# 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
$\left\{\begin{array}[c]{rl} -\Delta u + \nabla p & = f\ \mbox{ in } \Omega\\ \nabla \cdot u & = 0\ \mbox{ in } \Omega \end{array}\right.$

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).

### 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 Navier-Stokes for Newtonian fluids

The Stokes model is the basis for fluid mechanics models and is a simplication of the Navier-Stokes 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 well-posedness of these problems results from a so-called inf-sup condition which is not automatically transfered at the discrete level.

In practice In order to ensure that the finite element approximation is well-posed, we will need to choose approximation spaces that satisfy a compatibility condition that ensures that a discrete inf-sup condition is satisfied.

### 2.1. Abstract Continuous Setting

Denote

• $X$ and $M$ two Hilbert spaces.

• 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:

Problem Mixed 1

Look for $(u,p) \in X \times M$ such that

$\left\{ \begin{array}[c]{rl} a(u,v) + b(v,p) & = f(v), \quad \forall v \in X\\ b(u,q) & = g(q), \quad \forall q \in M \end{array} \right.$

#### 2.1.1. Definition of a saddle point problem

Definition

If the bilinear form $a$ is symmetric and positive on $X\times X$, we say that the mixed problem 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 :

### 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

$c((u,p),(v,q)) = a(u,v)+b(v,p)+b(u,q)$

and $h\in \mathcal{L}(V,\RR)$ such that

$h(v,q) = f(v)+g(q)$

then problem Problem Mixed 1 reads

Problem

Look for $(u,p) \in V$ such that

$\begin{array}[c]{rl} c((u,p), (v,q)) & = h(v,q), \quad \forall (v,q) \in V \end{array}$
Theorem

We suppose that $a$ is coercive over $X$, the Problem is well-posed if and only if the bilinear form $b$ satisfies the following inf-sup condition:

there exists $\beta > 0$ such that

$\inf_{q \in M} \sup_{v \in X} \frac{b(v,q)}{||v||_X ||q||_M} \geq \beta$
 Lax-Milgram provides only a sufficient condition for well-posedness. The form $c$ in Problem does not satisfy Lax-Milgram.

Let’s introduce the so-called Lagrangian $l \in \mathcal{L}(X\times M, \RR)$ defined by

$l(v,q) = \frac{1}{2} a(v,v) + b(v,q) - f(v) - g(q)$
Definition

We say that the point $(u,p)\in X\times M$ is a saddle point of $l$ if

$\forall (v,q) \in X\times M, \quad l(u,q) \leq l(u,p) \leq l(v,p)$
Proposition

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 Problem Mixed 1.

## 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:

Formulation of the Abstract Discrete Problem

Look for $(u_h,p_h) \in X_h \times M_h$ such that

$\left\{ \begin{array}[c]{rl} a(u_h,v_h) + b(v_h,p_h) & = f(v_h), \quad \forall v_h \in X_h\\ b(u_h,q_h) & = g(q_h), \quad \forall q_h \in M_h \end{array} \right.$
Theorem

We suppose that $a$ is coercive over $X$ and that $X_h \subset X$ and $M_h \subset M$.

Then the Formulation of the Abstract Discrete Problem is well-posed if and only if the following discrete inf-sup 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 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 Babuska-Brezzi (BB) or Ladyhenskaya-Babuska-Brezzi (LBB).

Regarding error analysis, we have the following lemma

lemma

Thanks to the Lemma of Céa applied to Saddle-Point Problems, the unique solution $(u,p)$ of problem Formulation of the Abstract Discrete Problem satisfies

$\begin{array}[c]{rl} ||u-u_h||_X & \leq c_{1h} \inf_{v_h \in X_h} ||u-v_h||_X + c_{2} \inf_{q_h \in M_h} ||p-q_h||_M\\ ||p-p_h||_M & \leq c_{3h} \inf_{v_h \in X_h} ||u-v_h||_X + c_{4h} \inf_{q_h \in M_h} ||p-q_h||_M \end{array}$

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.

### 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

$\begin{bmatrix} \mathcal{A} & \mathcal{B}^T\\ \mathcal{B} & 0 \end{bmatrix} \begin{bmatrix} U \\ P \end{bmatrix} = \begin{bmatrix} F\\ G \end{bmatrix}$

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

• $F_i=f(\phi_i)$

• $G_k=g(\psi_k)$

 Since $a$ is symmetric and coercive, $\mathcal{A}$ is symmetric positive definite The matrix of the system is symmetric but not positive The inf-sup condition  is equivalent to the fact that $\mathcal{B}$ is of maximum rank, i.e. $\ker(\mathcal{B}^T) = \{0 \}$. From theorem Theorem, the matrix of the system  is invertible
 When the inf-sup is not satisfied The counter examples when the inf-sup condition  is not satisfied(e.g. $\mathcal{B}$ is not maximum rank ) occur usually in two cases: Locking $\dim {M_h} > \dim {X_h}$: the space of pressure is too large for the matrix $\mathcal{B}$ to be maximum rank. In that case $\mathcal{B}$ is injective ($\ker(\mathcal{B}) = \{0\})$. We call this locking. Spurious modes there exists a vector $Q^* \neq 0$ in $\ker(\mathcal{B}^T)$. The discrete field$q^*_h$ in $M_h$, $q^*_h=\sum_{k=1}^{N_p} Q^*_k \psi_k$, associated is called a spurious mode. $q^*_H$ is such that $b(v_h,q^*_h)=0.$

We now introduce the Uzawa matrix as follows

Définition: Matrice d’Uzawa

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.

## 4. Mixed finite element for Stokes

### 4.1. Variational formulation

• 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.

Problem

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 recover the formulation of Problem Mixed 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.

### 4.2. Finite element approximation

Denote $X_h \subset [H^1_0(\Omega)]^d$ and $M_h \subset L^2_0(\Omega)$

Problem

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 Theorem is well-posed if and only if $X_h$ and $M_h$ are such that there exists $\beta_h > 0$
$\inf_{q_h \in M_h} \sup_{v_h \in X_h} \frac{\int_\Omega q_h \nabla \cdot v_h}{||v_h||_{X_h} ||q_h||_{M_h}} \geq \beta_h$

### 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

$\begin{array}[c]{rl} N_{\mathrm{cells}} - N_{\mathrm{edges}} + N_{vertices} &= 1-I\\ N^\partial_{\mathrm{vertices}} - N^\partial_{\mathrm{edges}} &= 0 \end{array}$

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, $U^*=0$.

#### 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

$\forall v_h \in [Q^1_{c,h}]^d, \quad \int_\Omega q^*_h \nabla \cdot v_h = 0$

and thus the associated $X_h$, $M_h$ do not satisfy the condition.

#### 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

$\forall v_h \in [P^1_{c,h}]^d, \quad \int_\Omega q^*_h \nabla \cdot v_h = 0$

and thus the associated $X_h$, $M_h$ do not satisfy the condition.

### 4.4. Mini-Element

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.

Definition: Mini-Element

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:

• $\hat{b} \in H^1_0(\hat{K})$

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

• $\hat{b}(\hat{C}) = 1, \quad \mbox{where} \quad \hat{C}$ 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

\begin{aligned} 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} \end{aligned}
lemma

The spaces $X_h$ and $M_h \cap L^2_0(\Omega)$ satisfy the compatibility condition  uniformly in $h$.

Theorem

Suppose that $(u,p)$, solution of Problem Mixed 1, is smooth enough, ie. $u \in [H^2(\Omega)]^d \cap [H_0^1(\Omega)]^d$ and $p\in H^1(\Omega) \cap L_0^2(\Omega)$.

Then there exists a constant $c$ such that for all $h >0$

$\| u- u_h \|_{1,\Omega} + \|p-p_h\|_{0,\Omega} \leq c h (\|u\|_{2,\Omega} + \|p\|_{1,\Omega})$

and if the Stokes problem is stabilizing then

$\|u-u_h\|_{0,\Omega} \leq c h^2 ( \|u\|_{2,\Omega} +\|p\|_{1,\Omega}).$
Definition: Stabilizing Stokes problem

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 $\RR^d, d=2,3$.

### 4.5. Taylor-Hood Element

The mini-element solved the compatibility condition problem, but the error estimation in equation is not optimal in the sense that

1. the pressure space is sufficiently rich to enable a $h^2$ convergence in the pressure error,

2. but the velocity space is not rich enough to ensure a $h^2$ convergence in the velocity error.

The idea of the Taylor-Hood 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}
lemma

The spaces $X_h$ and $M_h \cap L^2_0(\Omega)$ satisfy the compatibility condition  uniformly in $h$.

Theorem

Suppose that $(u,p)$, solution of problem Problem Mixed 1, is smooth enough, ie. $u \in [H^3(\Omega)]^d \cap [H_0^1(\Omega)]^d$ and $p\in H^2(\Omega) \cap L_0^2(\Omega)$.

Then there exists a constant $c$ such that for all $h >0$

$\| u- u_h \|_{1,\Omega} + \|p-p_h\|_{0,\Omega} \leq c h^2 (\|u\|_{3,\Omega} + \|p\|_{2,\Omega})$

and if the Stokes problem is stabilizing then

$\|u-u_h\|_{0,\Omega} \leq c h^3 ( \|u\|_{3,\Omega} +\|p\|_{2,\Omega}).$
Generalized Taylor-Hood element

We consider the mixed finite elements $\poly{P}_k/\poly{P}_{k-1}$ and $\poly{Q}_k/\poly{Q}_{k-1}$ 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^{k-1}_{c,h} \end{aligned}

On Hypercubes

\begin{aligned} \label{eq:chmixte:43} X_h &=& [Q^{k}_{c,h}]^d\\ M_h &=& Q^{k-1}_{c,h} \end{aligned}

We then have

$\|u-u_h\|_{0,\Omega} + h ( \| u- u_h \|_{1,\Omega} + \|p-p_h\|_{0,\Omega} ) \leq c h^{k+1} (\|u\|_{k+1,\Omega} +\|p\|_{k,\Omega})$

There are other stable discretization spaces

• Discrete inf-sup condition: dictates the choice of spaces

• Inf-sup stables spaces:

• $\mathbb Q_k$-$\mathbb Q_{k-2}$, $\mathbb Q_k$-$\mathbb Q^{disc}_{k-2}$

• $\mathbb P_k$-$\mathbb P_{k-1}$, $\mathbb P_k$-$\mathbb P_{k-2}$, $\mathbb P_k$-$\mathbb P^{disc}_{k-2}$

• Discrete inf-sup 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

$\begin{array}{r c l} \mathbf{u}(x,y) & = & \left(1 - e^{\lambda x } \cos (2 \pi y), \frac{\lambda}{2 \pi} e^{\lambda x } \sin (2 \pi y)\right)^T \\ p(x,y) & = & -\frac{e^{2 \lambda x}}{2} \\ \lambda & = & \frac{1}{2 \nu} - \sqrt{\frac{1}{4\nu^2} + 4\pi^2}. \end{array}$

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

$\mathbf{f} = \left( e^{\lambda x} \left( \left( \lambda^2 - 4\pi^2 \right) \nu \cos (2\pi y) - \lambda e^{\lambda x} \right), e^{\lambda x} \nu \sin (2 \pi y) (-\lambda^2 + 4 \pi^2) \right)^T$

Dirichlet boundary conditions are manufactured from the exact solution.

1. An euclidian space which is complete for the norm induced by the scalar product