Skip to content
Projects
Groups
Snippets
Help
Loading...
Help
Support
Submit feedback
Contribute to GitLab
Sign in
Toggle navigation
dune-functions
Project overview
Project overview
Details
Activity
Releases
Cycle Analytics
Repository
Repository
Files
Commits
Branches
Tags
Contributors
Graph
Compare
Charts
Issues
14
Issues
14
List
Boards
Labels
Milestones
Merge Requests
19
Merge Requests
19
CI / CD
CI / CD
Pipelines
Jobs
Schedules
Charts
Wiki
Wiki
Members
Members
Collapse sidebar
Close sidebar
Activity
Graph
Charts
Create a new issue
Jobs
Commits
Issue Boards
Open sidebar
staging
dune-functions
Commits
ddaef66f
Commit
ddaef66f
authored
Mar 21, 2017
by
Carsten Gräser
Browse files
Options
Browse Files
Download
Email Patches
Plain Diff
[doc][manual] Add toc and fix typo
parent
f9984477
Changes
1
Hide whitespace changes
Inline
Side-by-side
Showing
1 changed file
with
9 additions
and
5 deletions
+9
-5
doc/manual/dune-functions-manual.tex
doc/manual/dune-functions-manual.tex
+9
-5
No files found.
doc/manual/dune-functions-manual.tex
View file @
ddaef66f
...
...
@@ -14,8 +14,8 @@
\usepackage
{
xspace
}
\usepackage
[square,numbers,sort]
{
natbib
}
%
\usepackage[colorinlistoftodos]{todonotes}
\usepackage
[colorinlistoftodos,disable]
{
todonotes
}
\usepackage
[colorinlistoftodos]
{
todonotes
}
%
\usepackage[colorinlistoftodos,disable]{todonotes}
\usepackage
{
environ
}
\usepackage
{
enumitem
}
\usepackage
{
listings
}
...
...
@@ -91,6 +91,8 @@
\maketitle
\tableofcontents
\begin{abstract}
\dunemodule
{
dune-functions
}
is a
\dune
module that provides interfaces for functions and function space bases.
It forms one abstraction level above grids, shape functions, and linear algebra, but still sits below
...
...
@@ -156,7 +158,7 @@ with basis functions taking values in different Euclidean spaces,
\dunemodule
{
dune-functions
}
allows to systematically construct bases for function spaces with a
higher-dimensional range. These include vector-valued functions, mixed finite elements, and spaces
for multi-physics. The building blocks are typically scalar-valued basis functions, but sometimes vector-valued
ones like the N
\'
ed
\'
elec basis
is
used as well. The constructions are systematically described as tree structures.
ones like the N
\'
ed
\'
elec basis
are
used as well. The constructions are systematically described as tree structures.
This tree construction of finite element spaces has first been systematically worked out in~
\cite
{
muething:2015
}
.
Readers who are only interested in scalar finite element spaces may try to proceed directly to
...
...
@@ -187,7 +189,8 @@ basis vectors in $\R^2$. Then we define the product of the two bases as
&
=
\big\{
(b
_
1,0), (b
_
2,0),
\dots
, (b
_{
n
_
1
}
,0)
\big\}
\cup
\big\{
(0,c
_
1), (0,c
_
2),
\dots
, (0,c
_{
n
_
2
}
)
\big\}
.
\end{align*}
This is the natural basis of the space
$
\operatorname
{
span
}
B
\otimes
\operatorname
{
span
}
C
$
. Its basis functions take values in
$
\R
\otimes
\R
=
\R
^
2
$
.
This is the natural basis of the space
$
\operatorname
{
span
}
B
\otimes
\operatorname
{
span
}
C
$
.
Its basis functions take values in
$
\R
\otimes
\R
=
\R
^
2
$
.
More generally, if
$
B
$
and
$
C
$
are bases with ranges
$
\R
^{
m
_
1
}$
and
$
\R
^{
m
_
2
}$
, respectively, then
\begin{equation*}
...
...
@@ -203,7 +206,8 @@ $\R^{m_1} \otimes \R^{m_2} = \R^{m_1+m_2}$.
This construction allows to build vector-valued and mixed finite element spaces of arbitrary complexity.
For example, the space of
first-order Lagrangian finite elements with values in
$
\R
^
3
$
can be seen as the product
$
P
_
1
\otimes
P
_
1
\otimes
P
_
1
$
.
The simplest Taylor--Hood element is the product of
$
(
P
_
2
)
^
3
=
P
_
2
\otimes
P
_
2
\otimes
P
_
2
$
for the velocities with
$
P
_
1
$
for the pressure.
The simplest Taylor--Hood element is the product
$
(
P
_
2
)
^
3
\otimes
P
_
1
$
of
$
(
P
_
2
)
^
3
=
P
_
2
\otimes
P
_
2
\otimes
P
_
2
$
for the velocities with
$
P
_
1
$
for the pressure.
If more physical quantities need to be dealt with, more factor bases can be included easily. Note also that we have not
required that these spaces be defined with respect to the same grid (or any grid at all, for that matter).
...
...
Write
Preview
Markdown
is supported
0%
Try again
or
attach a new file
Attach a file
Cancel
You are about to add
0
people
to the discussion. Proceed with caution.
Finish editing this message first!
Cancel
Please
register
or
sign in
to comment