Approximation de problèmes mixtes
Ce document est en anglais et doit être traduit. 
1. Model Problems
We consider now model problems as systems of PDEs where several functions are unknowns and which don’t play the same roles mathematically and physically.
 Stokes
where \(u: \Omega \mapsto \RR^d\) is a velocity and \(p: \Omega \mapsto \RR\) is a pressure.
 Darcy
where \(\sigma: \Omega \mapsto \RR^d\) is a velocity and \(u: \Omega \mapsto \RR\) is a hydraulic charge(pressure).
1.1. Applications
We shall focus on Stokes, but the abstract setting of the next section is the same for Stokes and Darcy.
 Stokes and incompressible NavierStokes for Newtonian fluids

The Stokes model is the basis for fluid mechanics models and is a simplication of the NavierStokes equations where the viscous effects/terms are much bigger than the convective ones
The first equation results from the conservation of momentum and the second from the conservation of mass.
The wellposedness of these problems results from a socalled infsup condition which is not automatically transfered at the discrete level.
In practice In order to ensure that the finite element approximation is wellposed, we will need to choose approximation spaces that satisfy a compatibility condition that ensures that a discrete infsup condition is satisfied.
2. Saddle point problems
2.1. Abstract Continuous Setting
Denote

\(X\) and \(M\) two Hilbert spaces.^{[1]}

two linear forms \(f \in X'=\mathcal{L}(X, \RR)\) and \(g \in M'=\mathcal{L}(M, \RR)\)

\(a \in \mathcal{L}(X\times X, \RR)\) and \(b \in \mathcal{L}(X\times M, \RR)\) two bilinear forms
We are interested in the following abstract problem:
2.1.1. Definition of a saddle point problem
The structure of the problem is as follows

the space of solution is the same of the test space

the unknown \(p\) does not appear in the second equation

the unknown functions \(u\) and \(p\) are coupled via the same bilinear form \(b\) is the first and second equation.
The next question is :
2.2. Well posedness
2.2.1. Reformulation
Let’s rewrite the mixed problem 1.
Denote \(V=X\times M\) and introduce \(c \in \mathcal{L}(V\times V, \RR)\) such that
and \(h\in \mathcal{L}(V,\RR)\) such that
then problem Problem Mixed 1 reads
LaxMilgram provides only a sufficient condition for wellposedness. The form \(c\) in Problem does not satisfy LaxMilgram. 
Let’s introduce the socalled Lagrangian \(l \in \mathcal{L}(X\times M, \RR)\) defined by
3. Finite element approximation
3.1. Abstract Discrete Problem
We now turn to the approximation of the Problem Mixed 1 by a standard Galerkin method in a conforming way.
Denote the two spaces \(X_h \subset X\) and \(M_h \subset M\), we consider the following problem:
The compatibility condition problem Formulation of the Abstract Discrete Problem, to be well posed, requires that the spaces \(X_h\) and \(M_h\) satisfy the condition.
This is known as the BabuskaBrezzi (BB) or LadyhenskayaBabuskaBrezzi (LBB).
Regarding error analysis, we have the following lemma
The constants \(c_{1h}, c_{3h}, c_{4h}\) are as large as \(\beta_h\) is small. 
3.2. Linear system associated
The discretisation process leads to a linear system.
We denote

\(N_u = \dim {X_h}\)

\(N_p = \dim {M_h}\)

\(\{\phi_i\}_{i=1,...,N_u}\) a basis of \(X_h\)

\(\{\psi_k\}_{k=1,...,N_p}\) a basis of \(M_h\)

for all \(u_h = \sum_{i=1}^{N_u} u_i \phi_i\), we associate \(U \in \R{N_u}\), \(U=(u_1,\ldots,u_{N_u})^T\), the component vector of \(u_h\) is \(\{\phi_i\}_{i=1,\ldots,N_u}\)

for all \(p_h = \sum_{k=1}^{N_p} u_k \psi_k\), we associate \(P \in \R{N_p}\), \(P=(p_1,\ldots,p_{N_p})^T\), the component vector of \(p_h\) is \(\{\psi_k\}_{k=1,\ldots,N_p}\)
The matricial form of problem Formulation of the Abstract Discrete Problem reads
where the matrix \(\mathcal{A} \in \R{N_u,N_u}\) and \(\mathcal{B} \in \R{N_p,N_u}\) have the coefficients
and the vectors \(\mathcal{F} \in \R{N_u}\) and \(\mathcal{G} \in \R{N_p}\) have the coefficients

\(F_i=f(\phi_i)\)

\(G_k=g(\psi_k)\)

When the infsup is not satisfied
The counter examples when the infsup condition is not satisfied(e.g. \(\mathcal{B}\) is not maximum rank ) occur usually in two cases:
\[b(v_h,q^*_h)=0.\]

We now introduce the Uzawa matrix as follows
 Applications

The Uzawa matrix occurs when eliminating the velocity in system and get a linear system on \(P\):
then one application is to solve by solving iteratively and compute the velocity afterwards.
4. Mixed finite element for Stokes
4.1. Variational formulation
We start with the Wellposedness at the continuous level

We consider the model problem with homogeneous Dirichlet condition on velocity \(u = 0\) on \(\partial \Omega\)

We suppose the \(f \in [L^2(\Omega)]^d\) and \(g \in L^2(\Omega)\) with
Introduce
The condition comes from the divergence theorem applied to the divergence equation and the fact that \(u=0\) on the boundary
This is a necessary condition for the existence of a solution \((u,p)\) for the Stokes equations with these boundary conditions.
We turn now to the variational formulation.
The Stokes problem reads
We recover the formulation of Problem Mixed 1 with \(X=[H^1_0(\Omega)]^d\) and \(M=L^2_0(\Omega)\) and
Pressure up to a constant
The pressure is known up to a constant, that’s why we look for them in \(L^2_0(\Omega)\) to ensure uniqueness.

4.2. Finite element approximation
Denote \(X_h \subset [H^1_0(\Omega)]^d\) and \(M_h \subset L^2_0(\Omega)\)
This problem, thanks to theorem Theorem is wellposed if and only if \(X_h\) and \(M_h\) are such that there exists \(\beta_h > 0\) 
4.3. Bad finite elements for Stokes
In this section, we present two classical bad finite element approximations.
4.3.1. Finite element \(\poly{P}_1/\poly{P}_0\): locking
Thanks to the Euler relations, we have
where \(I\) is the number of holes in \(\Omega\).
We have that \(\dim {M_h} = N_{\mathrm{cells}}\),\(\dim {X_h} = 2 N^i_{\mathrm{vertices}}\) and so
so \(M_h\) is too rich for the condition and we have \(\ker(\mathcal{B}) = \{0\}\) such that the only discrete \(u_h^*\), with components \(U^*\), satisfying \(\mathcal{B} U^*\) is the null field, \(U^*=0\).
4.3.2. Finite element \(\poly{Q}_1/\poly{P}_0\): spurious mode
We can construct in that case a function \(q_h^*\) on a uniform grid which is equal alternatively 1, +1 (chessboard) in the cells of the mesh, then
and thus the associated \(X_h\), \(M_h\) do not satisfy the condition.
4.3.3. Finite element \(\poly{P}_1/\poly{P}_1\): spurious mode
We can construct in that case a function \(q_h^*\) on a uniform grid which is equal alternatively 1, 0, +1 at the vertices of the mesh, then
and thus the associated \(X_h\), \(M_h\) do not satisfy the condition.
4.4. MiniElement
The problem with the \(\poly{P}_1/\poly{P}_1\) mixed finite element is that the velocity is not rich enough.
A cure is to add a function \(v_h^*\) in the velocity approximation space to ensure that
where \(q_h^*\) is the spurious mode.
To do that we add the bubble function to the \(\poly{P}_1\) velocity space.
 Example

The function
where \((\hat{\lambda}_0, \ldots, \hat{\lambda}_d)\) denote the barycentric coordinates on \(\hat{K}\)
Denote now \(\hat{b}\) a bubble fonction on \(\hat{K}\), we set
and introduce
4.5. TaylorHood Element
The minielement solved the compatibility condition problem, but the error estimation in equation is not optimal in the sense that

the pressure space is sufficiently rich to enable a \(h^2\) convergence in the pressure error,

but the velocity space is not rich enough to ensure a \(h^2\) convergence in the velocity error.
The idea of the TaylorHood element is to enrich even more the velocity space to ensure optimal convergence in \(h\).
Here we will take \([\poly{P}_2\)^d] for the velocity and \(\poly{P}_1\) for the pressure.
Introduce
 Generalized TaylorHood element

We consider the mixed finite elements \(\poly{P}_k/\poly{P}_{k1}\) and \(\poly{Q}_k/\poly{Q}_{k1}\) which allows to approximate the velocity and pressure respectively with, on Simplices
On Hypercubes
We then have
There are other stable discretization spaces

Discrete infsup condition: dictates the choice of spaces

Infsup stables spaces:

\(\mathbb Q_k\)\(\mathbb Q_{k2}\), \(\mathbb Q_k\)\(\mathbb Q^{disc}_{k2}\)

\(\mathbb P_k\)\(\mathbb P_{k1}\), \(\mathbb P_k\)\(\mathbb P_{k2}\), \(\mathbb P_k\)\(\mathbb P^{disc}_{k2}\)

Discrete infsup constant independent of \(h\), but dependent on \(k\)

5. Test Cases for Stokes
5.1. Kovasznay
We consider the Kovasznay solution of the steady Stokes equations.
The exact solution reads as follows
The domain is defined as \(\domain = (0.5,1) \times (0.5,1.5)\) and \(\nu = 0.035\).
The forcing term for the momentum equation is obtained from the solution and is
Dirichlet boundary conditions are manufactured from the exact solution.