Ma thèse s'intitule complexes discret pour les fluides incompressible.
Les complexes sont donc naturellement un objet central.
2ième passage
L'objectif est d'introduire les principaux résultats obtenu,
en particulier je vais essayer de ne pas rentrer dans les aspects les plus technique.
Je ne parlerais pas du premier chapitre de ma thèse sur les lois de Biot Savart.
L'idée est alors d'écrire les equations à l'aide des opérateurs différentiels apparaissant dans le complex.
Two complexes
De Rham:
Stokes:
Subcomplexes
$V^0$
$V^1$
$V^2$
$V^3$
$V^0_h$
$V^1_h$
$V^2_h$
$V^3_h$
$d^0$
$d^1$
$d^2$
$d^0_h$
$d^1_h$
$d^2_h$
$\pi_h^0$
$\pi_h^1$
$\pi_h^2$
$\pi_h^3$
Bounded projections
There exist some projectors $\pi_h^i : V^i \rightarrow V_h^i$ that:
Are bounded for the graph norm: $\exists c > 0,$ $\forall v \in V^i,$ $\Vert \pi_h^i v\Vert_{V} \leq c \Vert v \Vert_{V}$
(or better for the $L^2$-norm).
Commute with the differential operators.
Approximation properties
Errors are typically proportional to
\begin{equation} E(v) := \inf_{v_h \in V_h} \Vert v - v_h \Vert .\end{equation}
Finite Element Exterior Calculus
General principles:
Objects: Discretes spaces forming a subcomplex $V^0_h$, $V^1_h$, $V^2_h$ and $V^3_h$.
Operators: $d_h := d_{\vert V_h}$ and $\ccancel[blue]{\delta_h} := d_h^\star$.
Operators: $d_h := d_{\vert V_h}$ and $\delta_h := d_h^\star$.
$\delta_h$ is global and hard to analyse, it is replaced by $\omega^{i-1}$ such that:
\begin{equation}
\langle \omega^{i-1}, \sigma^{i-1} \rangle = \langle u^i, d^{i-1}_h \sigma^{i-1} \rangle, \forall \sigma^{i-1} \in V^{i-1}_h .
\end{equation}
Concrete spaces:
The method is independent of the exact choice of spaces.
For the De Rham complex two families of arbitrary degree on simplicial and cubical meshes already exist.
From
D. N. Arnold. Finite Element Exterior Calculus, 2018
Tools
Hodge decomposition:
Isomorphism of the cohomology:
Scheme for incompressible Navier-Stokes equations
Find $(\bvec{\omega},\bvec{u}, p, \phi) \in V^1_h \times V^2_h \times V^3_h \times \mathfrak{H}^3_h$ such that
$\forall (\bvec{\tau},\bvec{v}, q, \chi) \in V^1_h \times V^2_h \times V^3_h \times \mathfrak{H}^3_h$,
Pressure robustness: $\bvec{\omega}$ and $\bvec{u}$ do not change when changing the source by a gradient $\bvec{f} \rightarrow \bvec{f} + \nabla g$.
Energy: using a Crank-Nicolson discretization $\frac{\partial \bvec{u}^{n + \frac{1}{2}}}{\partial t} \approx \frac{\bvec{u}^{n+1} - \bvec{u}^n}{\Delta t}$ it holds:
$\Vert \bvec{u}^{n+1} \Vert^2 = \Vert \bvec{u}^n \Vert^2 - 2 \nu \Delta t \Vert \bvec{\omega}^{n + \frac{1}{2}} \Vert^2$.
Sketch of proofs
Well-posedness of the underlying Stokes problem:
Generalization with lower order terms introducing two functions $l_3 : V^1 \rightarrow V^2$ and $l_5 : V^2 \rightarrow V^2$:
Continuous well-posedness with a compactness argument.
Discrete Inf-sup stability with test functions constructed from the continuous solution operator.
Applications
Convergence towards an exact solution
\begin{equation}
u = \begin{pmatrix}
-a (e^{a x}\sin(a y + d z) + e^{a z}\cos(a x + d y))e^{-d^2 t}\\
-a (e^{a y}\sin(a z + d x) + e^{a x}\cos(a y + d z))e^{-d^2 t}\\
-a (e^{a z}\sin(a x + d y) + e^{a y}\cos(a z + d x))e^{-d^2 t}
\end{pmatrix}
\end{equation}
An element $\uvec{u}$ of the discrete space $\ul{X}_h$ is a collection of local functions over the cells, faces, edges and vertices:
\begin{equation}
\uvec{u} := \lbrace ((u_V)_{V \in \Vh}, (u_E)_{E \in \Eh}, (u_F)_{F \in \Fh}, (u_T)_{T \in \Th}) \rbrace.
\end{equation}
Interpolator: $\ul{I}_h : X^k \rightarrow \ul{X}_h^k$.
Typically designed to commute with the differential operator.
Reconstruction: $R : \ul{X}_h^k \rightarrow X^k$.
Typically a right inverse of the interpolator.
Example on the De Rham complex
The last operator is $0$, and does not constrain the local spaces.
The penultimate operator is $\DIV$, the related Stokes formula is:
\begin{equation}
\int_\Omega \DIV \bvec{\omega}\, q = - \int_\Omega \bvec{\omega} \cdot \GRAD q + \int_{\partial \Omega} \bvec{\omega} \cdot \bvec{n} \, q.
\end{equation}
The image space suggests $\DIV \bvec{\omega} \in \Poly{k}(T) \implies q \in \Poly{k}(T)$.
Two terms on the RHS:
$\int_\Omega \bvec{\omega} \GRAD q$ requires the moments on $\GRAD \Poly{k}(T)$ (completed to $\NE{k}(T)$).
$\int_{\partial \Omega} \bvec{\omega} \cdot \bvec{n} \, q$ requires the moments on $\Poly{k}(F)$.
The local spaces for the discrete $L^2(\Omega)$ are:
\begin{align}
V \rightarrow&\, \emptyset \\
E \rightarrow&\, \emptyset \\
F \rightarrow&\, \emptyset \\
T \rightarrow&\, \Poly{k}(T) ,
\end{align}
and for the discrete $\bvec{H}(\text{div},\Omega)$:
\begin{align}
V \rightarrow&\, \emptyset \\
E \rightarrow&\, \emptyset \\
F \rightarrow&\, \Poly{k}(F) \\
T \rightarrow&\, \Goly{k-1}(T) \oplus \Goly{c,k}(T) .
\end{align}
The discrete differential operator $D_T$ on a cell $T$ is such that
\begin{equation}
\int_T D_T \uvec{w}_T \, q = - \int_T \bvec{\omega}_{\Goly{},T} \cdot \GRAD q + \sum_{F \in \FT} \wTF \int_F \omega_F \, q , \quad \forall q \in \Poly{k}(T).
\end{equation}
From
D. A. Di Pietro and J. Droniou. An Arbitrary-Order Discrete de Rham
Complex on Polyhedral Meshes: Exactness, Poincaré Inequalities, and Con-
sistency, 2021
Stokes complex
Stokes formula for $\nabla$:
\begin{equation}
\int_\Omega \nabla \bvec{\omega} \tdot \bvec{\sigma} = - \int_\Omega \bvec{\omega} \cdot \nabla \cdot \bvec{\sigma} + \int_{\partial \Omega} \bvec{\omega} \cdot (\bvec{\sigma}\, \bvec{n})\ .
\end{equation}
We need:
New polynomial spaces like $\Roly{k}(T)$ on tensor-valued functions.
The data of all components on faces.
We keep the degree of freedom of the fully discrete De Rham complex and add those required by $\nabla$.
A natural space for the adjoint of $\nabla: \NE{k+1}(T) \rightarrow \bPoly{k}(T;\Real^{3,3})$
\begin{equation}
\Rolybc{c,k} = \left \lbrace \begin{matrix}
yz C_1 + y \gamma - z \beta \\
xz C_2 - x \gamma + z \lambda \\
xy C_3 + x \beta - y \lambda
\end{matrix},\quad
\begin{matrix}
\lambda \in \Poly{k-2}[Y,Z],\; \beta \in \Poly{k-2}[X,Z],\; \gamma \in \Poly{k-2}[X,Y], \\
C_i \in \Poly{k-3}[X,Y,Z],\; C_1 + C_2 + C_3 = 0.
\end{matrix}
\right \rbrace
\end{equation}
Degrees of freedom of the fully Discrete De Rham
$\uHgradh$:
\begin{align}
V \rightarrow&\, \Real \\
E \rightarrow&\, \Poly{k-1}(E) \\
F \rightarrow&\, \Poly{k-1}(F) \\
T \rightarrow&\, \Poly{k-1}(T)
\end{align}
$\uHcurlh$:
\begin{align}
V \rightarrow&\, \emptyset \\
E \rightarrow&\, \Poly{k}(E) \\
F \rightarrow&\, \RT{k}(F) \\
T \rightarrow&\, \RT{k}(T)
\end{align}
$\uHvh$ :
\begin{align}
V \rightarrow&\, \emptyset \\
E \rightarrow&\, \emptyset \\
F \rightarrow&\, \Poly{k}(F) \\
T \rightarrow&\, \NE{k}(T)
\end{align}
$\uLsh$:
\begin{align}
V \rightarrow&\, \emptyset \\
E \rightarrow&\, \emptyset \\
F \rightarrow&\, \emptyset \\
T \rightarrow&\, \Poly{k}(T)
\end{align}
Adding the missing components on the faces of $\uHvh$ gives $F \rightarrow \begin{matrix} \Poly{k}(F) \\ \bPoly{k}(F; \Real^2) \end{matrix}$.
They must have an pre-image by $\ul{d}_h$ (here by $\uvec{C}_h$).
The extrinsic components must be included with the same spaces in $\uHcurlh$.
The Stokes formula for the computable components is:
\begin{equation}
\int_F \VROT u_F \cdot \bvec{r}_F = \int_F u_F \ROT \bvec{r}_F + \sum_{E \in \EF} \wFE \int_E (\bvec{u}_E \cdot \nF) (\bvec{r}_F \cdot \tE) .
\end{equation}
Hence we must add all components on the edges of $\uHcurlh$.
The local spaces of $\uHcurlh$ now include:
\begin{equation}
V \rightarrow\, \emptyset ,\
E \rightarrow\, \begin{matrix} \Poly{k}(E)\\ \bPoly{k}(E;\Real^2) \end{matrix} ,\
F \rightarrow\, \begin{matrix} \RT{k}(F) \\ \bPoly{k}(F;\Real^2) \\ \Poly{k-1}(F) \end{matrix} ,\
T \rightarrow\, \RT{k}(T) .
\end{equation}
The new components of $\uHcurlh$ must have a pre-image by $\ul{d}_h$ (here $\uvec{G}_h$).
They decompose into extrinsic and computable components.
The local spaces of $\uHgradh$ now include:
\begin{equation}
V \rightarrow\, \begin{matrix} \Real \\ \Real^3 \end{matrix} ,\
E \rightarrow\, \begin{matrix} \Poly{k-1}(E)\\ \bPoly{k}(E;\Real^2) \end{matrix} ,\
F \rightarrow\, \begin{matrix} \Poly{k-1}(F)\\ \Poly{k-1}(F) \end{matrix} ,\
T \rightarrow\, \Poly{k-1}(T).
\end{equation}
Problem:
$\uHvh$ is not smooth enough to get Poincaré inequality (equivalently the discrete $\nabla$ does not carry enough information).
We had to add a continuous $1$-skeleton on $\uHvh$ and its pre-image.
The full picture is:
$\uHgradh$:
\begin{align}
V \rightarrow&\, \begin{matrix} \Real \\ \Real^3 \end{matrix} \\[1em]
E \rightarrow&\, \begin{matrix} \Poly{k-1}(E)\\ \bPoly{k}(E;\Real^2) \end{matrix} \\[1em]
F \rightarrow&\, \begin{matrix} \Poly{k-1}(F)\\ \Poly{k-1}(F) \end{matrix} \\[1em]
T \rightarrow&\, \Poly{k-1}(T)
\end{align}
$\uHcurlh$:
\begin{align}
V \rightarrow&\, \begin{matrix} \Real^3 \\ \Real^3 \end{matrix} \\[1em]
E \rightarrow&\, \begin{matrix} \bPoly{k+1}(E;\Real^3)\\ \bPoly{k}(E;\Real^3) \end{matrix} \\[1em]
F \rightarrow&\, \RT{k}(F) \\[1em]
T \rightarrow&\, \RT{k}(T)
\end{align}
$\uHvh$ :
\begin{align}
V \rightarrow&\, \Real^3 \\[1em]
E \rightarrow&\, \bPoly{k+1}(E;\Real^3) \\[1em]
F \rightarrow&\, \begin{matrix} \RT{k}(F) \\ \bPoly{k}(F;\Real^2) \\ \Poly{k-1}(F) \end{matrix} \\[1em]
T \rightarrow&\, \NE{k}(T)
\end{align}
$\uLsh$:
\begin{align}
V \rightarrow&\, \emptyset \\[1em]
E \rightarrow&\, \emptyset \\[1em]
F \rightarrow&\, \emptyset \\[1em]
T \rightarrow&\, \Poly{k}(T)
\end{align}
Poincaré inequality
For all $\uvec{w}_h \in \uHvh$ such that $\sum_{T \in \Th} \int_{T} \pna \uvec{w}_T = 0$
\begin{equation}
\opnNa[h]{\uvec{w}_h} \lesssim \opnLt[h]{\uNah \uvec{w}_h}.
\end{equation}
Right-inverse for the divergence
If the mesh is quasi-uniform then for all $\ul{p}_h \in \uLsh$ there is $\uvec{w}_h \in \uHvh$ such that $\Dh \uvec{w}_h = \ul{p}_h$ and
\begin{equation}
\opnNa[h]{\uvec{w}_h} + \opnLt[h]{\uNah \uvec{w}_h} \lesssim \normLs[h]{\ul{p}_h}.
\end{equation}
Adjoint consistency
\begin{equation}
\AdjG(\bvec{V},\uvec{w}_h) := \sum_{T \in \Th} \left (\spLt{\uIL \bvec{V}}{\uNaT \uvec{w}_T} + \int_T \TDIV \bvec{V} \cdot \pna \uvec{w}_T \right )
\end{equation}
For all $\bvec{V} \in \bvec{C}^0(\overline{\Omega}) \cap \bvec{H}^1_0(\Omega)$ such that $\bvec{V} \in \bvec{H}^{k+2}(\Th)$
and all $\uvec{w}_h \in \uHvh$, it holds:
\begin{equation}
\seminorm{\AdjG(\bvec{V},\uvec{w}_h)} \lesssim h^{k+1} \left ( \seminorm[\bvec{H}^{k+1}]{\bvec{V}} + \seminorm[\bvec{H}^{k+2}]{\bvec{V}} \right )
\left ( \opnNa[h]{\uvec{w}_h} + \opnLt[h]{\uNah \uvec{w}_h} \right ).
\end{equation}
\begin{equation}
\AdjL(\bvec{v},\uvec{w}_h) := \sum_{T \in \Th} \left ( \int_T \TLAPLACIAN \bvec{v} \cdot \pna \uvec{w}_T + \spLt{\uNaT \uIH\bvec{v}}{\uNaT \uvec{w}_T} \right ) .
\end{equation}
For all $\bvec{v} \in \bvec{H}^2(\Omega) \cap \bvec{C}^1(\overline{\Omega})$ such that $\TGRAD \bvec{v} \cdot \nOmega = 0$
and $\bvec{v} \in \bvec{H}^{k+2}(\Th)$
and for all $\uvec{w}_h \in \uHvh$, it holds:
\begin{equation}
\seminorm{\AdjL(\bvec{v},\uvec{w}_h)} \lesssim h^{k+1} \seminorm[\bvec{H}^{(k+2,3)}]{\bvec{v}}
\opnLt[h]{\uNah \uvec{w}_h}.
\end{equation}