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Backofen, Rainer
amdis
Commits
349b91af
Commit
349b91af
authored
Jun 01, 2010
by
Praetorius, Simon
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list of operators
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349b91af
\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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