mathdefs.sty

% This package provides useful (and recurrent) math definitions, as well as some
% shortcuts for environments such as figure
% The package contains also math macros by Jerome Bonelle for the original CDO
%
% Author: Riccardo Milani
%
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{mathdefs}[2018/02/28]

\RequirePackage{amssymb}
%\RequirePackage{amsmath}% Already loaded by mathtools
\RequirePackage{mathtools}
\RequirePackage{amsthm}
\RequirePackage{stmaryrd}% for double square brackets
\RequirePackage{xspace}
%\RequirePackage{empheq}% braces inside subequations
%
% New commands
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                   SECTION: JB MACROS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%=======================
% Math shortcuts
%=======================
\newcommand{\vect}[1]{\underline{#1}}
\newcommand{\tensor}[1]{\underline{\underline{#1}}}
\newcommand{\mesure}[1]{\lvert #1 \rvert}
\newcommand{\tange}[1]{\vect{\tau}_{#1}}
\newcommand{\norm}{\vect{n}}
\newcommand{\norma}[1]{\norm_{#1}}
\newcommand{\incid}[2]{\iota_{#1,#2}}
\newcommand{\mean}[1]{\overline{#1}}
%\newcommand{\zmean}{ \smallint\negmedspace{\scriptscriptstyle=0}}
\newcommand{\zmean}{\perp\mathbf{1}}
%\newcommand{\jump}[1]{\lvert\negmedspace\!\!< #1  >\!\!\negmedspace\rvert}
\newcommand{\integ}[1]{\int\limits_{#1}}%\,}
%\newcommand{\dual}[1]{ {#1}^{\ast} }
\newcommand{\twistm}[1]{ \tilde{#1} }
\newcommand{\twist}[1]{ \widetilde{#1} }
\newcommand{\transpose}[1]{ {#1}^{\textsc{t}} }
\newcommand{\dif}{\,\mathrm{d}}
\newcommand{\Leq}{\lesssim}
\newcommand{\pond}[1]{\varpi_{#1}}
\newcommand{\usc}[1]{^{\textsc{#1}}}
\newcommand{\EB}{\mathrm{\textsc{EB}}}
%
% Sets
\newcommand{\tq}{\,|\,}
\newcommand{\Set}[1]{\lbrace #1 \rbrace}
\newcommand{\Factor}[1]{\left( #1 \right)}
\newcommand{\Factorb}[1]{\left[ #1 \right]}
\newcommand{\ar}[3]{\ensuremath{\begin{bmatrix} #1 \\ #2 \\ #3 \end{bmatrix}}}
%
\newcommand{\Ddi}{\boldsymbol{\delta}}
\newcommand{\In}{\mathrm{\textsc{i}}}
\newcommand{\Bd}{\mathrm{\textsc{b}}}
\newcommand{\pb}{\mathrm{pb}}
\newcommand{\bc}{\mathrm{bc}}
\newcommand{\harm}{\mathsf{h}}
\newcommand{\vb}{\mathsf{vb}}
\newcommand{\cb}{\mathsf{cb}}
\newcommand{\bnd}{\partial}
\newcommand{\Dom}{\Omega}
\newcommand{\pOm}{\bnd\Dom}
\newcommand{\closure}{\overline{\bnd}}
%
\newcommand{\rhs}{\mathsf{RHS}}
\newcommand{\mom}{\mathsf{mom}}
\newcommand{\mas}{\mathsf{mas}}
\newcommand{\vor}{\mathsf{vor}}
\newcommand{\BC}{\mathsf{B \! C}}
\newcommand{\Src}{\mathsf{S}}
\newcommand{\SrcP}{\Src_{\mathsf{p}}}
\newcommand{\SrcD}{\Src_{\mathsf{d}}}


%=======================
% Variables
%=======================
% Continuous setting
\newcommand{\Oex}{\mathrm{1}}
\newcommand{\Vit}{u}
\newcommand{\Vex}{\vect{\Vit}}
\newcommand{\Vort}{\omega}
\newcommand{\Wex}{\vect{\Vort}}
\newcommand{\Pex}{p}
\newcommand{\Gex}{\vect{g}}
\newcommand{\Gdual}{\Gex^\zeta}
\newcommand{\Fex}{\vect{\phi}}
\newcommand{\Fdual}{\vect{\phi}^\zeta}
\newcommand{\Sex}{s}
\newcommand{\Wsex}{\vect{\Vort}^{\ast}}
\newcommand{\Wsdi}{\Wdi^{\ast}}
\newcommand{\Psex}{\Pex^{\ast}}
\newcommand{\Foex}{\vect{f}}
\newcommand{\Fosex}{\vect{f}^{\ast}}
\newcommand{\Posex}{\theta}
\newcommand{\Povex}{\vect{\psi}}
% % Test functions
\newcommand{\Vtest}{\vect{v}}
\newcommand{\Vttest}{\vect{w}}
\newcommand{\Ptest}{q}
\newcommand{\Pttest}{r}

% Discrete setting
\newcommand{\Odi}{\boldsymbol{1}}
\newcommand{\Vdi}{\mathbf{\Vit}}
\newcommand{\Wdi}{\boldsymbol{\Vort}}
\newcommand{\Pdi}{\mathbf{\Pex}}
\newcommand{\Psdi}{\Pdi^{\ast}}
\newcommand{\Gdi}{\mathbf{g}}
\newcommand{\Fdi}{\boldsymbol{\phi}}
\newcommand{\Sdi}{\mathbf{\Sex}}
\newcommand{\Sgdi}{\boldsymbol{\sigma}}
\newcommand{\Fodi}{\mathbf{f}}
\newcommand{\Posdi}{\boldsymbol{\theta}}
\newcommand{\Povdi}{\boldsymbol{\psi}}
\newcommand{\Taudi}{\boldsymbol{\tau}}
\newcommand{\Xdi}{\mathbf{x}}
\newcommand{\Ydi}{\mathbf{y}}

% Auxiliary variables
\newcommand{\Adi}{\mathbf{a}}
\newcommand{\Bdi}{\mathbf{b}}
\newcommand{\Qdi}{\mathbf{q}}
\newcommand{\Vvdi}{\mathbf{w}}
\newcommand{\Zdi}{\boldsymbol{\zeta}}

% Phenomenological parameters
\newcommand{\rhop}{(\rho)}
\newcommand{\rhom}{\rho^{{\scriptscriptstyle -1}}}
\newcommand{\rhomp}{(\rho)^{{\scriptscriptstyle -1}}}
\newcommand{\Visc}{\nu}
\newcommand{\Viscp}{(\Visc)}
\newcommand{\Viscm}{\Visc^{{\scriptscriptstyle -1}}}
\newcommand{\Viscpm}{(\Visc)^{{\scriptscriptstyle -1}}}
\newcommand{\CondTh}{\kappa}
\newcommand{\CondThp}{(\CondTh)}
\newcommand{\CondThm}{\CondTh^{{\scriptscriptstyle -1}}}
\newcommand{\CondThpm}{(\CondTh)^{{\scriptscriptstyle -1}}}
\newcommand{\TeTh}{\tensor{\CondTh}}
\newcommand{\TeThav}{\tensor{\widehat \CondTh}} % CondTh Lipschitz
\newcommand{\alpham}{\alpha^{{\scriptscriptstyle -1}}}
\newcommand{\alphap}{(\alpha)}
\newcommand{\alphapm}{(\alpha)^{{\scriptscriptstyle -1}}}

%=======================
% Mesh entities (also called chain spaces)
%=======================
\newcommand{\vtx}{\mathrm{v}}
\newcommand{\edge}{\mathrm{e}}
\newcommand{\face}{\mathrm{f}}
\newcommand{\cell}{\mathrm{c}}
\newcommand{\x}{\mathrm{x}}

\newcommand{\vtxd}{\twistm{\vtx}}
\newcommand{\edged}{\twistm{\edge}}
\newcommand{\faced}{\twistm{\face}}
\newcommand{\celld}{\twistm{\cell}}
\newcommand{\yd}{\twistm{\mathrm{y}}}
\newcommand{\Kd}{\twist{K}}

\newcommand{\pt}{\vect{x}}
\newcommand{\nent}[2]{\mathrm{n}_{#1,#2}}
\newcommand{\edgev}{\vect{\edge}}
\newcommand{\edgedv}{\vect{\edged}}
\newcommand{\facev}{\vect{\face}}
\newcommand{\facedv}{\vect{\faced}}

% Set of mesh entities
\newcommand{\Mesh}{\mathrm{M}}
\newcommand{\xh}{\mathrm{X}}
\newcommand{\vtxh}{\mathrm{V}}
\newcommand{\edgeh}{\mathrm{E}}
\newcommand{\faceh}{\mathrm{F}}
\newcommand{\cellh}{\mathrm{C}}
%
\newcommand{\Meshd}{\twist{\Mesh}}
\newcommand{\yhd}{\twist{\mathrm{Y}}}
\newcommand{\vtxhd}{\twist{\vtxh}}
\newcommand{\edgehd}{\twist{\edgeh}}
\newcommand{\facehd}{\twist{\faceh}}
\newcommand{\cellhd}{\twist{\cellh}}

% Sous-maillage
\newcommand{\svol}{\mathfrak{s}} % subsimplex
\newcommand{\Svol}[1]{\svol^{\textsc{#1}}}
\newcommand{\Smesh}{\mathfrak{S}}
\newcommand{\pvol}{\mathfrak{p}} % partition
\newcommand{\Pvol}{\mathfrak{P}}
\newcommand{\Part}[1]{\Pvol_{#1}}
\newcommand{\Pmesh}[1]{\Part{#1}}
\newcommand{\PVmesh}{\Pmesh{\textsc{v},\cell}}
\newcommand{\PEmesh}{\Pmesh{\textsc{e},\cell}}
\newcommand{\PFmesh}{\Pmesh{\textsc{f},\cell}}
\newcommand{\PEFmesh}{\Pmesh{\textsc{ef},\cell}}
\newcommand{\tef}{\mathfrak{t}_{\edge\face}}
\newcommand{\pef}{\pvol_{\edge\face,\cell}}
\newcommand{\pfc}{\pvol_{\face,\cell}}
%
\newcommand{\frakc}{\mathfrak{c}}
\newcommand{\frakt}{\mathfrak{t}}
\newcommand{\frake}{\mathfrak{e}}
\newcommand{\frakv}{\mathfrak{v}}
\newcommand{\frakC}{\mathfrak{C}}
\newcommand{\frakT}{\mathfrak{T}}
\newcommand{\frakE}{\mathfrak{E}}
\newcommand{\frakV}{\mathfrak{V}}
%
\newcommand{\Step}{h}
\newcommand{\StepMesh}{\Step_{\scriptscriptstyle{\Mesh}}}
\newcommand{\Stepc}{\Step_{\cell}}

%=======================
% DoF spaces (also called cochain spaces)
%=======================
\newcommand{\vtxsp}{\mathcal{V}}
\newcommand{\edgesp}{\mathcal{E}}
\newcommand{\facesp}{\mathcal{F}}
\newcommand{\cellsp}{\mathcal{C}}

\newcommand{\vtxspd}{\twist{\vtxsp}}
\newcommand{\edgespd}{\twist{\edgesp}}
\newcommand{\facespd}{\twist{\facesp}}
\newcommand{\cellspd}{\twist{\cellsp}}
\newcommand{\xsp}{\mathcal{X}}
\newcommand{\ysp}{\mathcal{Y}}
\newcommand{\yspd}{\twist{\mathcal{Y}}}

\newcommand{\PrimalDual}[2]{{\scriptscriptstyle{#1}}{\textstyle \twistm{{\scriptscriptstyle{#2}}}}}
\newcommand{\EpFd}{\PrimalDual{\edgesp}{\facesp}}
\newcommand{\VpCd}{\PrimalDual{\vtxsp}{\cellsp}}
\newcommand{\FpEd}{\PrimalDual{\facesp}{\edgesp}}
\newcommand{\CpVd}{\PrimalDual{\cellsp}{\vtxsp}}
\newcommand{\XpYd}{\PrimalDual{\xsp}{\ysp}}
\newcommand{\DualPrimal}[2]{{\textstyle \twistm{{\scriptscriptstyle{#1}}}}{\scriptscriptstyle{#2}}}
\newcommand{\VdCp}{\DualPrimal{\vtxsp}{\cellsp}}
\newcommand{\EdFp}{\DualPrimal{\edgesp}{\facesp}}
\newcommand{\FdEp}{\DualPrimal{\facesp}{\edgesp}}
\newcommand{\CdVp}{\DualPrimal{\cellsp}{\vtxsp}}
\newcommand{\YdXp}{\DualPrimal{\ysp}{\xsp}}

\newcommand{\PrimalDualSub}[3]{ {\scriptscriptstyle{#1_{#3}}}{\textstyle \twistm{{\scriptscriptstyle{#2}}}_{\scriptscriptstyle{#3}}} }
\newcommand{\EpFdc}{\PrimalDualSub{\edgesp}{\facesp}{\cell}}
\newcommand{\VpCdc}{\PrimalDualSub{\vtxsp}{\cellsp}{\cell}}
\newcommand{\FpEdc}{\PrimalDualSub{\facesp}{\edgesp}{\cell}}
\newcommand{\CpVdc}{\PrimalDualSub{\cellsp}{\vtxsp}{\cell}}
\newcommand{\XpYdc}{\PrimalDualSub{\xsp}{\ysp}{\cell}}
\newcommand{\DualPrimalSub}[3]{ {\textstyle \twistm{{\scriptscriptstyle{#1}}}_{\scriptscriptstyle{#3}}}{\scriptscriptstyle{#2_{#3}}} }
\newcommand{\FdEpc}{\DualPrimalSub{\facesp}{\edgesp}{\cell}}
\newcommand{\EdFpc}{\DualPrimalSub{\edgesp}{\facesp}{\cell}}
\newcommand{\YdXpc}{\DualPrimalSub{\ysp}{\xsp}{\cell}}

%=======================
% Operators
%=======================
\newcommand{\Ker}{\operatorname{Ker}}
\newcommand{\Ran}{\operatorname{Im}}
\newcommand{\Harm}[1]{\operatorname{Harm}#1}
\DeclareMathOperator{\ima}{im}
\DeclareMathOperator{\trace}{tr}
\DeclareMathOperator{\card}{card}
%\DeclareMathOperator{\Dim}{dim}
\DeclareMathOperator{\domain}{domain}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\grd}{\vect{grad}}
\DeclareMathOperator{\divm}{div}
\DeclareMathOperator{\curl}{\vect{curl}}
\DeclareMathOperator{\grdt}{\tensor{\mathrm{grad}}}
\DeclareMathOperator{\divv}{\vect{div}}
\DeclareMathOperator{\lapvec}{\vect{\Delta}}

%=======================
% Discrete Operators
%=======================
\newcommand{\opid}{\mathsf{Id}}
\newcommand{\TeId}{\tensor{\opid}}
%
% Discrete differential operators
\newcommand{\opgrad}{\mathsf{GRAD}}
\newcommand{\opdiv}{\mathsf{DIV}}
\newcommand{\opcurl}{\mathsf{CURL}}
\newcommand{\opgradd}{\twist{\opgrad}}
\newcommand{\opdivd}{\twist{\opdiv}}
\newcommand{\opcurld}{\twist{\opcurl}}
%
% Reduction or de Rham map
\newcommand{\opred}{\mathsf{R}}
\newcommand{\opRed}[1]{\opred_{#1}}
%
% Projection maps
\newcommand{\opproj}{\mathsf{T}} %modif: before P
\newcommand{\opProj}[1]{\opproj_{#1}}
\newcommand{\PVc}{\opProj{\vtxsp,\cell}}
\newcommand{\PEc}{\opProj{\edgesp,\cell}}
\newcommand{\PFc}{\opProj{\facesp,\cell}}
%
\newcommand{\optrans}{\mathsf{T}}
\newcommand{\opT}[1]{\optrans_{#1}}
%
% Hodge operators
\newcommand{\opHodge}{\mathsf{H}}
\newcommand{\opHodgec}{\hat{\opHodge}}
\newcommand{\opHodgeFull}[3]{\opHodge_{#1}^{\scriptscriptstyle{#2 #3}}}
\newcommand{\Hperp}{\perp\opHodge}
%
% Reconstruction operators
\newcommand{\conf}{{\rm conf}}
%
\newcommand{\oprec}{\mathtt{L}}
\newcommand{\opcrec}{\mathtt{C}}
\newcommand{\opsrec}{\mathtt{S}}
\newcommand{\opRec}[1]{\oprec_{#1}}
\newcommand{\opReco}[1]{\oprec_{#1}}
\newcommand{\opCRec}[1]{\opcrec_{#1}}
\newcommand{\opSRec}[1]{\opsrec_{#1}}
\newcommand{\opShRec}[1]{\hat{\opsrec}_{#1}}
\newcommand{\opRecv}[1]{\vect{\oprec}_{#1}}
\newcommand{\opCRecv}[1]{\vect{\opcrec}_{#1}}
\newcommand{\opSRecv}[1]{\vect{\opsrec}_{#1}}
\newcommand{\opShRecv}[1]{\hat{\vect{\opsrec}}_{#1}}
% Deprecated
\newcommand{\opRecc}[1]{\oprec_{#1}^\conf}
\newcommand{\opRecvc}[1]{\vect{\oprec}_{#1}^\conf}
%
% Fonctions de reconstruction
\newcommand{\freco}{\ell}
\newcommand{\frecov}{\vect{\freco}}
\newcommand{\Freco}[2]{\freco_{#1}^{\textsc{#2}}}
\newcommand{\Frecov}[2]{\frecov_{#1}^{\textsc{#2}}}
\newcommand{\frecoc}{\freco^\conf}
\newcommand{\frecovc}{\frecov^\conf}
\newcommand{\fform}{\varphi}
\newcommand{\Fform}[1]{\fform^{\textsc{#1}}}
\newcommand{\Fformv}[1]{\vect{\fform}^{\textsc{#1}}}
%
% Approximation maps
\newcommand{\opapx}{\mathtt{A}}
\newcommand{\opapxv}{\vect{\opapx}}
\newcommand{\opApx}[1]{\opapx_{#1}}
\newcommand{\opApxv}[1]{\opapxv_{#1}}
% Deprecated
\newcommand{\opApxc}[1]{\opapx_{#1}^\conf}
\newcommand{\opApxvc}[1]{\opapxv_{#1}^\conf}
%
\newcommand{\Commut}[2]{\lfloor #1, \, #2 \rceil}
%
% Generic operators
\newcommand{\opA}{\mathsf{A}}
\newcommand{\opB}{\mathsf{B}}
\newcommand{\opC}{\mathsf{C}}
\newcommand{\opD}{\mathsf{D}}
\newcommand{\opBt}{\transpose{\opB}}
\newcommand{\opCt}{\transpose{\opC}}
\newcommand{\opDt}{\transpose{\opD}}
%
%=======================
% Stokes
%=======================
%
\newcommand{\Wt}{\Wex_{\tau}^{\bc}}
\newcommand{\Un}{\Vit_\nu^{\bc}}
\newcommand{\Ut}{\Vex_\tau^{\bc}}
\newcommand{\Pbc}{\Pex^{\bc}}
\newcommand{\UnWt}{\textbf{(UnWt)}\@\xspace}
\newcommand{\UtPr}{\textbf{(UtPr)}\@\xspace}
\newcommand{\PL}{\cblue{(PL)}\@\xspace}
\newcommand{\DL}{\cred{(DL)}\@\xspace}
%
%=======================
% Functional analysis
%=======================
\newcommand{\Hcurl}[1]{H(\vect{\rm curl};#1)}
\newcommand{\Hdiv}[1]{H({\rm div};#1)}
\newcommand{\Hilbert}[1]{H^{#1}}
\newcommand{\Sobolev}[2]{W^{#1,#2}}
\newcommand{\Lebesgue}[1]{L^{#1}}
\newcommand{\Hilbertv}[1]{\vect{H}^{#1}}
\newcommand{\Sobolevv}[2]{\vect{W}^{#1,#2}}
\newcommand{\Lebesguev}[1]{\vect{L}^{#1}}

\newcommand{\Ldisc}[2]{#1,#2}
\newcommand{\Real}{{\mathbb{R}}}
\newcommand{\Poly}{{\mathbb{P}}}
\newcommand{\Polyv}[2]{[\Poly_{#1}(#2)]^{3}}

\newcommand{\CSo}[2]{C_{#2}^{_{(#1)}}}
\newcommand{\CPo}[1]{C_{\textsc{p}}^{_{(#1)}}}
\newcommand{\CPW}[1]{C_{\textsc{pw}}^{_{(#1)}}}
\newcommand{\CPoO}[1]{C_{\textsc{p},\Omega}^{_{(#1)}}}
\newcommand{\PoC}{[ \Poly_0(\cellh) ]^3}
\newcommand{\HiC}{[H^1(\cellh)]^3}
\newcommand{\LeO}{[L^4(\Omega)]^3}

\newcommand{\normBV}[1]{\normc{#1}_{\textsc{bv}}}

% Domaine et co-domaine
\newcommand{\Space}[1]{S_{#1}} % De dimension infinie
\newcommand{\SpaceGrd}{\Space{\vtxsp}^{\mathrm{g}}(\Omega)}
\newcommand{\SpaceCurl}{\Space{\edgesp}^{\mathrm{c}}(\Omega)}
\newcommand{\SpaceDiv}{\Space{\facesp}^{\mathrm{d}}(\Omega)}
\newcommand{\Spacev}[1]{\vect{S}_{#1}} % De dimension infinie
\newcommand{\SPol}[1]{P_{#1}} % De dimension finie (espace de polynomes)
\newcommand{\SPolv}[1]{\vect{P}_{#1}} % De dimension finie (espace de polynomes à valeur vectorielle)

% Norms, scalar products
\newcommand{\pscont}[2]{\left[ #1,#2 \right]}
\newcommand{\psdico}[2]{\left[ #1,#2 \right]}%{\left(#1 , #2 \right)}
%\newcommand{\psdisc}[2]{ \big[ \negmedspace \big[ #1,#2 \big] \negmedspace \big] } % original
%\newcommand{\psdisc}[2]{ \ldbrack #1,#2 \rdbrack } % new % ldbrack Needs mathbx which causes problems
\newcommand{\psdisc}[2]{ \llbracket #1,#2 \rrbracket } % new
\newcommand{\proddual}[2]{\left\langle #1,\, #2 \right\rangle}
\newcommand{\normc}[1]{\lvert\!\lvert #1 \rvert\!\rvert}
\newcommand{\normd}[1]{\lvert\!\lvert\!\vert #1 \rvert\!\rvert\!\rvert}
%\newcommand{\jump}[1]{ \ldbrack #1 \rdbrack } % new % ldbrack Needs mathbx which causes problems
\newcommand{\jump}[1]{ \llbracket #1 \rrbracket } % new

%
% Analysis of numerical results
%
\newcommand{\rate}{\mathrm{\textsc{r}}}
\newcommand{\Errd}[1]{\mathbf{Er}_{#1}}
\newcommand{\Errc}[1]{Er_{#1}}
\newcommand{\nsys}{\mathrm{n_{\textsc{s}ys}}}
\newcommand{\nite}{\mathrm{n_{\textsc{i}te}}}
\newcommand{\nnz}{\mathrm{\textsc{nnz}}}
\newcommand{\CompCo}{\boldsymbol{\chi}}
\newcommand{\CompCoR}{\boldsymbol{\chi}_{\mathrm{\textsc{r}el}}}
\newcommand{\stencil}{\mathrm{St}}
%
\newcommand{\Hex}{\texttt{Hex}\@\xspace}
\newcommand{\PrT}{\texttt{PrT}\@\xspace}
\newcommand{\PrG}{\texttt{PrG}\@\xspace}
\newcommand{\CB}{\texttt{CB}\@\xspace}
\newcommand{\K}{\texttt{Ker}\@\xspace}
%
%=======================
% Macros for environment
%=======================
% Equations
\newcommand{\eq}[1]{\begin{equation} #1 \end{equation}}
\newcommand{\eqn}[1]{\begin{equation*} #1 \end{equation*}}
\newcommand{\eqb}[1]{\begin{equation}\boxed{ #1 }\end{equation}}
\newcommand{\eqnb}[1]{\begin{equation*}\boxed{ #1 }\end{equation*}}
%
% Figure
%
\newcommand{\Figure}[4]{%
\begin{figure}[#1]
\centering
\includegraphics[width=#2\textwidth]{#3}
\caption{#4}
\end{figure}
}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                   SECTION: MR MACROS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Miscellaneous
\newcommand{\bO}[1]{\ensuremath{\mathcal{O}\left(#1\right)}}
% Vector, hybrid, dual
\newcommand{\dsc}{\ensuremath{\mathrm{h}}}
\newcommand{\dual}[1]{\ensuremath{\widetilde{#1}}}
\newcommand{\hyb}[1]{\ensuremath{\widehat{#1}}}
\newcommand{\vct}[1]{\ensuremath{\underline{#1}}}
\newcommand{\tens}[1]{\ensuremath{\underline{\underline{#1}}}}
\newcommand{\hybv}[1]{\ensuremath{\widehat{\vct{#1}}}}
\newcommand{\hVex}{\ensuremath{\hyb{\Vex}}}
\newcommand{\hVtest}{\ensuremath{\hyb{\Vtest}}}
\newcommand{\hVttest}{\ensuremath{\hyb{\Vttest}}}
% Frequently used variables modifiers
%\newcommand{\ut}{\ensuremath{\tilde{u}}}
%\newcommand{\uh}{\ensuremath{\hat{u}}}
% Time discretisation
\newcommand{\tmst}[1]{\sp{#1}} % \sp = ^
\newcommand{\np}[1][1]{\tmst{n+#1}}
\newcommand{\nm}[1][1]{\tmst{n-#1}}
\newcommand{\deltat}{\Delta t} %\tau
% Iterations
\newcommand{\itst}[1]{\sb{#1}} % \sb = _
\newcommand{\kp}[1][1]{\itst{k+#1}}
\newcommand{\km}[1][1]{\itst{k-#1}}
% Geometry
\newcommand{\meas}[1]{\ensuremath{\left\lvert #1\right\rvert}}
\newcommand{\ifc}{\ensuremath{\incid{\face}{\cell}}}
\newcommand{\nf}{\ensuremath{\norma{\face}}}
\newcommand{\nfc}{\ensuremath{\norma{\face\cell}}}
\newcommand{\restr}[1]{{}_{\lvert #1}}
% Sets shortcuts
\newcommand{\Fc}{\ensuremath{\faceh_{\cell}}}
\newcommand{\finf}{\ensuremath{\face \in \Fc}}
\newcommand{\finh}{\ensuremath{\face \in \faceh}}
\newcommand{\cinc}{\ensuremath{\cell \in \cellh}}
\newcommand{\cinf}{\ensuremath{\cell \in \cellh_{\face}}}
\newcommand{\uc}{\ensuremath{_{\cell}}}
\newcommand{\uf}{\ensuremath{_{\face}}}
\newcommand{\ud}{\ensuremath{_{\dsc}}}
\newcommand{\uh}{\ensuremath{\ud}}
% Discrete settings
\newcommand{\R}{\ensuremath{\mathbb{R}}}
\newcommand{\Pk}[1][k]{\ensuremath{\mathbb{P}^{#1}}}
% % Spaces
% % % Hybrid, pressure-velocity
\newcommand{\vsp}{\ensuremath{\mathcal{U}}}
\newcommand{\vvsp}{\ensuremath{\vct{\vsp}}}         % Vectorial Cell   Velocity Space
\newcommand{\hvsp}{\ensuremath{\hybv{\vsp}}}        %           Hybrid
\newcommand{\spsp}{\ensuremath{\mathcal{P}}}        % Scalar    (Cell) Pressure Space
% % % Original CDO spaces
\newcommand{\psp}[1]{\ensuremath{\mathcal{#1}}}             % Primal CDO Spaces
\newcommand{\dsp}[1]{\ensuremath{\dual{\mathcal{#1}}}}      %   Dual CDO Spaces
% % Reductions / de Rahm maps
% % % Hybrid, velocity
\newcommand{\prj}{\pi}
%\newcommand{\red}{\ensuremath{\mathsf{R}}}
\newcommand{\red}{\ensuremath{\prj}}
\newcommand{\Red}[1]{\ensuremath{\red_{#1}}}
\newcommand{\vRed}[1]{\ensuremath{\vct{\red}_{#1}}}
\newcommand{\Redc}{\ensuremath{\vRed{\cell}}}
\newcommand{\Redf}{\ensuremath{\vRed{\face}}}
\newcommand{\Redh}{\ensuremath{\hybv{\red}_{\cell}}}
\newcommand{\Redg}{\ensuremath{\hybv{\red}_{\dsc}}}
% % % Pressure
\newcommand{\Redp}{\ensuremath{\Red{\cell}}}
\newcommand{\Redph}{\ensuremath{\Red{\dsc}}}
% Differential operators
% % Continuous
\newcommand{\pd}[2]{\frac{\partial{#1}}{\partial{#2}}}
\newcommand{\ddt}[1]{\pd{#1}{t}}
\newcommand{\dive}{\ensuremath{\vct{\nabla}\cdot}}
\newcommand{\grad}{\ensuremath{\vct{\nabla}\,}}
\newcommand{\rot}{\ensuremath{\vct{\nabla}\times}}
\newcommand{\graddiv}{\ensuremath{\grad\dive}}
\newcommand{\lapl}{\ensuremath{\vct{\Delta}\,}}
% % Discrete / CDO
\newcommand{\dddt}[1]{\frac{#1^{n+1}-#1^{n}}{\deltat}}
\newcommand{\dddtp}[1]{\frac{#1^{n+1}-#1^{n}}{\deltat}}
\newcommand{\dddtm}[1]{\frac{#1^{n}-#1^{n-1}}{\deltat}}
\newcommand{\Gh}{\ensuremath{\mathrm{G}}}%{\ensuremath{\dual{\mathrm{G}}^{H}_{\cell}\,}}
\newcommand{\vGh}{\ensuremath{\vct{\Gh}}}%{\ensuremath{\dual{\vct{\mathrm{G}}}^{H}_{\cell}\,}}
\newcommand{\tGh}{\ensuremath{\tens{\Gh}}}%{\ensuremath{\dual{\tens{\mathrm{G}}}^{H}_{\cell}\,}}
\newcommand{\Dh}{\ensuremath{\mathrm{D}}}%{\ensuremath{\mathrm{D}^{H}_{\cell}\,}}}
\newcommand{\Dhh}{\ensuremath{\mathrm{D}_{\cellsp}}}%{\ensuremath{\mathrm{D}^{H}_{\cellsp}\,}}}
% Scalar, inner, duality products, norms
\newcommand{\abs}[1]{\ensuremath{\left\lvert #1\right\rvert}} % Same as \meas
\newcommand{\nrm}[1]{\ensuremath{\left\lVert #1\right\rVert}}
\newcommand{\dsb}[2]{\ensuremath{\llbracket #1,\,#2 \rrbracket}} % Double square brackets
%
\endinput