \(H=1/8, 1/16\), and 1 / 32 and refine \(\tilde{h}\) in each iteration. In Fig. 4, we plot the \(H^1\) norm between the reference and numerical solutions w.r.t the mesh size in \(\omega \). We see indeed that the error saturates at a threshold value that depends on

*H*.

*Minimization with interface control*

For this experiment, we compare the coupling done with the cost function of case 1 and of case 2 on an elliptic problem with \(\omega \subseteq \varOmega \), i.e., when the boundaries of \(\omega \) and \(\varOmega \) intersect (see Fig. 1, right picture).

### Experiment 2

*Y*-periodic in the fast variable, and the homogeneous tensor \(a^0_2\) can be explicitly derived as

*H*.

*Minimization with interface control on non-matching grids*

For the last experiment, we combine the two previous effects. The fastest coupling is obtained by performing by considering the minimization with of the cost of case 2 with interpolation of the two meshes in the overlap, whereas the slowest coupling is obtained by the minimization with the cost function of case 1 using identical meshes in the overlap.

### Experiment 3

## 5 Conclusion

- (i)
to use a cost function over the boundary of the overlapping region;

- (ii)
to consider independent meshes in the overlap and an interpolation procedure between the fine and coarse meshes in the overlapping region.

## 6 Appendix

*a*, the set of

*a*-harmonic functions is denoted by \(\mathcal {H}(\omega _1)\) and consists of functions \(u\in L^2(\omega _1)\cap H^1_{\text {loc}}(\omega _1)\) such that

### Theorem 6.1

We note that elliptic problems with a non-null right-hand side and problems where \(\partial \omega \cap \varGamma \ne \emptyset \), can also be considered and we refer to [19] for details. We give next a bound of the \(L^2\) norm over \(\omega \) by the \(L^2\) norm over the overlap \(\omega _0\).

### Lemma 6.2

*C*is a constant depending on \(\alpha , \beta ,\) and the Poincaré constant is associated with \(\omega _1\) and \(\omega _2\), respectively.

### Proof

see [7, Lemma 2.1]. \(\square \)

### Lemma 6.3

### Lemma 6.4

*C*is independent of

*h*.

We now give the discrete strong Cauchy–Schwarz inequality, and to simplify the notations, we omit the \(\varepsilon \) dependency in \(v_1\).

### Lemma 6.5

## Declarations

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## Authors’ Affiliations

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