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Commit 90a5cb38 authored by Praetorius, Simon's avatar Praetorius, Simon
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documentation of operator terms updated

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\documentclass[10pt,a4paper]{article}
\usepackage[a4paper,top=1.5cm,bottom=1.5cm]{geometry}
\usepackage{fancyhdr}
% \usepackage[utf8x]{inputenc}
% \usepackage{ucs}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{array}
\usepackage{longtable}
\pagestyle{fancy}
\fancyhf{}
\fancyhead[R]{\today}
\renewcommand{\headrulewidth}{0pt}
\begin{document}\small
\setlength{\LTleft}{-2.5cm}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\begin{longtable}{p{0.275\textwidth}|p{1\textwidth}}
\hline
\multicolumn{2}{c}{\scriptsize Zero-Order-Terms}\\
\hline
$c\;u$ & \texttt{Simple\_ZOT}($c\in\mathbb{R}$) \\
$f(\vec{x})\;u$ & \texttt{CoordsAtQP\_ZOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$f(v)\;u$ & \texttt{VecAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\rightarrow\mathbb{R}$) \\
$f(v, \vec{x})\; u$ & \texttt{VecAndCoordsAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$f(v)\;g(w)\;u$ & \texttt{MultVecAtQP\_ZOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\rightarrow\mathbb{R}$, $g:\mathbb{R}\rightarrow\mathbb{R}$) \\
$f(v, w)\;u$ & \texttt{Vec2AtQP\_ZOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$) \\
$f(v_1, v_2, v_3)\;u$ & \texttt{Vec3AtQP\_ZOT}($v_1,v_2,v_3\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$) \\
$f(\nabla v)\;u$ & \texttt{FctGradient\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$f(\nabla v, \vec{x})\;u$ & \texttt{FctGradientCoords\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$f(v, \nabla v)\;u$ & \texttt{VecAndGradAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$f(v, \nabla v, \vec{x})\;u$ & \texttt{VecGradCoordsAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$f(v, \nabla v, w)\;u$ & \texttt{Vec2AndGradAtQP\_ZOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}\rightarrow\mathbb{R}$) \\
$f(v, \nabla w)\;u$ & \texttt{VecAndGradVecAtQP\_ZOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$f(v_1, v_2 \nabla v_3)\;u$ & \texttt{Vec2AndGradVecAtQP\_ZOT}($v_1,v_2,v_3\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$f(v, \nabla w_1, \nabla w_2)\;u$ & \texttt{VecAndGradVec2AtQP\_ZOT}($v,w_1,w_2\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$f(v,w, \nabla v, \nabla w)\;u$ & \texttt{Vec2AndGrad2AtQP\_ZOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$f(\{v_i\}_i)\;u$ & \texttt{VecOfDOFVecsAtQP\_ZOT}(\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle \rangle$, $f:$\small{vector}$\langle\mathbb{R}\rangle\rightarrow\mathbb{R}$) \\
$f(\{\nabla v_i\}_i)\;u$ & \texttt{VecOfGradientsAtQP\_ZOT}(\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $f:$\small{vector}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}$) \\
$f(v, \{\nabla w_i\}_i)\;u$ & \texttt{VecAndVecOfGradientsAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$,\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $f:\mathbb{R}\times$\small{vector}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}$) \\
$\partial_1 v_1\,[+\partial_2 v_2 + \partial_3 v_3]\;u$ & \texttt{VecDivergence\_ZOT}($v_1\,[,v_2,v_3]\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$) \\
$f(\{v_i\}_i, \{\nabla w_j\}_j, \vec{x})\;u$ & \texttt{General\_ZOT}(\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $f:\mathbb{R}^n\times$\small{vector}$\langle\mathbb{R}\rangle\times$\small{vector}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}$) \\
\hline
%==============================================
\multicolumn{2}{c}{\scriptsize First-Order-Terms, sign in strong formulation: + (for flag: GRD\_PHI)}\\
\hline
$\vec{1} \cdot \nabla u$ & \texttt{Simple\_FOT}() \\
$c\,\vec{1} \cdot \nabla u$ & \texttt{FactorSimple\_FOT}($c\in\mathbb{R}$) \\
$\vec{b} \cdot \nabla u$ & \texttt{Vector\_FOT}($b\in\mathbb{R}^n$) \\
$v\cdot w\cdot\vec{b}\cdot\nabla u$ & \texttt{Vec2AtQP\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $b\in\mathbb{R}^n$) \\
$f(v)\,\vec{b} \cdot \nabla u$ & \texttt{VecAtQP\_FOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\rightarrow\mathbb{R}$, $b\in\mathbb{R}^n$) \\
$f(\vec{x})\,\vec{1} \cdot \nabla u$ & \texttt{CoordsAtQP\_FOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$f(\vec{x})\,\vec{b} \cdot \nabla u$ & \texttt{VecCoordsAtQP\_FOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$, $b\in\mathbb{R}^n$) \\
$f(\vec{x})\cdot v\cdot\vec{b}\cdot\nabla u$ & \texttt{FctVecAtQP\_FOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$,$f:\mathbb{R}^n\rightarrow\mathbb{R}$,$b\in\mathbb{R}^n$) \\
$v_1\cdot f(v_2,v_3)\,\vec{b} \cdot \nabla u$ & \texttt{Vec3FctAtQP\_FOT}($v_1,v_2,v_3\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $b\in\mathbb{R}^n$) \\
$f(v,w,\nabla v)\,\vec{b} \cdot \nabla u$ & \texttt{Vec2AndGradAtQP\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$, $b\in\mathbb{R}^n$) \\
$F(v) \cdot \nabla u$ & \texttt{VectorFct\_FOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}\rightarrow\mathbb{R}^n$) \\
$F(\nabla v) \cdot \nabla u$ & \texttt{VectorGradient\_FOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
$F(\vec{x}) \cdot \nabla u$ & \texttt{VecFctAtQP\_FOT}($F:\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
$F(v, \nabla w) \cdot \nabla u$ & \texttt{VecGrad\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
$F(\nabla v, \nabla w) \cdot \nabla u$ & \texttt{FctGrad2\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
$F(v_1, v_2,\nabla v_3) \cdot \nabla u$ & \texttt{Vec2Grad\_FOT\footnote[1]{* available on request}}($v_1,v_2,v_3\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
$F(\vec{v}) \cdot \nabla u$ & \texttt{WorldVecFct\_FOT\footnotemark[1]}($\vec{v}\in${\scriptsize WorldVector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
$F(\{v_i\}_i, \{\nabla w_j\}_j, \vec{x}) \cdot \nabla u$ & \texttt{General\_FOT}(\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $F:\mathbb{R}^n\times$\small{vector}$\langle\mathbb{R}\rangle\times$\small{vector}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}^n$) \\
\hline
%==========================================================
\multicolumn{2}{c}{\scriptsize Second-Order-Terms, sign in strong formulation: -}\\
\hline
$\Delta u$ & \texttt{Simple\_SOT}() \\
$c \cdot \Delta u$ & \texttt{Simple\_SOT}($c\in\mathbb{R}$) \\
$\nabla\cdot (f(\vec{x}) \nabla u)$ & \texttt{CoordsAtQP\_SOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$\nabla\cdot (f(v) \nabla u)$ & \texttt{VecAtQP\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\rightarrow\mathbb{R}$) \\
$\nabla\cdot (f(v, \vec{x}) \nabla u)$ & \texttt{VecAndCoordsAtQP\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$\nabla\cdot (f(v, w) \nabla u)$ & \texttt{Vec2AtQP\_SOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$) \\
$\nabla\cdot (f(\nabla v) \nabla u)$ & \texttt{FctGradient\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$\nabla\cdot (f(v, \nabla v) \nabla u)$ & \texttt{VecAndGradAtQP\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$\nabla\cdot (f(v, \nabla v, \vec{x}) \nabla u)$ & \texttt{VecGradCoordsAtQP\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$\nabla\cdot (f(v,\nabla w) \nabla u)$ & \texttt{VecGrad\_SOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
$\partial_i (c\,\partial_j(u))$ & \texttt{FactorIJ\_SOT}($i,j\in\mathbb{N}$, $c\in\mathbb{R}$) \\
$\partial_i (f(\vec{x})\,\partial_j(u))$ & \texttt{CoordsAtQP\_IJ\_SOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$, $i,j\in\mathbb{N}$) \\
$\partial_i (f(v)\,\partial_j(u))$ & \texttt{VecAtQP\_IJ\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\rightarrow\mathbb{R}$, $i,j\in\mathbb{N}$) \\
$\nabla \cdot (A \nabla u)$ & \texttt{Matrix\_SOT}($A\in\mathbb{R}^{n\times n}$) \\
$\nabla \cdot (A(v) \nabla u)$ & \texttt{MatrixFct\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $A:\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$, $div:\mathbb{R}^{n\times n}\rightarrow\mathbb{R}^{n}$) \\
$\nabla \cdot (A\cdot f(v,w) \nabla u)$ & \texttt{MatrixVec2\_SOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $A\in\mathbb{R}^{n\times n}$) \\
$\nabla \cdot (A(v,w) \nabla u)$ & \texttt{MatrixVec2Fct\_SOT\footnotemark[1]}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $A:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$) \\
$\nabla \cdot (A(\nabla v) \nabla u)$ & \texttt{MatrixGradient\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $A:\mathbb{R}^n\rightarrow\mathbb{R}^{n\times n}$, $div:\mathbb{R}^{n\times n}\rightarrow\mathbb{R}^{n}$) \\
$\nabla \cdot (A(v, \nabla v) \nabla u)$ & \texttt{VecMatrixGradientAtQP\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $A:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}^{n\times n}$, $div:\mathbb{R}^{n\times n}\rightarrow\mathbb{R}^{n}$) \\
$\nabla \cdot (A(\nabla v, \vec{x}) \nabla u)$ & \texttt{MatrixGradientAndCoords\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $A:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^{n\times n}$, $div:\mathbb{R}^{n\times n}\rightarrow\mathbb{R}^{n}$) \\
$\nabla \cdot (A(\{v_i\}_i, \{\nabla w_j\}_j, \vec{x}) \nabla u)$ & \texttt{General\_SOT}(\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $ A:\mathbb{R}^n\times$\small{vector}$\langle\mathbb{R}\rangle\times$\small{vector}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}^{n\times n}$, $div:\mathbb{R}^{n\times n}\rightarrow\mathbb{R}^{n}$) \\
\end{longtable}
\end{document}
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\documentclass[10pt,a4paper]{article}
\usepackage[a4paper,top=1.5cm,bottom=1.5cm]{geometry}
\usepackage{fancyhdr}
% \usepackage[utf8x]{inputenc}
% \usepackage{ucs}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{array}
\usepackage{longtable}
\pagestyle{fancy}
\fancyhf{}
\parindent0cm
\parskip1ex plus.3ex minus.3ex
\fancyhead[R]{\today}
\renewcommand{\headrulewidth}{0pt}
\begin{document}\small
\setlength{\LTleft}{-2.5cm}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\section*{Operator terms}
\begin{longtable}{>{\begin{math}}p{0.275\textwidth}<{\end{math}}|p{1\textwidth}}
\hline
\multicolumn{2}{c}{\scriptsize Zero-Order-Terms}\\
\hline
\langle c\;\phi,\psi\rangle & \texttt{Simple\_ZOT}($c\in\mathbb{R}$) \\
\langle f(\vec{x})\;\phi,\psi\rangle & \texttt{CoordsAtQP\_ZOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(v)\;\phi,\psi\rangle & \texttt{VecAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\rightarrow\mathbb{R}$) \\
\langle f(v, \vec{x})\; \phi,\psi\rangle & \texttt{VecAndCoordsAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(v)\;g(w)\;\phi,\psi\rangle & \texttt{MultVecAtQP\_ZOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\rightarrow\mathbb{R}$, $g:\mathbb{R}\rightarrow\mathbb{R}$) \\
\langle f(v, w)\;\phi,\psi\rangle & \texttt{Vec2AtQP\_ZOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$) \\
\langle f(v_1, v_2, v_3)\;\phi,\psi\rangle & \texttt{Vec3AtQP\_ZOT}($v_1,v_2,v_3\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$) \\
\langle f(\nabla v)\;\phi,\psi\rangle & \texttt{FctGradient\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(\nabla v, \vec{x})\;\phi,\psi\rangle & \texttt{FctGradientCoords\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(v, \nabla v)\;\phi,\psi\rangle & \texttt{VecAndGradAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(v, \nabla v, \vec{x})\;\phi,\psi\rangle & \texttt{VecGradCoordsAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(v, \nabla v, w)\;\phi,\psi\rangle & \texttt{Vec2AndGradAtQP\_ZOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}\rightarrow\mathbb{R}$) \\
\langle f(v, \nabla w)\;\phi,\psi\rangle & \texttt{VecAndGradVecAtQP\_ZOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(v_1, v_2 \nabla v_3)\;\phi,\psi\rangle & \texttt{Vec2AndGradVecAtQP\_ZOT}($v_1,v_2,v_3\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(v, \nabla w_1, \nabla w_2)\;\phi,\psi\rangle & \texttt{VecAndGradVec2AtQP\_ZOT}($v,w_1,w_2\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(v,w, \nabla v, \nabla w)\;\phi,\psi\rangle & \texttt{Vec2AndGrad2AtQP\_ZOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(\{v_i\}_i)\;\phi,\psi\rangle & \texttt{VecOfDOFVecsAtQP\_ZOT}(\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle \rangle$, $f:$\small{vector}$\langle\mathbb{R}\rangle\rightarrow\mathbb{R}$) \\
\langle f(\{\nabla v_i\}_i)\;\phi,\psi\rangle & \texttt{VecOfGradientsAtQP\_ZOT}(\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $f:$\small{vector}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}$) \\
\langle f(v, \{\nabla w_i\}_i)\;\phi,\psi\rangle & \texttt{VecAndVecOfGradientsAtQP\_ZOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$,\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $f:\mathbb{R}\times$\small{vector}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}$) \\
\langle \partial_1 v_1\,[+\partial_2 v_2 + \partial_3 v_3]\;\phi,\psi\rangle & \texttt{VecDivergence\_ZOT}($v_1\,[,v_2,v_3]\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$) \\
\langle f(\{v_i\}_i, \{\nabla w_j\}_j, \vec{x})\;\phi,\psi\rangle & \texttt{General\_ZOT}(\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $f:\mathbb{R}^n\times$\small{vector}$\langle\mathbb{R}\rangle\times$\small{vector}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}$) \\
\hline
\end{longtable}
\begin{longtable}{>{\begin{math}}p{0.275\textwidth}<{\end{math}}|>{\begin{math}}p{0.275\textwidth}<{\end{math}}|p{.7\textwidth}}
\hline
%==============================================
\multicolumn{3}{c}{\scriptsize First-Order-Terms}\\
\hline
\text{\scriptsize GRD\_PHI} & \text{\scriptsize GRD\_PSI} & \\
\hline
\langle \vec{1} \cdot \nabla \phi,\psi\rangle & \langle \vec{1} \phi,\nabla \psi\rangle & \texttt{Simple\_FOT}() \\
\langle c\,\vec{1} \cdot \nabla \phi,\psi\rangle & \langle c\,\vec{1} \phi,\nabla \psi\rangle & \texttt{FactorSimple\_FOT}($c\in\mathbb{R}$) \\
\langle \vec{b} \cdot \nabla \phi,\psi\rangle & \langle \vec{b} \phi,\nabla \psi\rangle & \texttt{Vector\_FOT}($b\in\mathbb{R}^n$) \\
\langle v\cdot w\cdot\vec{b}\cdot\nabla \phi,\psi\rangle & \langle v\cdot w\cdot\vec{b} \phi,\nabla \psi\rangle & \texttt{Vec2AtQP\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $b\in\mathbb{R}^n$) \\
\langle f(v)\,\vec{b} \cdot \nabla \phi,\psi\rangle & \langle f(v)\cdot \vec{b} \phi,\nabla \psi\rangle &\texttt{VecAtQP\_FOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\rightarrow\mathbb{R}$, $b\in\mathbb{R}^n$) \\
\langle f(\vec{x})\,\vec{1} \cdot \nabla \phi,\psi\rangle & \langle f(\vec{x})\cdot \vec{1} \phi,\nabla \psi\rangle &\texttt{CoordsAtQP\_FOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(\vec{x})\,\vec{b} \cdot \nabla \phi,\psi\rangle & \langle f(\vec{x})\cdot \vec{b} \phi,\nabla \psi\rangle &\texttt{VecCoordsAtQP\_FOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$, $b\in\mathbb{R}^n$) \\
\langle f(\vec{x})\cdot v\cdot\vec{b}\cdot\nabla \phi,\psi\rangle & \langle f(\vec{x})\cdot v\cdot\vec{b} \phi,\nabla \psi\rangle &\texttt{FctVecAtQP\_FOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$,$f:\mathbb{R}^n\rightarrow\mathbb{R}$,$b\in\mathbb{R}^n$) \\
\langle v_1\cdot f(v_2,v_3)\,\vec{b} \cdot \nabla \phi,\psi\rangle & \langle v_1\cdot f(v_2,v_3)\cdot\vec{b} \phi,\nabla \psi\rangle &\texttt{Vec3FctAtQP\_FOT}($v_1,v_2,v_3\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $b\in\mathbb{R}^n$) \\
\langle f(v,w,\nabla v)\,\vec{b} \cdot \nabla \phi,\psi\rangle & \langle f(v,w,\nabla v)\cdot\vec{b} \phi,\nabla \psi\rangle &\texttt{Vec2AndGradAtQP\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$, $b\in\mathbb{R}^n$) \\
\langle F(v) \cdot \nabla \phi,\psi\rangle & \langle F(v)\,\phi, \nabla \psi\rangle &\texttt{VectorFct\_FOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}\rightarrow\mathbb{R}^n$) \\
\langle F(\nabla v) \cdot \nabla \phi,\psi\rangle & \langle F(\nabla v)\,\phi, \nabla \psi\rangle &\texttt{VectorGradient\_FOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
\langle F(\vec{x}) \cdot \nabla \phi,\psi\rangle & \langle F(\vec{x}) \, \phi,\nabla \psi\rangle &\texttt{VecFctAtQP\_FOT}($F:\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
\langle F(v, \nabla w) \cdot \nabla \phi,\psi\rangle & \langle F(v, \nabla w) \, \phi,\nabla \psi\rangle &\texttt{VecGrad\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
\langle F(\nabla v, \nabla w) \cdot \nabla \phi,\psi\rangle & \langle F(\nabla v, \nabla w) \, \phi,\nabla \psi\rangle &\texttt{FctGrad2\_FOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
%\langle F(v_1, v_2,\nabla v_3) \cdot \nabla \phi,\psi\rangle & \langle F(v_1, v_2,\nabla v_3) \, \phi,\nabla \psi\rangle &\texttt{Vec2Grad\_FOT\footnote[1]{* available on request}}($v_1,v_2,v_3\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $F:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
%\langle F(\vec{v}) \cdot \nabla \phi,\psi\rangle & \langle F(\vec{v}) \, \phi,\nabla \psi\rangle &\texttt{WorldVecFct\_FOT\footnotemark[1]}($\vec{v}\in${\scriptsize WorldVector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$) \\
\langle F(\{v_i\}_i, \{\nabla w_j\}_j, \vec{x}) \cdot \nabla \phi,\psi\rangle & \langle F(\{v_i\}_i, \{\nabla w_j\}_j, \vec{x}) \, \phi,\nabla \psi\rangle &\texttt{General\_FOT}(\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vector}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, \\ & & $F:\mathbb{R}^n\times$\small{vector}$\langle\mathbb{R}\rangle\times$\small{vector}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}^n$) \\
\hline
\end{longtable}
\newpage
\begin{longtable}{>{\begin{math}}p{0.275\textwidth}<{\end{math}}|p{\textwidth}}
\hline
%==========================================================
\multicolumn{2}{c}{\scriptsize Second-Order-Terms}\\
\hline
\langle c \cdot \nabla \phi, \nabla \psi\rangle & \texttt{Simple\_SOT}($[c\in\mathbb{R}]$) \\
\langle f(\vec{x}) \nabla \phi,\nabla \psi\rangle & \texttt{CoordsAtQP\_SOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(v) \nabla \phi,\nabla \psi\rangle & \texttt{VecAtQP\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $[f:\mathbb{R}\rightarrow\mathbb{R}]$) \\
\langle f(v, \vec{x}) \nabla \phi,\nabla \psi\rangle & \texttt{VecAndCoordsAtQP\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(v, w) \nabla \phi,\nabla \psi\rangle & \texttt{Vec2AtQP\_SOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $[f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}]$) \\
\langle f(\nabla v) \nabla \phi,\nabla \psi\rangle & \texttt{FctGradient\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(v, \nabla v) \nabla \phi,\nabla \psi\rangle & \texttt{VecAndGradAtQP\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(v, \nabla v, \vec{x}) \nabla \phi,\nabla \psi\rangle & \texttt{VecGradCoordsAtQP\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle f(v,\nabla w) \nabla \phi,\nabla \psi\rangle & \texttt{VecGrad\_SOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$) \\
\langle c\,\partial_j(\phi),\partial_i(\psi)\rangle & \texttt{FactorIJ\_SOT}($i,j\in\mathbb{N}$, $c\in\mathbb{R}$) \\
\langle f(\vec{x})\,\partial_j(\phi),\partial_i(\psi)\rangle & \texttt{CoordsAtQP\_IJ\_SOT}($f:\mathbb{R}^n\rightarrow\mathbb{R}$, $i,j\in\mathbb{N}$) \\
\langle f(v)\,\partial_j(\phi),\partial_i(\psi)\rangle & \texttt{VecAtQP\_IJ\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\rightarrow\mathbb{R}$, $i,j\in\mathbb{N}$) \\
\langle f(v_1,v_2)\,\partial_j(\phi),\partial_i(\psi)\rangle & \texttt{Vec2AtQP\_IJ\_SOT}($v_1,v_2\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $i,j\in\mathbb{N}$) \\
\langle B \nabla \phi,\nabla \psi\rangle & \texttt{Matrix\_SOT}($B\in\mathbb{R}^{n\times n}$) \\
\langle A(v) \nabla \phi,\nabla \psi\rangle & \texttt{MatrixFct\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $A:\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$, $[div^*]$) \\
\langle B\cdot f(v,w) \nabla \phi,\nabla \psi\rangle & \texttt{MatrixVec2\_SOT}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $B\in\mathbb{R}^{n\times n}$) \\
%\langle A(v,w) \nabla \phi,\nabla \psi\rangle & \texttt{MatrixVec2Fct\_SOT\footnotemark[1]}($v,w\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $A:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$) \\
\langle A(\nabla v) \nabla \phi,\nabla \psi\rangle & \texttt{MatrixGradient\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $A:\mathbb{R}^n\rightarrow\mathbb{R}^{n\times n}$, $[div^*]$) \\
\langle A(v, \nabla v) \nabla \phi,\nabla \psi\rangle & \texttt{VecMatrixGradientAtQP\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $A:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}^{n\times n}$, $[div^*]$) \\
\langle A(\nabla v, \vec{x}) \nabla \phi,\nabla \psi\rangle & \texttt{MatrixGradientAndCoords\_SOT}($v\in${\scriptsize DOFVector}$\langle\mathbb{R}\rangle$, $A:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^{n\times n}$, $[div^*]$) \\
\langle A(\vec{x}, \{v_i\}_i, \{\nabla w_j\}_j) \nabla \phi,\nabla \psi\rangle & \texttt{General\_SOT}(\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $ A:\mathbb{R}^n\times$\small{vec}$\langle\mathbb{R}\rangle\times$\small{vec}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}^{n\times n}$, $[div^*]$) \\
\langle A(\vec{x}, \vec{n}, \{v_i\}_i, \{\nabla w_j\}_j) \nabla \phi,\nabla \psi\rangle & \texttt{GeneralParametric\_SOT}(\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$,\small{vec}$\langle${\scriptsize DOFVector}$\langle\mathbb{R}\rangle\rangle$, $ A:\mathbb{R}^n\times\mathbb{R}^n\times$\small{vec}$\langle\mathbb{R}\rangle\times$\small{vec}$\langle\mathbb{R}^n\rangle\rightarrow\mathbb{R}^{n\times n}$, $[div^*]$) \\
\hline
\end{longtable}
\section*{Comments}
\begin{itemize}
\item All operators are listed in the files \texttt{ZeroOrderTerm.h}, \texttt{FirstOrderTerm.h} and \texttt{SecondOrderTerm.h}.
\item The following definitions/shortcuts are used to reduce typing:
$L_2$-Scalar product: $\langle\cdot,\cdot\rangle$, trialfunction: $\phi$, testfunction $\psi$, coefficients $c\in\mathbb{R},\;\vec{1},\vec{b},\vec{x}\in\mathbb{R}^n,$ with $(\vec{1})_i = 1$, $B\in\mathbb{R}^{n\times n}$, functors $f:(\ldots)\rightarrow\mathbb{R}$, $F:(\ldots)\rightarrow\mathbb{R}^n$ and $A:(\ldots)\rightarrow\mathbb{R}^{n\times n}$.
\item Some mathematical notations are used to describe data-structures: $\mathbb{R}$ means \texttt{double}, $\mathbb{R}^n$ means \texttt{WorldVector<double>} and $\mathbb{R}^{n\times n}$ means \texttt{WorldMatrix<double>}.
\item $f, F, A$ can be implemented as \texttt{(*)AbstractFunction$\langle$ReturnType, InputType1, InputType2, ...$\rangle$}, where \texttt{(*)}$\in$\{$\emptyset$, \texttt{Binary}, \texttt{Tertiary}, \texttt{Quart}\} depending on the number of input arguments.
\item The data-structure \texttt{DOFVector<*>} is always a pointer to a DOFVector.
\item Optional arguments are depicted in square brackets $[*]$, where constants $c = 1$ by default, functions are \texttt{NULL}-pointers by default and are treated as identity functors or simple multiplication functors.
\item The argument $div:=\mathbb{R}^{n\times n}\rightarrow\mathbb{R}^n$ is only interesting for error estimators and optional. Is should implement the divergence of the matrix function in the operator.
\item The argument \small{vec}$\langle*\rangle$ should be implemented as \texttt{std::vector$\langle*\rangle$}.
\item In the last Second-Order-Operator \texttt{GeneralParametric\_SOT}, the second argument $\vec{n}$ to $A$ is the elementnormal, especially for surface meshes.
\end{itemize}
\end{document}
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