Crouzeix-Raviart multiscale finite element methods 

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Theoretical framework of multiscale finite element methods

This chapter deals with the theoretical framework of multiscale finite element methods. Before introducing multiscale finite element methods, we first recall some classical theories of numerical analysis. These theories guarantee the existence and uniqueness of a solution to flow problems, in particular, Stokes problems and Oseen problems. Meanwhile, we recall these flow problems and define their variational formulations. The existence and uniqueness of a solution to these problems are guaranteed by previously presented theories. Afterwards, we present the Crouzeix-Raviart finite element and the finite element formulations of flows problems as well as their well-posedness. After these preliminary preparations, we finally introduce the main idea of the original multiscale finite element methods proposed by [87]. This introduction makes it easier to understand Crouzeix-Raviart multiscale finite element methods which will be presented in the next chapter. At last, the cost of multiscale finite element methods and traditional finite element methods are compared in order to show the good performance of the first one.
Outline Sections 2.1 to 2.3 introduces the theories of numerical analysis of an abstract problem and variational formulations of flow problems. Section 2.4 describes briefly the finite element method and the Crouzeix-Raviart finite element. Section 2.5 addresses the classical multiscale finite element method. Section 2.6 evaluates the the cost and performance of multiscale finite element methods.

Analysis of an abstract variational problem

This section recalls briefly some classical theories of numerical analysis of an abstract prob-lem. For more details about these theories, the reader can refer to many books on finite element methods, such as [67, 73].
Let V and M be two Hilbert spaces [9]. The scalar product defined on these spaces are denoted respectively by ( ; )V and ( ; )M . The norms associated to these scalar products are denoted respectively by k kV and k kM . Let V 0 and M0 be the dual spaces of V and M and let k kV 0 and k kM 0 be the associated dual norms. The dual space V 0 (respectively M0) is the space of linear forms defined on V (respectively M). We denote by h ; i the product of an element in the Hilbert space and an element of its dual space. Let f and g be element of V 0 and M0 respectively, i.e. f and g are two linear forms.
Let a(:; 🙂 and b(:; 🙂 be two continuous bilinear forms:
a(:; 🙂 : V V ! R; b(:; 🙂 : V M ! R
Then we consider the following variational problem: Given f 2 V 0 and g 2 M0, find (u; p) 2
V M such that (b (u; q) = qh
g; q M (2.1)
a (u; v) + b (v; p) = f; vi 8v 2 V
h i 8 2
Our purpose is to derive the necessary and sufficient conditions so that the problem (2.1) is well-posed (the problem has one and only one solution). Let W be a subspace of V and W is defined by:
W = fv 2 V j 8q 2 M; b (v; q) = 0g
We suppose firstly that g = 0. The initial problem (2.1) can be rewritten as: Find u 2 W such that
a (u; v) = hf; vi 8v 2 W (2.2)
The well-posedness of problem (2.2) is ensured by the following theorem (see theorem 1.7. of [73]) which is due to Lax & Milgram’s theorem [101].
Theorem 2.1.1. We assume that
– a(:; 🙂 is continuous, i.e., there exists a constant such that ja (u; v) j kukW kvkW 8u; v 2 W
– a(:; 🙂 is elliptic on W , i.e., there exists a constant such that a (v; v) kvk2W 8v 2 W
Then the problem (2.2) has one and only one solution u 2 W . Moreover, the mapping f ! u is an isomorphism from W 0 onto W .
The work of [67] proposes a more general theorem than Lax & Milgram’s theorem for the bilinear form a(:; :). We recall the theorem below:
Theorem 2.1.2 (Banach-Necasˇ-Babuška (BNB)). Let X be a Banach space and let Y be a reflexive Banach space. Let a be a bilinear form: X Y ! R and f 2 Y 0. Then the problem
(2.2) is well-posed if and only if:
9 >0; inf sup a (w; v)
kwkX kvkY
w2X v2Y
8v 2 Y; (8w 2 X; a(w; v) = 0) ) (v = 0)
Moreover, the following a priori estimate holds: 8f 2 Y 0; kukX 1 kfkY 0
This theorem is proved in [67] and it is stated that Theorem 2.1.1 is a consequence of Theo-rem 2.1.2. We will apply this theorem directly in what follows.
Now we need to consider how to treat the case g 6= 0 and discuss the existence and uniqueness of p to problem (2.1). To do this, we introduce the inf-sup condition introduced by Babuška-Brezzi [28, 32].
Theorem 2.1.3 (Babuška-Brezzi). The three following properties are equivalent:
1, there exists a constant > 0 such that
inf sup b(v; q)
v q
q 2 M;q=0 v2V k kV k
6 kM
2, there exists an isomorphism B0 from M onto W which verifies:
b(v; q) = v; B0q V;V 0 ; B0q V 0 kqkM 8q 2 M
The space W is defined as: W = h V v W; h; v = 0 .
3, there exists an isomorphism B from W ? onto M0 which verifies that b(v; q) = hBv; qiM0 ;M ; kBvkM0 kvkV 8v 2 W ?
The orthogonal space W ? of W is defined as W ? = fv 2 V j 8w 2 W; (v; w) = 0g.
This theorem is proved in [73] and we apply it directly in what follows.
Theorem 2.1.4. Assume that
1, a(:; 🙂 is a bilinear form continuous on V V .
2, b(:; 🙂 is a bilinear form continuous on V M.
3, a(:; 🙂 is V –elliptic, i.e. there exists a constant > 0 such that a(v; v) kvk2V 8v 2 V
4, b(:; 🙂 verifies the condition inf-sup: there exists a constant > 0 such that
inf b(v; q)
v q
q 2 M;q=0 sup k kV k kM
6 v2V
Then problem (2.1) is well-posed and it has one unique solution u 2 V; p 2 M for any f 2 V 0; g 2 M0.
This theorem can be proved easily using Theorem 2.1.3 and we apply directly this theorem in what follows.

The variational formulation of Stokes problem

Let be a connected and bounded open set in Rd, d = 2 or 3, with a Lipschitz-continous boundary. The Stokes problem with homogeneous Dirichlet boundary condition is: find the velocity u : ! Rd and the pressure p : ! R solutions to:
u + rp = f in (2.3)
div u = g in (2.4)
u = 0 on @
where is the dynamic viscosity, f is a given force and g is a given function. In particular, g = 0 for incompressible flows.
We introduce the following Sobolev spaces [9]:
– L2( ) is the space of square integrable functions.
– L20( ) = R p 2 L2 ( ) p = 0 is a subspace of L2( )
– H1 ( ) d = nv 2 L2 ( ) d rv 2 L2 ( ) do
– H01 ( ) d is a subspace of H 1( ) d and it contains functions whose trace is zero on @
The scalar product of L2( ) and the associated norm are denoted respectively by (:; 🙂 and by k kL2 .
(u; v) = Z u v; kukL2 = (u; u)2
The norm for the space H01( ) d is defined as kuk0 = Z jruj2 1=2
We introduce the following bilinear forms: Z aSt(u; v) = Z ru : rv; b(v; q) = q div v and linear forms: Z
F (v) = hf; vi = Z f v; G (q) = hg; qi = gq 8vh 2 Vh; 8qh 2 Mh
Now we try to write the variational formulation of Stokes equations (2.3)–(2.4). Multiplying (2.3) by a test function v 2 H01( ) d and integrating over
Similarly, multiplying (2.4) by a function q 2 L2 ( ) and integrating on the domain , we have divu q = Z Z gq
With the notations above, the variational formulation of Stokes equations (2.3)–(2.4) can be u H1 ( ) d p L2 ( ) that
written in the abstract form: find 2 0 and 2 0 0 such d
(b(u; q) = G (q) 8q 2 L02 ( ) 8 2 (2.5)
aSt(u; v) + b(v; p) = F (v) v H1 ( )
In conclusion, the Stokes problem is one prototype example of problem (2.1), by choosing
V = H01 ( ) d and M = L20 ( ). The space W reads:
W = nv 2 H01 ( ) d 8q 2 L02 ( ) ; b(v; q) = 0o = nv 2 H01 ( ) d div v = 0 o
The existence and uniqueness of a solution to system (2.5) is guaranteed by Theorem 2.1.4.
We see that each of the hypothesis of Theorem 2.1.4 is verified: and H01 ( ) d L02 ( ). H01 d H01 d
– bilinear forms aSt(:; 🙂 and b(:; 🙂 are continuous respectively on ( ) ( )
– aSt(:; 🙂 is elliptic on V . We recall the Poincaré inequality: there exists a C > 0 such that 8v 2 H01 ( ) d ; aSt (v; v) = Z jrvj2 C Z jvj2 (2.6)
Since the semi-norm krvkL2( ) is equivalent to the full H1 norm by the Poincaré inequal-ity, we obtain easily that aSt(:; 🙂 is elliptic on V .
– the bilinear form b(:; 🙂 verifies the inf-sup condition. Since the gradient operator is an iso-morphism from V ? onto L2( ) and the divergence operator is an isomorphism from L20 ( ) R nto V , Theorem 2.1.3 implies that b(:; 🙂 verifies the inf-sup condition of Theorem 2.1.4.
We introduce another bilinear form cSt(:; 🙂 continuous on (V M)2: cSt((u; p); (v; q)) = Z ru : rv Z p div v Z q div u (2.7)
The variational formulation of (2.3)–(2.4) is equivalent to: find (u; p) 2 V M such that cSt((u; p); (v; q)) = F (v) 8 (v; q) 2 V M (2.8)
Let X = V M with V = H01 ( ) d and M = L02 ( ). Theorem 2.1.2 implies that problem (2.8) has a unique solution if the bilinear form cSt(:; 🙂 satisfies the following inf-sup property:
inf sup cSt ((u; p) ; (v; q)) (2.9)
ku; pkX kv; qkX
(u;p)2X (v;q)2X
with a constant > 0.

The variational formulation of Oseen problem

Let be a connected and bounded open set in Rd, d = 2 or 3, with a Lipschitz-continous boundary. The steady-state Oseen problem with homogeneous Dirichlet boundary condition is to find the velocity u : ! Rd and the pressure p : ! R solutions to:
u + (Uo r) u + rp = f in (2.10)
div u = g in (2.11)
u = 0 on @ (2.12)
where is the dynamic viscosity, Uo is a known velocity, is the flow density, f is a given force and g is a given function. In particular, g = 0 for incompressible flows.
We introduce the bilinear form aOs(:; 🙂 for the Oseen problem: aOs (u; v) = ( ru : rv + (Uo r) u v) (2.13)
Thus the variational formulation of the Oseen problem is to find u 2 L20 ( ) such that aOs(u; v) + b(v; p) = F (v) v H1 ( ) d
(b(u; q) = G (q) 8q 2 L02 ( ) 8 2 0
H01 ( ) d and p 2
(2.14)
By choosing V = H1 ( ) d and M = L2 ( ), we apply Theorem 2.1.4 to guarantee the existence and uniqueness of a solution to (2.14). It is clear that the hypothesis 1, 2
Theorem 2.1.4 are verified by bilinear forms aOs(:; 🙂 and b(:; :).
We introduce another continuous bilinear form which is equivalent to aOs(:; :): aOs(u; v) = Zru : rv + 2 (Uo ru) v 2 (Uo rv) u 2 uv div Uo with u = 0 on @ (boundary condition (2.12)).
Since the two bilinear forms aOs(:; 🙂 and baOs(:; 🙂 are equivalent and it is easier to prove that the hypothesis 3 holds using baOs(:; :), we now prove that baOs is coercive on V . It is easy to observe that aOs(u; u) = Z ru : ru 2 uu div Uo by assuming that div Uo 0.
Since the semi-norm krvkL2( ) is equivalent to the full H1 norm by Poincaré inequality (2.6), we obtain that aSt(:; 🙂 is elliptic on V .
Thus the hypothesis 3 holds for aOs(:; 🙂 and aOs(:; :). formulation of Oseen
Consequently Theorem 2.1.4 guarantees that the following variational problem is well-posed.
aOs(u; v) + b(v; p) = F (v) v H1 ( ) d
(b(u; q) = G (q) q L2 ( ) 8 2 0 (2.15)
b 8 2 0
We introduce also the bilinear form bcOs(:; :): cOs((u; p); (v; q)) = Z ru : rv + 2 (Uo ru) v 2 (Uo rv) u
The variational formula of the Oseen problem can be written as: find (u; p) 2 V M such that
Os ((u; p); (v; q)) = F (v) 8 (v; q) 2 V M
By choosing X = V M, Theorem 2.1.2 guarantees the existence and uniqueness of a solution to this problem.

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The variational formulation of Navier-Stokes problem

Let be a connected and bounded open set in Rd, d = 2 or 3, with a Lipschitz-continous bound-ary. The steady-state Navier-Stokes problem with homogeneous Dirichlet boundary condition is to find the velocity u : ! Rd and the pressure p : ! R solutions to:
u + (u r) u + rp = f in (2.16)
div u = g in (2.17)
u = 0 on @
where is the dynamic viscosity and is the flow density, f is a given force and g is a given function. In particular, g = 0 for incompressible flows.
We introduce the following nonlinear forms:
aNS (u; v) = Z ( ru : rv + (u r) u v) Z Z
cNS((u; p); (v; q)) = Z ( ru : rv + (u r) u v) p div v q div u
The generalization of the abstract variational problem analyzed in section 2.1 to nonlinear problems can be found in [73]. The family of nonlinear problems contains the Navier-Stokes problem in particular. The analysis of a nonlinear abstract variational problem is much more complicated than a linear abstract problem. Thus we choose not to present the numerical analysis of Navier-Stokes problem in this thesis.

The finite element method

The finite element method is a very popular method to solve Partial Differential Equations (PDEs). It approximates the continuous space of solution of PDEs by a finite dimensional space. To show the main idea of the method, we present the approximation of the abstract variational problem analyzed in section 2.1.
Let h denote a discretization parameter tending to zero. For each h, let Vh and Mh two finite dimensional spaces that Vh and Mh. We introduce two bilinear forms ah(:; 🙂 and bh(:; 🙂 on Vh Vh and Vh Mh respectively. We now approximate the problem (2.1) by the discrete problem: Given f 2 V 0 and g 2 M0, find (uh; ph) 2 Vh Mh such that bh (uh; qh) = g; qh q hMh i 8 2 Vh (2.18) ah (uh; vh) + bh (vh; ph) = f; vh vh h i 8 2
The existence and uniqueness of a solution to system (2.18) is guaranteed by the following discrete inf-sup condition Theorem 2.4.1, which is a discrete version of Theorem 2.1.4:
Theorem 2.4.1. Assume that
1, ah(:; 🙂 is a bilinear form continuous on Vh Vh.
2, bh(:; 🙂 is a bilinear form continuous on Vh Mh.
3, ah(:; 🙂 is Vh-elliptic, i.e. there exists a constant h > 0 such that ah(vh; vh) h kvk2Vh 8vh 2 Vh
4, bh(:; 🙂 verifies inf-sup condition: there exists a constant h > 0 such that inf sup bh(vh; qh) h
2 k v hkVh k q hkMh qh Mh;qh=0 6 vh2Vh
Then problem (2.18) is well-posed and it has one unique solution uh 2 Vh; ph 2 Mh for any f 2 V 0; g 2 M0.

The Crouzeix-Raviart finite element

Let be a connected and bounded open set in Rd with d = 2 or 3. We denote Th a discretization of by triangles (d = 2) or tetrahedrons (d = 3) noted as K. The i-th face of Th is noted as fi and the middle of the face fi is noted as xi. The barycenter of the triangle or tetrahedron is noted as gK . The set of faces in the discretization is noted as Eh. The Crouzeix-Raviart finite element shown in Figure 2.1 was first introduced in [52]. The velocity unknown of this element is in the barycenter of each face and the pressure unknown is in the barycenter of each element.
We note Vh and Mh as the approximation space of the velocity uh and the pressure ph re-spectively. The spaces Vh and Mh are defined by V = L2 ( ) d v ( (K))d ; v x ; K
h nvh 2 2 K hjK 2 P1 h is continuous at points i 8 2 Tho
Mh = qh 2 L0 ( ) qh 2 P0(K); 8 K 2 T h
The approximation space Vh is not included in V = H0 ( ) approximation space Mh is included in the continuous space M = L02 ( ).
Let fa0; ; adg be the vertices of K, fi be the face of K opposite to ai and ni be the outward normal to fi. The associated barycentric coordinates ( 0; ; d) are defined by: : x ! (x) = 1 (x ai) ni for 0 i d
i i (aj ai) ni

Table of contents :

Acknowledgements
1 Introduction 
1.1 Challenges and motivation
1.2 Flow problems in heterogeneous media
1.3 The homogenization theory
1.3.1 Problem setting
1.3.2 Two-scale asymptotic expansions
1.4 Literature overview
1.4.1 Numerical homogenization methods
1.4.2 The Multiscale Finite Element Methods (MsFEMs)
1.5 Main contributions of the thesis
1.6 Contents of the thesis
2 Theoretical framework of multiscale finite element methods 
2.1 Analysis of an abstract variational problem
2.2 The variational formulation of Stokes problem
2.3 The variational formulation of Oseen problem
2.3.1 The variational formulation of Navier-Stokes problem
2.4 The finite element method
2.4.1 The Crouzeix-Raviart finite element
2.4.2 A finite element formulation of Stokes problem
2.5 The original multiscale finite element method
2.5.1 Discretization of the domain
2.5.2 Computation of multiscale basis functions
2.5.3 The coarse-scale formulation
2.5.4 Reconstruction of fine-scale solutions
2.6 The cost of multiscale finite element methods
3 Crouzeix-Raviart multiscale finite element methods 
3.1 Introduction
3.1.1 Discretization of the heterogeneous domain
3.1.2 Multiscale functional spaces
3.2 The Crouzeix-Raviart multiscale finite element method defined by Stokes equations
3.2.1 The construction of the approximation space XSt
3.2.2 The local problem defined by Stokes equations
3.2.3 The basis function of the space V St
3.2.4 The coarse-scale problems and stabilized formulations
3.2.5 The reconstruction of fine-scale features
3.3 The Crouzeix-Raviart multiscale finite element method defined by Oseen equations
3.3.1 The construction of the approximation space XOs
3.3.2 The local problem defined by Oseen equations
3.3.3 The proof of the well-posedness of the local problem
3.3.4 The basis functions of the space V Os
3.3.5 The coarse-scale problem and stabilized formulations
3.3.6 The reconstruction of fine-scale features
3.4 The Crouzeix-Raviart multiscale finite element method defined by adding solutions of local Stokes and Oseen problems
3.5 The Crouzeix-Raviart multiscale finite element method enriched by bubble functions
3.5.1 Bubble functions defined by Stokes equations
3.5.2 Theoretical analysis of the construction of XSt+b
3.5.3 The local problems defined by Stokes equations
3.5.4 The contribution of bubble functions
3.6 The high-order Crouzeix-Raviart multiscale finite element method defined by Stokes equations
3.6.1 The construction of the approximation space bX
3.6.2 The local problems defined by Stokes equations
3.6.3 The basis functions of the space bV St
3.6.4 The choices of weighting functions and finite elements
3.6.5 The coarse-scale problem
3.6.6 The reconstruction of fine-scale features
3.7 The high-order Crouzeix-Raviart multiscale finite element method defined by Oseen equations
3.7.1 The local problems defined by Oseen equations
3.7.2 The basis functions of the space bV Os
3.7.3 The choices of weighting functions and finite elements
3.7.4 The coarse-scale problem
3.7.5 The reconstruction of fine-scale features
4 Technical aspects of Crouzeix-Raviart multiscale finite element methods 
4.1 The finite volume element method
4.1.1 The Crouzeix-Raviart finite element and the control volume
4.2 The discretization of local problems
4.2.1 The conservation of mass
4.2.2 The conservation of momentum equation
4.2.3 The discretization of the Oseen term
4.2.4 The discretization of the velocity integral boundary condition
4.2.5 The discretization of the temporal inertial term
4.3 The solution of local problems
4.4 The discretization of coarse-scale problems
4.4.1 Discretization of coarse-scale Oseen problems
4.4.2 The computation of matrices
4.4.3 Discretization of the nonlinear convection term
4.5 The solution of coarse-scale problems
4.6 Technical aspects of high-order multiscale methods
4.6.1 The P1-nonconforming/P1 finite element
4.6.2 The discretization of local problems
4.6.3 The discretization of coarse-scale problems
4.6.4 The solution of local and coarse-scale problems with a direct solver
4.6.5 The validation of solutions of local problems
5 The multiscale simulation chain 
5.1 Parallelisms in the simulation chain
5.2 Pre-processings in the SALOME platform
5.2.1 The SALOME platform
5.2.2 The GEOM and MESH modules
5.2.3 Parallelisms of the generation of meshes
5.2.4 Generation of the coarse mesh and conforming fine meshes
5.2.5 Generation of the coarse mesh and nonconforming fine meshes
5.2.6 Treatment of tangent points
5.2.7 A special algorithm for periodic heterogeneous media
5.3 Implementations in TrioCFD
5.3.1 TRUST and TrioCFD
5.3.2 Preparations of data files for numerical simulations
5.3.3 PROJECT_LOCAL_PB for local problems
5.3.4 PROJECT_MAT for matrix assembly
5.3.5 PROJECT_COARSE_PB for coarse-scale problems
5.3.6 PROJECT_POS for the reconstruction of fine-scale solutions
5.4 Post-processings in VisIt
5.4.1 The visualization tool VisIt
5.4.2 The visualization of fine-scale solutions in VisIt
6 Numerical simulations 
6.1 Notations
6.2 Simulations in a two-dimensional homogeneous medium
6.3 Numerical convergence in the periodic case
6.4 Simulations in two-dimensional non-periodic heterogeneous media
6.4.1 Applications to Stokes flows
6.4.2 Applications to Oseen flows
6.4.3 Applications to Navier-Stokes flows
6.5 Simulations in two-dimensional periodic heterogeneous media
6.5.1 Numerical convergence of Crouzeix-Raviart MsFEMs with respect to H
6.5.2 Error analysis with respect to the heterogeneity
6.5.3 Applications to highly heterogeneous media
6.6 Simulations in three-dimensional media
6.6.1 Flows in a homogeneous medium
6.6.2 Applications to a non-periodic heterogeneous medium
6.6.3 Applications to a periodic heterogeneous medium
7 Conclusions 
7.1 Theoretical aspects
7.1.1 The Crouzeix-Raviart multiscale method defined by Stokes equations
7.1.2 The Crouzeix-Raviart multiscale method defined by Oseen equations
7.1.3 The Crouzeix-Raviart multiscale method defined by both Stokes and Oseen local solutions
7.1.4 The Crouzeix-Raviart multiscale method enriched by bubble functions
7.1.5 The high-order Crouzeix-Raviart multiscale finite element method
7.1.6 Comparison of Crouzeix-Raviart multiscale finite element methods
7.2 The multiscale simulation chain SALOME-TrioCFD-VisIt
7.2.1 The intra- and extra-cellular parallelisms
7.2.2 Generation of meshes in SALOME
7.2.3 Implementations in TrioCFD
A Source files and data files in the multiscale simulation chain
A.1 A Python script for generating meshes in SALOME
A.2 An example of data file of TrioCFD
A.3 A Python script for visualization in VisIt
B Solution of local problems in high-order multiscale methods
B.1 The well-posedness of discrete local problems
B.2 Multiscale basis functions
C Résumé en français
Bibliography

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