Commit 349b91af by Praetorius, Simon

list of operators

parent ffa046af
 \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{Laplace\_SOT}() \\ $c \cdot \Delta u$ & \texttt{FactorLaplace\_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} \ No newline at end of file
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