$\newcommand\ccancel[2][black]{{\color{#1}{\cancel{\color{black}{#2}}}}}$ $\newcommand\bvec[1]{\boldsymbol{#1}}$ $\newcommand{\Real}{\mathbb{R}}$ $\DeclareMathOperator{\DIFF}{d}$ $\DeclareMathOperator{\DIV}{div}$ $\DeclareMathOperator{\TDIV}{\bf \nabla \cdot}$ $\DeclareMathOperator{\TLAPLACIAN}{\bf \Delta}$ $\DeclareMathOperator{\GRAD}{\bf grad}$ $\DeclareMathOperator{\TGRAD}{\bf \nabla}$ $\DeclareMathOperator{\CURL}{\bf curl}$ $\DeclareMathOperator{\ROT}{rot_F}$ $\DeclareMathOperator{\VROT}{\bf rot_F}$ $\DeclareMathOperator{\tdot}{\bf :}$ $\DeclareMathOperator{\Tr}{Tr}$ $\DeclareMathOperator{\ID}{{\bf I}_{3,3}}$ $\newcommand{\uvec}[1]{\underline{\bvec{#1}}}$ $\newcommand{\ul}[1]{\underline{#1}}$ $ \newcommand{\Hh}{\mathcal{H}} \newcommand{\Mh}{\mathcal{M}_h} \newcommand{\Th}{\mathcal{T}_h} \newcommand{\Fh}{\mathcal{F}_h} \newcommand{\Eh}{\mathcal{E}_h} \newcommand{\Xh}{\mathcal{X}_h} \newcommand{\Vh}{\mathcal{V}_h} \newcommand{\FT}{\mathcal{F}_T} \newcommand{\ET}{\mathcal{E}_T} \newcommand{\EF}{\mathcal{E}_F} \newcommand{\FE}{\mathcal{F}_E} \newcommand{\VT}{\mathcal{V}_T} \newcommand{\VF}{\mathcal{V}_F} \newcommand{\VE}{\mathcal{V}_E} \newcommand{\Ech}{\mathcal{E}_h} \newcommand{\tE}{\bvec{t}_E} \newcommand{\nFE}{\bvec{n}_{FE}} \newcommand{\nF}{\bvec{n}_F} \newcommand{\nOmega}{\bvec{n}_{\Omega}} \newcommand{\wFE}{\omega_{FE}} \newcommand{\wTF}{\omega_{TF}} % Polynomial spaces \newcommand{\Poly}[2][]{\mathcal{P}_{#1}^{#2}} \newcommand{\Qoly}[2][]{\mathcal{Q}_{#1}^{#2}} \newcommand{\bQoly}[2][]{\boldsymbol{\mathcal{Q}}_{#1}^{#2}} \newcommand{\bPoly}[2][]{\boldsymbol{\mathcal{P}}_{#1}^{#2}} \newcommand{\Roly}[1]{\boldsymbol{\mathcal{R}}^{#1}} \newcommand{\Rolyb}[1]{\widehat{\boldsymbol{\mathcal{R}}}^{#1}} \newcommand{\Rolybc}[1]{\overline{\boldsymbol{\mathcal{R}}}^{#1}} \newcommand{\Goly}[1]{\boldsymbol{\mathcal{G}}^{#1}} \newcommand{\Golyb}[1]{{\boldsymbol{\mathcal{G}}}^{#1}} \newcommand{\EPoly}[1]{\widetilde{\boldsymbol{\mathcal{P}}}_{n,E}^{#1}(E)} \newcommand{\NE}[1]{\boldsymbol{\mathcal{N}}^{#1}} \newcommand{\NEb}[1]{{\boldsymbol{\mathcal{N}}}^{#1}} \newcommand{\RT}[1]{\boldsymbol{\mathcal{RT}}^{#1}} \newcommand{\RTb}[1]{\overline{\boldsymbol{\mathcal{RT}}}^{#1}} \newcommand{\FRTb}[1]{\widetilde{\boldsymbol{\mathcal{P}}}^{#1}(F)} \newcommand{\uHgradh}{\underline{X}_{\GRAD,h}^k} \newcommand{\uHcurlh}{\underline{\bvec{X}}_{\CURL,h}^k} \newcommand{\uHvh}{\underline{\bvec{X}}_{\TGRAD,h}^k} \newcommand{\uLsh}{{\mathcal{P}}^k(\Th)} $ $ \newcommand{\vertiii}[1]{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert #1 \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}} \newcommand{\norm}[2][]{\left \|#2 \right \|_{#1}} \newcommand{\seminorm}[2][]{\left |#2 \right |_{#1}} \newcommand{\pna}{P_{\TGRAD,T}^{k+1}} \newcommand{\uIH}[1][T]{\uvec{I}_{\TGRAD,#1}^k} \newcommand{\opnNa}[2][T]{\vertiii{#2}_{\TGRAD,#1}} \newcommand{\opnLt}[2][T]{\vertiii{#2}_{\bvec{L}^2,#1}} \newcommand{\uNah}{\underline{\TGRAD}_h^{k+1}} \newcommand{\uNaT}{\underline{\TGRAD}_T^{k+1}} \newcommand{\Dh}{\underline{D}_h^{k}} \newcommand{\DT}{\underline{D}_T^{k}} \newcommand{\normLs}[2][T]{\Vert #2 \Vert_{L^2,#1}} \newcommand{\spLt}[3][T]{\left (#2, #3 \right )_{\bvec{L}^2,#1}} \newcommand{\stLt}[3][T]{\text{s}_{\bvec{L}^2,#1} \left (#2, #3 \right )} \newcommand{\uIL}[1][T]{\uvec{I}_{\bvec{L}^2,#1}^k} \newcommand{\lproj}[2]{\pi_{\mathcal{P},#2}^{#1}} \newcommand{\AdjG}{\tilde{\mathcal{E}}_{\TGRAD,h}} \newcommand{\AdjL}{\tilde{\mathcal{E}}_{\TLAPLACIAN,h}} \newcommand{\normHNa}[1]{\Vert #1 \Vert_{\mu, \TGRAD, 1, h}} $

Discrete complexes for fluid dynamics

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.

Exterior calculus

Proxy over a $3$-dimensional domain

Example (exterior derivative of a $1$-form):

Dire que c'est un complexe

Navier-Stokes equations

\begin{equation*} \begin{aligned} \frac{\partial \bvec{u}}{\partial t} - \nu \Delta \bvec{u} + \nabla p + (\bvec{u} \cdot \nabla)\bvec{u} =&\, f,\\ \nabla \cdot \bvec{u} =&\, 0. \end{aligned} \end{equation*}

Using the identities:
  • $(\bvec{u} \cdot \nabla) \, \bvec{u} = \frac{1}{2} \nabla (\bvec{u} \cdot \bvec{u}) - \bvec{u} \times (\CURL \bvec{u})$.
  • $\Delta \bvec{u} = \GRAD \DIV \bvec{u} - \CURL (\CURL \bvec{u}) = - \CURL (\CURL \bvec{u})$.

\begin{equation*} \begin{aligned} \frac{\partial \bvec{u}}{\partial t} + \nu \CURL (\CURL \bvec{u}) + \GRAD P + (\CURL \bvec{u}) \times \bvec{u} =&\, f,\\ \DIV \bvec{u} =&\, 0, \end{aligned} \end{equation*} where $P := \frac{1}{2} \bvec{u} \cdot \bvec{u} + p$ is the Bernoulli pressure.

Through proxies:

We define: \begin{equation} \delta := (-1)^{n(k-1)+1} \star \DIFF \star . \end{equation}

  • If $\bvec{u} \in \Lambda^1(\Omega)$ and $p \in \Lambda^0(\Omega)$, \begin{equation*} \begin{aligned} \frac{\partial \bvec{u}}{\partial t} + \nu \delta \DIFF \bvec{u} + \DIFF P + \star \big( ( \star \DIFF \bvec{u} ) \wedge \bvec{u} \big) =&\, f,\\ \delta \bvec{u} =&\, 0. \end{aligned} \end{equation*}
  • If $\bvec{u} \in \Lambda^2(\Omega)$ and $p \in \Lambda^3(\Omega)$, \begin{equation*} \begin{aligned} \frac{\partial \bvec{u}}{\partial t} + \nu \DIFF \delta \bvec{u} + \delta P + \delta \bvec{u} \wedge \star \bvec{u} =&\, f,\\ \DIFF \bvec{u} =&\, 0. \end{aligned} \end{equation*}
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}

From D. N. Arnold. Finite Element Exterior Calculus, 2018

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 $\forall (\bvec{\tau},\bvec{v}, q, \chi) \in V^1_h \times V^2_h \times V^3_h \times \mathfrak{H}^3_h$,

  • Well-posedness.
  • Convergence of the linearized problem.
  • Preservation of structure:
    • Divergence free: $\Vert \nabla \cdot \bvec{u} \Vert = 0$.
    • 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: $\Vert \bvec{u}^{n+1} \Vert^2 = \Vert \bvec{u}^n \Vert^2 - 2 \nu \Delta t \Vert \bvec{\omega}^{n + \frac{1}{2}} \Vert^2$.

Applications

Polyhedral methods

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} \Rolyb{c,k}(T) = \begin{pmatrix} \DIV^{-1} \\ \DIV^{-1} \\ \DIV^{-1} \end{pmatrix} \circ \GRAD \Poly{0,k}(T),\quad \Rolybc{c,k}(T) = \lbrace W \in (\Roly{c,k}(T)^\intercal)^3, \Tr W = 0 \rbrace . \end{equation}

Explicit description available:

\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 $\uvec{C}_h$).

Decomposition of the curl operator

\begin{equation} \CURL \begin{pmatrix} u_1 \\ u_2 \\ u_n \end{pmatrix} = \begin{pmatrix} \partial_2 u_F - \partial_{n_F} u_2 \\ \partial_{n_F} u_1 - \partial_1 u_F \\ \partial_1 u_2 - \partial_2 u_1 \end{pmatrix} \end{equation}

\begin{equation} \underbrace{\begin{pmatrix} 0 \\ 0 \\ \partial_1 u_2 - \partial_2 u_1 \end{pmatrix}}_{\textcolor{red}{Intrinsic}} + \underbrace{\begin{pmatrix} \partial_2 u_F \\ - \partial_1 u_F \\ 0 \end{pmatrix}}_{\color{red}{Computable}} + \underbrace{\partial_{n_F} \begin{pmatrix} - u_2 \\ u_1 \\ 0 \end{pmatrix}}_{\color{red}{Extrinsic}} \end{equation}

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 add the intrinsic $\nabla$ on faces and edges: \begin{equation} \nabla_F : \begin{pmatrix} u_1 \\ u_2 \\ u_F \end{pmatrix} \rightarrow \begin{pmatrix} \partial_1 u_1 & \partial_2 u_1 \\ \partial_1 u_2 & \partial_2 u_2 \\ \partial_1 u_F & \partial_2 u_F \end{pmatrix} , \end{equation} \begin{equation} \nabla_E : \begin{pmatrix} u_E \\ u_1 \\ u_2 \end{pmatrix} \rightarrow \begin{pmatrix} \partial_E u_E \\ \partial_E u_1 \\ \partial_E u_2 \end{pmatrix}. \end{equation}

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}

Summary

Main results on $\nabla$

Example on the Stokes problem

Continuous solution

\begin{equation} \begin{aligned} -\mu \Delta \bvec{u} + \GRAD p =&\, \bvec{f},\\ \DIV \bvec{u} =&\, 0 . \end{aligned} \end{equation}

Weak formulation

\begin{equation} \mu \int_\Omega \GRAD \bvec{u} \tdot \GRAD \bvec{w} - \int_\Omega p\, \DIV \bvec{w} + \int_\Omega \DIV \bvec{u}\, q = \int_\Omega \bvec{f} \cdot \bvec{w} . \end{equation}

Discrete formulation

\begin{equation} \mu \left ( \uNah \uvec{v}_h, \uNah \uvec{w}_h \right )_{\bvec{L}^2,h} - \sum_{T \in \Th} \int_T \DT \uvec{w} \, p_T + \sum_{T \in \Th} \int_T \DT \uvec{v}_T \, q_T = \sum_{T \in \Th} \int_T \pna \uvec{w}_T \cdot \bvec{f} \end{equation}

Energy norm

\begin{equation} \normHNa{\uvec{v}_h} := \left ( \mu \norm[\nabla,h]{\uvec{v}_h}^2 + \mu \left ( \uNah \uvec{v}_h, \uNah \uvec{v}_h \right )_{\bvec{L}^2,h} \right )^{1/2} \end{equation}

Error estimate

\begin{equation} \begin{aligned} \mathcal{E}_h (\uvec{w}_h, \ul{q}_h) :=&\, \sum_{T \in \Th} \int_T \pna \uvec{w}_T \cdot \bvec{f} + \sum_{T \in \Th} \int_T \DT \uvec{w} \, \lproj{T}{k} p \\ &\quad - \mu \left ( \uNah (\uIH[h] \bvec{u}), \uNah \uvec{w}_h \right )_{\bvec{L}^2,h} - \sum_{T \in \Th} \int_T \DT (\uIH[h] \bvec{u})\, q_T \\ =&\, \AdjG(p \ID, \uvec{w}_h) - \stLt{\uNaT \uvec{w}_T}{\uIL (p \ID)} - \mu \AdjL(\bvec{u},\uvec{w}_h)\\ \lesssim&\, \, h^{k+1} \left ( \mu^{\frac{1}{2}} \seminorm[\bvec{H}^{(k+2,3)}(\Th)]{\bvec{u}} + \mu^{-\frac{1}{2}} \seminorm[H^{k+1}(\Th)]{p} + \mu^{-\frac{1}{2}} \seminorm[H^{k+2}(\Th)]{p} \right ) \normHNa{\uvec{w}_h} \end{aligned} \end{equation}

Thank you