Approximation de problèmes mixtes
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
\[ \left\{\begin{array}[c]{rl} \Delta u + \nabla p & = f\ \mbox{ in } \Omega\\ \nabla \cdot u & = 0\ \mbox{ in } \Omega \end{array}\right. \] where \(u: \Omega \mapsto \RR^d\) is a velocity and \(p: \Omega \mapsto \RR\) is a pressure.
 Darcy
\[ \left\{\begin{array}[c]{rl} \sigma + \nabla u & = f\ \mbox{ in } \Omega\\ \nabla \cdot \sigma & = g\ \mbox{ in } \Omega \end{array}\right. \] where \(\sigma: \Omega \mapsto \RR^d\) is a velocity and \(u: \Omega \mapsto \RR\) is a hydraulic charge(pressure).
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
\[ \left\{\begin{array}[c]{rl} \rho( \frac{\partial u}{\partial t} + u \cdot \nabla u)  \nu \Delta u + \nabla p & = f\ \mbox{ in } \Omega\\ \nabla \cdot u & = 0\ \mbox{ in } \Omega \end{array}\right. \] 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 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.
Saddle point problems
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:
Look for \((u,p) \in X \times M\) such that
\[ \left\{ \begin{array}[c]{rl} a(u,v) + b(v,p) & = f(v), \quad \forall v \in X\\ b(u,q) & = g(q), \quad \forall q \in M \end{array} \right. \]
Definition of a saddle point problem
If the bilinear form \(a\) is symmetric and positive on \(X\times X\), we say that [prob:chmixte:1] is 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 :
Well posed problem
Reformulation
Let’s rewrite Problem [prob:chmixte:1].
Denote \(V=X\times M\) and introduce \(c \in \mathcal{L}(V\times V, \RR)\) such that
\[ cu,p),(v,q = a(u,v)+b(v,p)+b(u,q) \] and \(h\in \mathcal{L}(V,\RR)\) such that
\[ h(v,q) = f(v)+g(q) \] then problem [prob:chmixte:1] reads
Look for \((u,p) \in V\) such that
\[ \begin{array}[c]{rl} cu,p), (v,q & = h(v,q), \quad \forall (v,q) \in V \end{array} \]
We suppose that \(a\) is coercive over \(X\), the [prob:chmixte:2] is wellposed if and only if the bilinear form \(b\) satisfies the following infsup condition:
there exists \(\beta > 0\) such that
\[ \inf_{q \in M} \sup_{v \in X} \frac{b(v,q)}{v_X q_M} \geq \beta \]
LaxMilgram provides only a sufficient condition for wellposedness 
The form \(c\) in [prob:chmixte:2] does not satisfy LaxMilgram. 
Let’s introduce the socalled Lagrangian \(l \in \mathcal{L}(X\times M, \RR)\) defined by
\[ l(v,q) = \frac{1}{2} a(v,v) + b(v,q)  f(v)  g(q) \]
We say that the point \((u,p)\in X\times M\) is a saddle point of \(l\) if
\[ \forall (v,q) \in X\times M, \quad l(u,q) \leq l(u,p) \leq l(v,p) \]
Under the hypothesys oF [thr:chmixte:1], the Lagrangian \(l\) defined by has a unique saddle point. Moreover this saddle point is the unique solution of problem [prob:chmixte:1].
Finite element approximation
Abstract Discrete Problem
We now turn to the approximation of the problem [prob:chmixte: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:
Look for \((u_h,p_h) \in X_h \times M_h\) such that
\[ \left\{ \begin{array}[c]{rl} a(u_h,v_h) + b(v_h,p_h) & = f(v_h), \quad \forall v_h \in X_h\\ b(u_h,q_h) & = g(q_h), \quad \forall q_h \in M_h \end{array} \right. \]
We suppose that \(a\) is coercive over \(X\) and that \(X_h \subset X\) and \(M_h \subset M\).
Then the [prob:chmixte:3] is wellposed if and only if the following discrete infsup condition is satisfied:
there exists \(\beta_h > 0\) such that
\[ \inf_{q_h \in M_h} \sup_{v_h \in X_h} \frac{b(v_h,q_h)}{v_h{X_h} q_h{M_h}} \geq \beta_h \]
The compatibility condition problem [prob:chmixte:3], 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
Thanks to the Lemma of Céa applied to SaddlePoint Problems, the unique solution \((u,p)\) of problem [prob:chmixte:3] satisfies
\[ \begin{array}[c]{rl} uu_hX & \leq c{1h} \inf_{v_h \in X_h} uv_hX + c{2} \inf_{q_h \in M_h} qq_hM\\ pp_h_X & \leq c{3h} \inf_{v_h \in X_h} uv_hX + c{4h} \inf_{q_h \in M_h} qq_h_M \end{array} \] where

\(c_{1h} = (1+\frac{a_{X,X}}{\alpha})(1+\frac{b_{X,M}}{\beta_h})\) with \(\alpha\) the coercivity constant of \(a\) over X.

\(c_{2} = \frac{b_{X,M}}{\alpha}\)

\(c_{3h} = c_{1h} \frac{a_{X,X}}{\beta_h}\), \(c_{4h} = 1+ \frac{b_{X,M}}{\beta_h}+\frac{a_{X,X}}{\beta_h}\)
The constants \(c_{1h}, c_{3h}, c_{4h}\) are as large as \(\beta_h\) is small. 
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 [prob:chmixte:3] reads
\[ \begin{bmatrix} \mathcal{A} & \mathcal{B}^T\\ \mathcal{B} & 0 \end{bmatrix} \begin{bmatrix} U \\ P \end{bmatrix} = \begin{bmatrix} F\\ G \end{bmatrix} \]
where the matrix \(\mathcal{A} \in \R{N_u,N_u}\) and \(\mathcal{B} \in \R{N_p,N_u}\) have the coefficients
\[ \mathcal{A}_{ij} = a(\phi_j,\phi_i), \quad \mathcal{B}_{ki} = b(\phi_i,\psi_k) \]
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
The matrix
\[ \mathcal{U} = \mathcal{B} \mathcal{A}^{1} \mathcal{B}^T \] is called the Uzawa matrix. It is symmetric positive definite from the properties of \(\mathcal{A}\), \(\mathcal{B}\)
 Applications

The Uzawa matrix occurs when eliminating the velocity in system and get a linear system on \(P\):
\[ \mathcal{U} P = \mathcal{B} \mathcal{A}^{1} F  G \] then one application is to solve by solving iteratively and compute the velocity afterwards.
Mixed finite element for Stokes
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 \[ \int_\Omega g = 0 \] Introduce
\[ L^2_0(\Omega) = \Big\{ q \in L^2(\Omega): \int_\Omega q = 0 \Big\} \]
The condition comes from the divergence theorem applied to the divergence equation and the fact that \(u=0\) on the boundary
\[ \int_\Omega g = \int_\Omega \nabla \cdot u = \int_{\partial \Omega} u \cdot n = 0 \] 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
Look for \((u,p) \in [H^1_0(\Omega)\)^d \times L^2_0(\Omega)] such that
\[ \left\{ \begin{array}[c]{rl} \int_\Omega \nabla u : \nabla v \int_\Omega p \nabla \cdot v & = \int_\Omega f \cdot v, \quad \forall v \in [H^{1_0(\Omega)]}d\\ \int_\Omega q \nabla \cdot u & =  \int_\Omega g q, \quad \forall q \in L^2_0(\Omega) \end{array} \right. \]
We retrieve the problem [prob:chmixte:1] with \(X=[H^1_0(\Omega)\)^d] and \(M=L^2_0(\Omega)\) and
\[ \begin{array}[c]{rlrl} a(u,v) &= \int_\Omega \nabla u : \nabla v,& \quad b(v,p) &= \int_\Omega p \nabla \cdot v,\\ \quad f(v) &= \int_\Omega f \cdot v,& \quad g(q) &=  \int_\Omega g q \end{array} \]
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.

Finite element approximation
Denote \(X_h \subset [H^1_0(\Omega)\)^d] and \(M_h \subset L^2_0(\Omega)\)
Look for \((u_h,p_h) \in X_h \times M_h\) such that
\[ \left\{ \begin{array}[c]{rl} \int_\Omega \nabla u_h : \nabla v_h + \int_\Omega p_h \nabla \cdot v_h & = \int_\Omega f \cdot v_h, \quad \forall v_h \in X_h\\ \int_\Omega q_h \nabla \cdot u_h & = \int_\Omega g q_h, \quad \forall q_h \in M_h \end{array} \right. \]
This problem, thanks to theorem [thr:chmixte:2] is wellposed if and only if \(X_h\) and \(M_h\) are such that there exists \(\beta_h > 0\) 
\[ \inf_{q_h \in M_h} \sup_{v_h \in X_h} \frac{\int_\Omega q_h \nabla \cdot v_h}{v_h{X_h} q_h{M_h}} \geq \beta_h \]
Some counter examples: bad finite element for Stokes
In this section, we present two classical bad finite element approximations.
Finite element \(\poly{P}_1/\poly{P}_0\): locking
Thanks to the Euler relations, we have
\[ \begin{array}[c]{rl} N_{\mathrm{cells}}  N_{\mathrm{edges}} + N_{vertices} &= 1I\\ N^\partial_{\mathrm{vertices}}  N^\partial_{\mathrm{edges}} &= 0 \end{array} \]
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
\[ \dim {M_h}  \dim {X_h} = N_{\mathrm{cells}}  2 N^i_{\mathrm{vertices}} = N^\partial_{\mathrm{edges}}  2 > 0 \]
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, \(U^*=0\).
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
\[ \forall v_h \in [Q^{1_{c,h}]}d, \quad \int_\Omega q^*_h \nabla \cdot v_h = 0 \] and thus the associated \(X_h\), \(M_h\) do not satisfy the condition.
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
\[ \forall v_h \in [P^{1_{c,h}]}d, \quad \int_\Omega q^*_h \nabla \cdot v_h = 0 \] and thus the associated \(X_h\), \(M_h\) do not satisfy the condition.
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
\[ \int_\Omega q^_h \nabla \cdot v_h^ \neq 0 \] where \(q_h^*\) is the spurious mode.
To do that we add the bubble function to the \(\poly{P}_1\) velocity space.
Recall the construction of finite elements on a reference convex \(\hat{K}\). We say that \(\hat{b}: \hat{K} \mapsto \RR\) is a bubble function if:

\(\hat{b} \in H^1_0(\hat{K})\)

\(0 \leq \hat{b}(\hat{x}) \leq 1, \quad \forall \hat{x} \in \hat{K}\)

\(\hat{b}(\hat{C}) = 1, \quad \mbox{where} \hat{C}\) is the barycenter of \(\hat{K}\)
 Example

The function
\[ \hat{b} = (d+1)^{d+1} \Pi_{i=0}^d\ \hat{\lambda}_i \] 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
\[ \hat{P} = [\poly{P}_1(\hat{K}) \oplus \mathrm{span} (\hat{b})]^d, \] and introduce
X_h &=& \Big\{ v_h \in [C^0(\bar{\Omega})]^d : \forall K \in \mathcal{T}_h, v_h \circ T_K \in \hat{P}; v_{h_{\partial \Omega}} = 0 \Big\}\\ M_h &=& P^1_{c,h}
The spaces \(X_h\) and \(M_h \cap L^2_0(\Omega)\) satisfy the compatibility condition uniformly in \(h\).
Suppose that \((u,p)\), solution of [prob:chmixte:1], is smooth enough, ie. \(u \in [H^2(\Omega)\)^d \cap [H^{1_0(\Omega)]}d] and \(p\in H^1(\Omega) \cap L^2_0(\Omega)\).
Then there exists a constant \(c\) such that for all \(h >0\)
\[ \ u u_h \{1,\Omega} + \pp_h\{0,\Omega} \leq c h (\u\{2,\Omega} + \p\{1,\Omega}) \] and if the Stokes problem is stabilizing then
\[ \uu_h\{0,\Omega} \leq c h^2 ( \u\{2,\Omega} +\p\_{1,\Omega}). \]
We say that the Stokes problem is stabilizing if there exists a constant \(c_S\) such that for all \(f \in [L^2(\Omega)\)^d], the unique solution \((u,p)\) of with \(g=0\) is such that:
\[ \u\{2,\Omega} + \p\{1,\Omega} \leq c_S \f\_{0,\Omega} \] A sufficient condition for stabilizing Stokes problem is that the \(\Omega\) is a polygonal convex in 2D or of class \(C^1\) in \(\RR^d, d=2,3\).
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 \[\begin{aligned} \label{eq:chmixte:39} X_h &=& [P^{2_{c,h}]}d\\ M_h &=& P^1_{c,h} \end{aligned} \]
The spaces \(X_h\) and \(M_h \cap L^2_0(\Omega)\) satisfy the compatibility condition uniformly in \(h\).
Suppose that \((u,p)\), solution of problem [prob:chmixte:1], is smooth enough, ie. \(u \in [H^3(\Omega)\)^d \cap [H^{1_0(\Omega)]}d] and \(p\in H^2(\Omega) \cap L^2_0(\Omega)\).
Then there exists a constant \(c\) such that for all \(h >0\)
\[ \ u u_h \{1,\Omega} + \pp_h\{0,\Omega} \leq c h^2 (\u\{3,\Omega} + \p\{2,\Omega}) \] and if the Stokes problem is stabilizing then
\[ \uu_h\{0,\Omega} \leq c h^3 ( \u\{3,\Omega} +\p\_{2,\Omega}). \]
 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 \[\begin{aligned} \label{eq:chmixte:42} X_h &=& [P^{{k}_{c,h}]}d\\ M_h &=& P^{k1}_{c,h} \end{aligned}\]] On Hypercubes \(\[\begin{aligned} \label{eq:chmixte:43} X_h &=& [Q^{k}_{c,h}\)^d\\ M_h &=& Q^{k1}_{c,h} \end{aligned} \] We then have
\[ \uu_h\{0,\Omega} + h ( \ u u_h \{1,\Omega} + \pp_h\{0,\Omega} ) \leq c h^{k+1} (\u\{k+1,\Omega} +\p\_{k,\Omega}) \]
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 \(k\)

Numerical validation: Test case
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 derived from the exact solution.