
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
[1] | Eric Chung, Yalchin Efendiev, Ke Shi, Shuai Ye . A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media. Networks and Heterogeneous Media, 2017, 12(4): 619-642. doi: 10.3934/nhm.2017025 |
[2] | Mario Ohlberger, Ben Schweizer, Maik Urban, Barbara Verfürth . Mathematical analysis of transmission properties of electromagnetic meta-materials. Networks and Heterogeneous Media, 2020, 15(1): 29-56. doi: 10.3934/nhm.2020002 |
[3] | Patrick Henning . Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems. Networks and Heterogeneous Media, 2012, 7(3): 503-524. doi: 10.3934/nhm.2012.7.503 |
[4] | Antoine Gloria Cermics . A direct approach to numerical homogenization in finite elasticity. Networks and Heterogeneous Media, 2006, 1(1): 109-141. doi: 10.3934/nhm.2006.1.109 |
[5] | Patrick Henning, Mario Ohlberger . The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks and Heterogeneous Media, 2010, 5(4): 711-744. doi: 10.3934/nhm.2010.5.711 |
[6] | Fabio Camilli, Claudio Marchi . On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks and Heterogeneous Media, 2011, 6(1): 61-75. doi: 10.3934/nhm.2011.6.61 |
[7] | Thomas Abballe, Grégoire Allaire, Éli Laucoin, Philippe Montarnal . Application of a coupled FV/FE multiscale method to cement media. Networks and Heterogeneous Media, 2010, 5(3): 603-615. doi: 10.3934/nhm.2010.5.603 |
[8] | Nils Svanstedt . Multiscale stochastic homogenization of monotone operators. Networks and Heterogeneous Media, 2007, 2(1): 181-192. doi: 10.3934/nhm.2007.2.181 |
[9] | Frederike Kissling, Christian Rohde . The computation of nonclassical shock waves with a heterogeneous multiscale method. Networks and Heterogeneous Media, 2010, 5(3): 661-674. doi: 10.3934/nhm.2010.5.661 |
[10] | Liselott Flodén, Jens Persson . Homogenization of nonlinear dissipative hyperbolic problems exhibiting arbitrarily many spatial and temporal scales. Networks and Heterogeneous Media, 2016, 11(4): 627-653. doi: 10.3934/nhm.2016012 |
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
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
−div(a(x,∇ϕξ))=0 |
with the boundary conditions
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
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
−div(a(x,∇u))=f(x)inD,u=gon∂D, | (2) |
where
The corresponding weak formulation is: (
∫Da(x,∇u)⋅∇v=∫Dfv,∀v∈W1,p0(D). |
The well-posedness of (
‖u‖1,p(D)=(∫Dκ(x)|∇u|pdx)1/p. |
Next, we describe the finite element approximation of the solution. We let
The discrete fine-scale problem is defined in the following: (
∫Da(x,∇uh)⋅∇v=∫Dfv,∀v∈Vh0(D). |
Additionally, we introduce a coarse discretization
ωi=⋃{Kj∈TH; yi∈¯Kj}, | (3) |
where
First, we introduce the concept of
Definition 2.1. Let
−div(a(x,∇˜u))=0inK, |
where
Remark 1. The
∫Kκ(x)|∇˜u|pdx=minv∈W1,pu(K)∫Kκ(x)|∇v|pdx, |
where
Remark 2. In this context, all
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
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
−div(a(x,∇u))=finD, |
with
−div(a(x,∇Nξ))=0inK, |
with boundary condition
a∗(ξ)=1|K|∫Ka(y,∇Nξ)dy. |
The coarse-grid equation is given by
−div(a∗(x,∇u∗))=finD, |
with
FNH(→c)=∫Da∗(x,∇∑ckϕk)⋅∇ϕjdx=∑K∈D∫Ka∗(∑ck∇ϕk)⋅∇ϕjdx, |
At this step, we denote
FNH(→c)=∑K∈D∫Ka∗(ξ)⋅∇ϕjdx=∑K∈D∫K(1|K|∫Ka(x,∇Nξ)dx)⋅∇ϕjdx=∫D1|K|(∫Ka(x,∇Hp(∑ck∇ϕk⋅x))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
F(→c)=∫Da(x,∇Hp(∑i,kχicωikϕωik)⋅∇ϕωijdx=∫Dfϕωijdx, |
where
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
∫Da(x,∇Hp(∑iLi∑k=1χicωikϕωik)⋅∇ϕωijdx=∫Dfϕωijdxfor any j, | (4) |
where
Remark 3. The GMsFEM numerical solution
∫D(a(x,∇u1)−a(x,∇u2))⋅∇vdx=0, |
for any test function. Since
∫Dκ(x)|∇u1−∇u2|pdx≤∫D(a(x,∇u1)−a(x,∇u2))⋅∇(u1−u2)dx=0, |
which guarantees
To propose our method, we first need to construct a set of partition of unity functions
−div(a(x,∇χi)=0in K∈ωi,χi=χ0ion ∂K,for all K∈ωi. |
We remark that using the second choice of partition of unity functions can, in general, provide better numerical performance.
Let
The first choice is to use all possible fine-grid functions in
δhj(xk)={1 for k=j0 for k≠j,∀xk∈Mh(ωi). |
Then the
−div(κ(x)∇ψωij)=0in ωi,ψωij=δhjon ∂ωi. | (5) |
The dimension of
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
{ϕωi1=cωi,λωi1=0,ϕωik=argminv∈VωisnapGωi(v)Gωiχ(v−Pk−1(v)),λωik=Gωi(ϕωik)Gωiχ(ϕωik−Pk−1(ϕωik)), for k≥2, | (6) |
where
The eigenfunctions
Vc=span{χiϕωik:k=1,⋅⋅⋅,Li;i=1,⋅⋅⋅,Nv}⊆W1,p(D). |
Recall that our solution assumes the form of
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=argminv∈XωikGωi(v)Gωiχ(v),λωik=Gωi(ϕωik)Gωiχ(ϕωik),fork≥2, | (7) |
where
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
Lemma 4.1.
κ(x)|∇u−∇v|p⪯(a(x,∇u)−a(x,∇v))⋅∇(u−v). | (8) |
Proof. We use Lemma 5.1 in Glowinski and Marrocco [35], which proves the following inequality:
(|z|p−2z−|y|p−2y,z−y)R2≥α|z−y|p. |
If we take
Lemma 4.2.
|a(x,∇u)−a(x,∇v))|⪯M(x,u,v)|∇u−∇v|, | (9) |
where
Proof. According to Lemma 5.3 in Glowinski and Marrocco [35], there holds the following inequality:
||z|p−2z−|y|p−2y|≤β|z−y|(|z|+|y|)p−2. |
If we take
The next two lemmas deal with the properties of
Lemma 4.3. Suppose
∫Kκ(x)|∇(˜u−˜v)|pdx⪯(∫Kκ(x)|∇˜w|pdx)qp(∫Kκ(x)|∇˜u|pdx+∫Kκ(x)|∇˜v|pdx)p−2p−1, | (10) |
where
Proof. Using Lemma 4.1 and integrating by parts, we immediately obtain the following inequality:
∫Kκ(x)|∇(˜u−˜v)|pdx⪯∫K(a(x,∇˜u)−a(x,∇˜v))⋅∇(˜u−˜v)=∫∂K(a(x,∇˜u)−a(x,∇˜v))⋅→n(˜u−˜v)ds−∫Kdiv(a(x,∇˜u)−a(x,∇˜v))(˜u−˜v)dx=∫∂K(a(x,∇˜u)−a(x,∇˜v))⋅→n(˜u−˜v)ds−0=∫∂K(a(x,∇˜u)−a(x,∇˜v))⋅→n˜wds−∫Kdiv(a(x,∇˜u)−a(x,∇˜v))˜wdx=∫∂K(a(x,∇˜u)−a(x,∇˜v))⋅∇˜wdx⪯∫KM(x,˜u,˜v)|∇˜u−∇˜v||∇˜w|dx, | (11) |
where on the last line we have used the continuity property of
∫Kκ(x)|∇(˜u−˜v)|pdx⪯(∫Kκ(x)|∇˜u−∇˜v|pdx)1p(∫Kκ(x)|∇˜w|pdx)1p(∫K(κ(x)−2p|M(x,˜u,˜v)|)pp−2dx)p−2p. |
Dividing both sides by
∫Kκ(x)|∇(˜u−˜v)|pdx⪯(∫Kκ(x)|∇˜w|pdx)qp(∫K(κ(x)−2p|M(x,˜u,˜v)|)pp−2dx)p−2p−1. | (12) |
Recall from Lemma 4.2 that
(∫K(κ(x)−2p|M(x,˜u,˜v)|)pp−2dx)p−2p−1=(∫Kκ(x)(|∇˜u|+|∇˜v|)pdx)p−2p−1⪯(∫Kκ(x)|∇˜u|pdx+∫Kκ(x)|∇˜v|pdx)p−2p−1. | (13) |
We substitute (13) into (12), and we then see that (10) is proved.
Lemma 4.4. For any
∫Kκ(x)|∇Hp(u0+v0)|pdx≤∫Kκ(x)|∇(Hp(u0)+Hp(v0))|pdx. | (14) |
Proof. Recall Remark 1, we have
∫Kκ(x)|∇Hp(u)|pdx=minv∈W1,pu(K)∫Kκ(x)|∇v|pdx. |
Therefore,
∫Kκ(x)|∇Hp(u0+v0)|pdx=minw∈W1,pu0+v0(K)∫Kκ(x)|∇w|pdx=minw1+w2∈W1,pu0+v0(K)∫Kκ(x)|∇(w1+w2)|pdx≤minw1∈W1,pu0(K),w2∈W1,pv0(K)∫Kκ(x)|∇(w1+w2)|pdx. |
Taking
∫Kκ(x)|∇Hp(u0+v0)|pdx≤∫Kκ(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
Lemma 4.5. Suppose
‖u−ums‖1,p(D)⪯‖u−Hp(vH)‖qp1,p(D)‖u‖p−2p−11,p(D)for any vH∈Vc, |
where
Proof. Using Lemma 4.1, we immediately obtain the following inequality:
∫Dκ(x)|∇(u−ums)|pdx⪯∫D(a(x,∇u)−a(x,∇ums))⋅∇(u−ums)=∫D(a(x,∇u)−a(x,∇ums))⋅∇(u−Hp(vH))⪯∫DM(x,u,ums)|∇u−∇ums||∇u−∇Hp(vH)|dx, |
for any
∫Dκ(x)|∇(u−ums)|pdx(∫Dκ(x)|∇(u−ums)|pdx)1p(∫Dκ(x)|∇(u−Hp(vH))|pdx)1p(∫D(κ(x)−2p|M(x,u,ums)|)pp−2dx)p−2p. |
Dividing both sides by
∫Dκ(x)|∇(u−ums)|pdx⪯(∫Dκ(x)|∇(u−Hp(vH))|pdx)qp×(∫D(κ(x)−2p|M(x,u,ums)|)pp−2dx)p−2p−1=(∫Dκ(x)|∇(u−Hp(vH))|pdx)qp×(∫D(κ(x)−2p⋅κ(x)(|∇u|+|∇ums|)p−2)pp−2dx)p−2p−1=(∫Dκ(x)|∇(u−Hp(vH))|pdx)qp(∫Dκ(x)(|∇u|+|∇ums|)pdx)p−2p−1⪯(∫Dκ(x)|∇(u−Hp(vH))|pdx)qp(∫Dκ(x)(|∇u|p+|∇ums|p)dx)p−2p−1⪯(∫Dκ(x)|∇(u−Hp(vH))|pdx)qp(∫Dκ(x)|∇u|pdx+∫Dκ(x)|∇ums|pdx)p−2p−1⪯(∫Dκ(x)|∇(u−Hp(vH))|pdx)qp(∫Dκ(x)|∇u|pdx)p−2p−1. |
It follows immediately
‖u−ums‖1,p(D)⪯‖u−Hp(vH)‖qp1,p(D)‖u‖p−2p−11,p(D). |
Lemma 4.6. Suppose
∫Kκ(x)|∇(u−Hp(u))|pdx⪯Hq∫K|f|qdx, | (15) |
where
Proof. It's clear that
−div(a(x,∇u)−a(x,∇Hp(u)))=f. |
Thus,
∫K(a(x,∇u)−a(x,∇Hp(u)))⋅∇(u−Hp(u))dx=∫K(u−Hp(u))fdx. |
By Lemma 4.1,
∫Kκ(x)|∇(u−Hp(u))|pdx⪯∫K|u−Hp(u)||f|dx⪯(∫K|u−Hp(u)|pdx)1p(∫K|f|qdx)1q. |
Using Poincaré's inequality, we get
∫Kκ(x)|∇(u−Hp(u))|pdx⪯H(∫K|∇(u−Hp(u))|pdx)1p(∫K|f|qdx)1q=κ−1p0H(∫Kκ0|∇(u−Hp(u))|pdx)1p(∫K|f|qdx)1q⪯H(∫Kκ(x)|∇(u−Hp(u))|pdx)1p(∫K|f|qdx)1q. |
Dividing both sides by
∫Kκ(x)|∇(u−Hp(u))|pdx⪯Hq∫K|f|qdx. |
Remark 5. This local error estimate proved in Lemma 4.6 immediately deduces the global error estimate:
‖u−Hp(u)‖p1,p(D)⪯Hq‖f‖qLq(D). | (16) |
Now, we come to the main convergence theorem.
Theorem 4.7. Suppose
‖u−ums‖1,p(D)⪯‖u‖p−2p−11,p(D){H1(p−1)2‖f‖1(p−1)2Lq(D)+(1Λ∗)1p(p−1)2‖u‖1p−11,p(D)}, | (17) |
where
Proof. We first define the interpolation of
I0u=argminv∈Vc{‖u−Hp(v)‖1,p(D)}. |
Since
‖u−ums‖1,p(D)⪯‖u‖p−2p−11,p(D)‖u−Hp(I0u)‖qp1,p(D)⪯‖u‖p−2p−11,p(D)(‖u−Hp(u)‖1,p(D)+‖Hp(u)−Hp(I0u)‖1,p(D))qp⪯‖u‖p−2p−11,p(D)(‖u−Hp(u)‖qp1,p(D)+‖Hp(u)−Hp(I0u)‖qp1,p(D))⪯‖u‖p−2p−11,p(D)(Hq2p2‖f‖q2p2Lq(D)+‖Hp(u)−Hp(I0u)‖qp1,p(D))=‖u‖p−2p−11,p(D)(H1(p−1)2‖f‖1(p−1)2Lq(D)+‖Hp(u)−Hp(I0u)‖1p−11,p(D)). | (18) |
In the following, we will estimate
Using Lemma 4.3, we have
‖Hp(u)−Hp(I0u)‖p1,p(D)⪯(∫Dκ(x)|∇Hp(u−I0u)|pdx)qp× | (19) |
(∫Dκ(x)|∇Hp(u)|pdx+∫Dκ(x)|∇Hp(I0u)|pdx)p−2p−1⪯(∫Dκ(x)|∇Hp(∑ωiχi(u−uωi0))|pdx)qp(∫Dκ(x)|∇u|pdx)p−2p−1 | (20) |
Applying the property of
‖Hp(u)−Hp(I0u)‖p1,p(D)⪯(∫D∑ωiκ(x)|∇Hp(χi(u−uωi0))|pdx)qp×(∫Dκ(x)|∇u|pdx)p−2p−1⪯(∑ωi∫ωiκ(x)|∇Hp(χi(u−uωi0))|pdx)qp(∫Dκ(x)|∇u|pdx)p−2p−1. | (21) |
Recall that
∫ωiκ(x)|∇Hp(χi(u−uωi0))|pdx⪯1λLi+1∫ωiκ(x)|∇Hp(u−uωi0)|pdx. | (22) |
Define
|Hp(u)−Hp(I0u)‖p1,p(D)⪯(∑ωi1λLi+1∫ωiκ(x)|∇Hp(u−uωi0)|pdx)qp× | (23) |
(∫Dκ(x)|∇u|pdx)p−2p−1⪯(1Λ∗∑ωi∫ωiκ(x)|∇Hp(u−uωi0)|pdx)qp(∫Dκ(x)|∇u|pdx)p−2p−1=(1Λ∗)qp(∑ωi∫ωiκ(x)|∇Hp(u−uωi0)|pdx)qp(∫Dκ(x)|∇u|pdx)p−2p−1⪯(1Λ∗)qp(∫Dκ(x)|∇Hp(u)|pdx)qp(∫Dκ(x)|∇u|pdx)p−2p−1. | (24) |
Using the energy minimization property of
‖Hp(u)−Hp(I0u)‖p1,p(D)⪯(1Λ∗)qp(∫Dκ(x)|∇u|pdx)qp(∫Dκ(x)|∇u|pdx)p−2p−1=(1Λ∗)qp(∫Dκ(x)|∇u|pdx)qp+p−2p−1=(1Λ∗)1p−1∫Dκ(x)|∇u|pdx. |
This gives
‖Hp(u)−Hp(I0u)‖1p−11,p(D)⪯(1Λ∗)1p(p−1)2‖u‖1p−11,p(D). | (25) |
Substituting (25) into (18), we obtain
‖u−ums‖1,p(D)⪯‖u‖p−2p−11,p(D){H1(p−1)2‖f‖1(p−1)2Lq(D)+(1Λ∗)1p(p−1)2‖u‖1p−11,p(D)}. |
Remark 6. We notice that
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
∑ωi∫ωiκ(x)|∇Hp(χi(u−Iωi0(u)))|pdx=∑ωi∫ωiκ(x)|∇Hp(~χi(˜u−Iωi0(˜u)))|pdx⪯∑ωi1λωiLi+1∫ωiκ(x)|∇Hp(˜u−Iωi0(˜u))|pdx⪯1Λ∗∑ω+i∫ω+iκ(x)|∇Hp(χ+i(u−Iωi0(u)))|pdx⪯1Λ∗∑ω+i1λω+iLi+1∫ω+iκ(x)|∇Hp(u−Iωi0(u))|pdx⪯1Λ∗1Λ+∗∑ω+i∫ω+iκ(x)|∇Hp(u−Iωi0(u))|pdx⪯1Λ∗1Λ+∗∫Dκ(x)|∇Hp(u)|pdx⪯1Λ∗1Λ+∗∫Dκ(x)|∇u|pdx, |
where
∑ωi∫ωiκ(x)|∇Hp(χi(u−Iωi0(u)))|pdx⪯1Λ∗1Λ+∗⋅⋅⋅1Λ+N∗∫Dκ(x)|∇u|pdx, |
where
In this part, we exhibit the process of numerically implementing the proposed method for
JD(u)=infv∈W1,pg(D)JD(v), | (26) |
where
It is easily established that
J′D(u)(w)=∫Dκ(x)|∇u|p−2∇u⋅∇wdx−∫Dfwdx∀w∈W1,p0(D). |
Hence, there exists a unique solution to (
JD(uh)=minvh∈Vh0(D)JD(vh). | (27) |
The well-posedness of (
Recall the discussion in Section 3.2.2, we can represent the GMsFEM solution by
Algorithm 1 A Quasi-Newton algorithm |
1: Initialization: An initial guess |
2: (1) Compute the gradient vector |
3: (2) Compute the stepsize |
4: (3) Set: |
5: (4) If |
6: for |
7: (1) Compute the gradient vector |
8: (2) Compute the approximation of the inverse of Hessian matrix: |
9: (3) Compute the stepsize |
10: (4) Set: |
11: (5) If |
12: end For |
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
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
Lperror=‖u−ums‖Lp(D)‖u‖Lp(D)×100%,Energy error=‖u−ums‖1,p(D)‖u‖1,p(D)×100%, | (28) |
where we recall that
For the first set of experiments, we take
Energy 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 % |
Energy 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 % |
Energy 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 % |
Energy 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 % |
By observing the columns in Table 1 (or the curves in Figure 4), we can clearly see that for each
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.
1 | 8.86e-4 | 1.13e3 |
2 | 2.59e-3 | 3.86e2 |
3 | 4.46e-3 | 2.24e2 |
4 | 1.55e2 | 6.44e-3 |
5 | 4.01e2 | 2.50e-3 |
Remark 7. We note that the choice of
Since
To verify that our proposed method is applicable to more situations, we examine other choices of permeability field
Energy error | ||
1 | 44.15 % | 1.42e3 |
2 | 36.44 % | 4.04e2 |
3 | 27.99 % | 2.35e2 |
4 | 6.77 % | 6.49e-3 |
5 | 5.30 % | 2.50e-3 |
We also consider a different high-contrast permeability field
Energy error | ||
1 | 47.08 % | 1.85e1 |
2 | 27.68 % | 4.64e0 |
3 | 20.81 % | 2.68e0 |
4 | 4.33 % | 2.26e-3 |
5 | 2.69 % | 1.01e-3 |
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
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
Remark 9. To illustrate how the error of DEIM affects the global error estimate of Theorem 4.8, we adopt the notation
∫DaDEIM(x,∇˜ums)⋅∇v=∫Dfv,∀v∈Vh0(D). |
Then we have
‖ums−˜ums‖p1,p(D)=∫Da(x,∇(ums−˜ums))⋅∇(ums−˜ums)≤∫Da(x,∇ums)⋅∇(ums−˜ums)−∫Da(x,∇˜ums)⋅∇(ums−˜ums)=∫D(ums−˜ums)f−∫Da(x,∇˜ums)⋅∇(ums−˜ums)=∫D(aDEIM(x,∇˜ums)−a(x,∇˜ums))⋅∇(ums−˜ums). |
We note that
‖ums−˜ums‖p1,p(D)≤δp−1‖ums−˜ums‖1,p(D), | (29) |
for some small
Combining (29) with Theorem 4.8, we obtain
‖u−˜ums‖1,p(D)≤‖u−ums‖1,p(D)+‖ums−˜ums‖1,p(D)⪯δ+‖u‖p−2p−11,p(D){H1(p−1)2‖f‖1(p−1)2Lq(D)+(1Λ∗)1p(p−1)2‖u‖1p−11,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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1. | Wee Chin Tan, Viet Ha Hoang, High dimensional finite element method for multiscale nonlinear monotone parabolic equations, 2019, 345, 03770427, 471, 10.1016/j.cam.2018.04.002 | |
2. | Shubin Fu, Eric Chung, Tina Mai, Generalized multiscale finite element method for a strain-limiting nonlinear elasticity model, 2019, 359, 03770427, 153, 10.1016/j.cam.2019.03.047 | |
3. | Xinliang Liu, Eric Chung, Lei Zhang, Iterated Numerical Homogenization for MultiScale Elliptic Equations with Monotone Nonlinearity, 2021, 19, 1540-3459, 1601, 10.1137/21M1389900 | |
4. | Barbara Verfürth, Numerical homogenization for nonlinear strongly monotone problems, 2022, 42, 0272-4979, 1313, 10.1093/imanum/drab004 | |
5. | Minam Moon, Generalized multiscale hybridizable discontinuous Galerkin (GMsHDG) method for flows in nonlinear porous media, 2022, 415, 03770427, 114441, 10.1016/j.cam.2022.114441 | |
6. | Weifeng Qiu, Ke Shi, Analysis on an HDG Method for the p-Laplacian Equations, 2019, 80, 0885-7474, 1019, 10.1007/s10915-019-00967-6 |
Algorithm 1 A Quasi-Newton algorithm |
1: Initialization: An initial guess |
2: (1) Compute the gradient vector |
3: (2) Compute the stepsize |
4: (3) Set: |
5: (4) If |
6: for |
7: (1) Compute the gradient vector |
8: (2) Compute the approximation of the inverse of Hessian matrix: |
9: (3) Compute the stepsize |
10: (4) Set: |
11: (5) If |
12: end For |
Energy 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 % |
Energy 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 % |
Energy 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 % |
Energy 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 % |
1 | 8.86e-4 | 1.13e3 |
2 | 2.59e-3 | 3.86e2 |
3 | 4.46e-3 | 2.24e2 |
4 | 1.55e2 | 6.44e-3 |
5 | 4.01e2 | 2.50e-3 |
Energy error | ||
1 | 44.15 % | 1.42e3 |
2 | 36.44 % | 4.04e2 |
3 | 27.99 % | 2.35e2 |
4 | 6.77 % | 6.49e-3 |
5 | 5.30 % | 2.50e-3 |
Energy error | ||
1 | 47.08 % | 1.85e1 |
2 | 27.68 % | 4.64e0 |
3 | 20.81 % | 2.68e0 |
4 | 4.33 % | 2.26e-3 |
5 | 2.69 % | 1.01e-3 |
Algorithm 1 A Quasi-Newton algorithm |
1: Initialization: An initial guess |
2: (1) Compute the gradient vector |
3: (2) Compute the stepsize |
4: (3) Set: |
5: (4) If |
6: for |
7: (1) Compute the gradient vector |
8: (2) Compute the approximation of the inverse of Hessian matrix: |
9: (3) Compute the stepsize |
10: (4) Set: |
11: (5) If |
12: end For |
Energy 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 % |
Energy 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 % |
Energy 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 % |
Energy 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 % |
1 | 8.86e-4 | 1.13e3 |
2 | 2.59e-3 | 3.86e2 |
3 | 4.46e-3 | 2.24e2 |
4 | 1.55e2 | 6.44e-3 |
5 | 4.01e2 | 2.50e-3 |
Energy error | ||
1 | 44.15 % | 1.42e3 |
2 | 36.44 % | 4.04e2 |
3 | 27.99 % | 2.35e2 |
4 | 6.77 % | 6.49e-3 |
5 | 5.30 % | 2.50e-3 |
Energy error | ||
1 | 47.08 % | 1.85e1 |
2 | 27.68 % | 4.64e0 |
3 | 20.81 % | 2.68e0 |
4 | 4.33 % | 2.26e-3 |
5 | 2.69 % | 1.01e-3 |