Skip to content
Snippets Groups Projects
Commit 9e5ee617 authored by Felix Schindler's avatar Felix Schindler
Browse files

began class description of functional stuff, also some reordering 

git-svn-id: https://dune.mathematik.uni-freiburg.de/svn/dune-fem-functionals/trunk@7 4028485c-44d9-4cde-a312-5a4635ee2db9
parent e82589d9
No related branches found
No related tags found
No related merge requests found
Hide whitespace changes
Inline Side-by-side
doc/latex/dune-fem-functionals.kilepr
+ 5
− 5
View file @ 9e5ee617
......@@ -17,7 +17,7 @@ MakeIndex=
QuickBuild=
[document-settings,item:dune-fem-functionals.tex]
Bookmarks=
Bookmarks=604,370
Encoding=UTF-8
Highlighting=LaTeX
Indentation Mode=normal
......@@ -36,14 +36,14 @@ order=-1
[item:dune-fem-functionals.tex]
archive=true
column=58
column=32
encoding=UTF-8
highlight=LaTeX
line=191
line=425
mode=LaTeX
open=true
order=0
[view-settings,view=0,item:dune-fem-functionals.tex]
CursorColumn=58
CursorLine=191
CursorColumn=32
CursorLine=425
doc/latex/dune-fem-functionals.pdf
+ 0
− 0
View file @ 9e5ee617
No preview for this file type
doc/latex/dune-fem-functionals.tex
+ 168
− 97
View file @ 9e5ee617
......@@ -75,8 +75,8 @@
\newcommand{\R}{\mathbb{R}}
\title{Draft on \dunefemfunctionals}
\author{Felix Albrecht (\url{felix.albrecht@uni-muenster.de}) and\\
Patrick Henning (\url{patrick.henning@uni-muenster.de})
\author{Felix Albrecht (\Code{felix.albrecht@uni-muenster.de}) and\\
Patrick Henning (\Code{patrick.henning@uni-muenster.de})
}
\date{\today}
......@@ -145,7 +145,7 @@
\dx
&&\text{for all } v \in H^1_0 \punkt
\end{align}
\end{definition}\theoremEndLine
\end{definition}
The weak formulation \eqref{equation::introduction::weak_formulation} gives rise to the introduction of
functionals and operators. A rigorous mathematical definition of these can be found in the next section.
......@@ -177,7 +177,7 @@
\dx\punkt
\notag
\end{align}
\end{definition}\theoremEndLine
\end{definition}
With these definitions at hand the weak formulation \eqref{equation::introduction::weak_formulation} can be
rewritten in the following way.
......@@ -198,27 +198,28 @@
\section{Analytical concept}
\subsection{Overview}
We start with an overview on the mathematical concept which is carried over to a corresponding programming
concept. The following notations and definitions are required for the subsequent sections.
We start with an overview on the mathematical concept which is carried over to a corresponding programming
concept. The following notations and definitions are required for the subsequent sections.
\begin{definition}[Function and Functionspace]\theoremNewline
Let ${n,d \in \N^{\geq 1}}$ be integers and ${\Omega \subseteq \mathbb{R}^n}$ a subset. A mapping
${v : \Omega \rightarrow \mathbb{R}^d}$ is called a \textnormal{function}. A vector space $V$ consiting only of
functions is called a \textnormal{functionspace}.
% does this make sense? is this not the definition of a vector space?
% If additionally
% \begin{align*}
% (\alpha v_1 + \beta v_2) \in V \qquad \text{for all }v_1,v_2 \in V \text{ and for all } \alpha,\beta \in \R
% \end{align*}
% holds, the space is called a \textnormal{linear functionspace}.
\end{definition}\theoremEndLine
\begin{definition}[Function and Functionspace]\theoremNewline
\label{definition::analytical_concept::function_functionspace}
Let ${n,d \in \N^{\geq 1}}$ be integers and ${\Omega \subseteq \mathbb{R}^n}$ a subset. A mapping
${v : \Omega \rightarrow \R^d}$ is called a \textnormal{function}, the set
\begin{align}
V :=
\big\{
v : \Omega \rightarrow \R^d
\big\}
\notag
\end{align}
is called a \textnormal{functionspace}. If $V$ is an $\R$--vector space, $V$ is called a \textnormal{linear functionspace}.
\end{definition}\theoremEndLine
\begin{definition}[Functional]\theoremNewline
\label{definition::analytical_concept::functional}
Let $V$ be a function space. A map
\begin{align}
F: V &\rightarrow \K\komma
F: V &\rightarrow \R\komma
\notag\\
v &\mapsto F[v]
\end{align}
......@@ -236,108 +237,178 @@
\end{definition}\theoremEndLine
\begin{definition}[Constraint]\theoremNewline
Let $V$ be a linear function space, $M \in \mathbb{N}_{>0}$ and $\{ F_1, ..., F_M \}$ a set of linear functionals on $V$. We define the corresponding vector of functionals $C$ by
\label{definition::analytical_concept::constraint}
Let $V$ be a linear function space, $M \in \mathbb{N}_{>0}$ and $\{ F_1, ..., F_M \}$ a set of linear
functionals on $V$. We define the corresponding vector of linear functionals $C$ by
\begin{align}
C: \{1,...,M\} \times V &\rightarrow \R \enspace \mbox{with} \enspace (i,v) \mapsto C[i][v] := F_i[v].
\end{align}
The condition:
\begin{align}
C[i][v] = 0 \enspace \forall 1 \le i \le M
\end{align}
is called a \textnormal{Constraint} for $v$.
\end{definition}\theoremEndLine
In particular every single linear functional implies a constraint.
The condition:
\begin{align}
C[i][v] = 0 \enspace \forall 1 \le i \le M
\end{align}
is called a \textnormal{Constraint} for $v$.
\end{definition}
In particular each linear functional implies a constraint.
\begin{definition}[Linear Subspace]\theoremNewline
Let $V$ be a linear function space and $C[i][\cdot]=0$ a constraint on $V$. Then we call
\label{definition::analytical_concept::linear_subspace}
Let $V$ be a linear function space and $C[\cdot][\cdot]=0$ a constraint on $V$. Then we call
\begin{align}
V_C := \{ v \in V| \hspace{4pt} C[i][v] = 0 \hspace{4pt}\forall i\in\{1,...,M\}\}
\end{align}
a \textnormal{linear subspace} of $V$ with respect to $C$.
\end{definition}\theoremEndLine
$V_C$ is a linear vector space, since the constraint functionals $C[i]$ are linear. Typically, $V_C$ becomes the space of test functions in our later problem.
\end{definition}
$V_C$ is a vector space itself, since the constraint functionals $C[i]$ are linear. Typically, $V_C$ becomes the
space of test functions in our later problem.
\begin{definition}[Affine Subspace]\theoremNewline
Let $V$ be a linear function space, $V_C$ a linear subspace and $g \in V$. Then we call
\begin{definition}[Affine space]\theoremNewline
\label{definition::analytical_concept::affine_space}
Let $V$ be a function space, $V_C$ a linear subspace and $g \in V$. Then we call
\begin{align}
V_{g} := \{ v+g| \hspace{4pt} v \in V_C \} \subset V
\end{align}
an \textnormal{affine subspace} of $V$ with respect to $g$ and $V_C$.
an \textnormal{affine space} with respect to $g$ and $V_C$.
\end{definition}\theoremEndLine
In general, $V_g$ is a nonlinear function space. It becomes the space of solution in our later problem.
In general, $V_g$ is only a subset of $V$ and not a subspace. It will be the space of solutions in our later
problem.
\begin{definition}[Operator]\theoremNewline
Let $V_C$ be a linear (constraint) function space with dual space $V^{\prime}_C$ and $V_g$ a constraint subspace. Then we call
\label{definition::analytical_concept::operator}
Let $V$ be a linear function space, ${V_C \subset V}$ a linear subspace, $V_C'$ its dual and ${V_g \subset V}$
an affine space. Then we call
\begin{align}
G : V_g \rightarrow V^{\prime}_C
\end{align}
an \textnormal{operator} on $V_g$.
\end{definition}\theoremEndLine
an \textnormal{operator} on $V_g$. If
\begin{align}
G(\alpha v + \beta w) = \alpha G(v) + \beta G(w)
\notag
\end{align}
holds for all ${\alpha\komma \beta \in \R}$ and for all ${v\komma w \in V_g}$ the operator $G$ is called
\textnormal{linear}.
\end{definition}
In the subsequent sections, we are dealing with the following problem.
\begin{problem}{Sample problem}\theoremNewline
\label{problem::analytical_concept::sample_problem}
Let $V$ be a linear space, ${V_C \subset V}$ a linear subspace, ${F \in V_C'}$ a functional and
${V_g \subset V}$ an affine space. Find ${u \in V_g}$, such that
\begin{align*}
G(u)[v] = F[v] &&\text{for all } v \in V_C.
\end{align*}
\end{problem}
In general, the functional $F$ on the right hand side of our problem is linear. The (differential) operator $G$
can be either linear or nonlinear.
\section{Examples}
\subsection{Elliptic PDE}
Let $\Omega \subset \mathbb{R}^d$ denote a polygonal bounded domain, $\mathcal{T}_H = \{ T_1, ..., T_N \}$ a
corresponding regular triangulation, $\mathcal{N}_H = \{ x_1, ...., x_{\tilde{N}}\}$ the set of nodes and
$\{ \Phi_1, ...., \Phi_{\tilde{N}}\}$ the associated Lagrange basis of order $1$. The (discrete) linear space of
solutions is given by
\begin{align*}
V :=
\Big\{
v_H \in C^0(\Omega)
\Big|
(v_H)_{|T}\in \mathbb{P}^1(T) \quad\forall T \in \mathcal{T}_H
\Big\}
\end{align*}
and the linear subspace of testfunctions by $V_0 := V \cap H^1_0(\Omega)$. Now, let us consider the
following discrete problem.
\begin{problem}\theoremNewline
For $g \in V$ find $u \in V$ with $u=g$ on $\partial \Omega$ and
\begin{align*}
\int\limits_{\Omega} \nabla u \cdot \nabla v \dx = \int\limits_{\Omega} f v \dx &&\text{for all }v \in V_0.
\end{align*}
(Note: $V_0$ can not be replaced by $V$).
\end{problem}
Putting this into the general framework above, we define the (linear) functional
${F: V_0 \rightarrow \R}$ by
\begin{align*}
F[v] := \int\limits_{\Omega} f v &&\text{for } v \in \mathring{V}.
\end{align*}
$V_0$ is a constraint subspace with the constraint $C[i][v] := v(x_i^b) = 0$ for any boundary node
$x_i^b$ (i.e. $\mathcal{N}_H \cap \partial \Omega = \{ x_1^b, ...., x_{\bar{N}}^b\}$). We can therefore identify
\begin{align*}
V_0 = \{ v \in V| \hspace{3pt} C[i][v] = 0 \enspace \forall 1\le i\le \bar{N} \} =: V_C.
\end{align*}
The affine space $V_{g_H}$is given by
\begin{align*}
V_{g_H} :=
\Big\{
v + g_H
\Big|
v \in V \enspace \mbox{and} \enspace g_H := \sum_{i=1}^{\tilde{N}} g(x_i) \Phi_i
\Big\}
\end{align*}
and the (differential) operator $G : V_{g_H} \rightarrow V_C^{\prime}$ by
\begin{align*}
G(u)[v] := \int_{\Omega} \nabla u \cdot \nabla v \dx.
\end{align*}
With these notations the original problem reads:
\begin{align*}
\text{Find} \enspace u \in V_{g_H} \enspace \mbox{with} \enspace G(u) = F \enspace \mbox{on} \enspace V_C.
\end{align*}
\section{Programming concept}
In this section we describe the general programming concept. For a detailed description of these classes, see
section ? and the doxygen documentation. These classes are realized in a namespace \Code{Functionals} in order to
avoid conflicts with other existing \dunefem-classes. We do not distinguish between interfaces and realizations
here, this is only intended as a conceptual overview.
\subsection{Functionals}
\begin{remark}[Class tree]\theoremItemizeNewline
\begin{itemize}
\item[] \CodeT{Functional}
\item[] \begin{itemize}
\item[$\rightarrow$] \CodeT{CombinedFunctional}
\item[$\rightarrow$] \CodeT{LinearFunctional}
\item[] \begin{itemize}
\item[$\rightarrow$] \CodeT{CombinedLinearFunctional}
\item[$\rightarrow$] \CodeT{CodimCFunctional}
\item[$\rightarrow$] \CodeT{IntegralFunctional}
\end{itemize}
\end{itemize}
\end{itemize}
\end{remark}
\begin{class}[\CodeT{Functional< FunctionSpace >}]\theoremNewline
This class represents a functional $F$ (see definition \ref{definition::analytical_concept::functional}). It
provides the basis for linear and nonlinear functionals. Its single method is \CodeT{operator()} which
represents ${F[v]}$.\\\\
\CodeT{RangeFieldType ret = operator( FunctionType function )}
\hrule
\hrule
\vspace{2mm}
\begin{tabular}{ll|l}
\textnormal{in:}
&\CodeT{FunctionType function}
& $v$\\
\hline
\textnormal{out:}
& \CodeT{RangeFieldType ret}
& ${F[v]}$
\end{tabular}
\end{class}
In the subsequent sections, we are dealing with the following problem:
\begin{problem}
For a linear space $V$, a linear (constraint) subspace $V_C$, a functional $F$ and an affine subspace $V_g$ of $V$, find $u \in V_g$ with
\begin{align*}
G(u)[v] = F[v] \enspace \forall v \in V_C.
\end{align*}
\end{problem}
In general, the functional $F$ on the right hand side of our problem is linear. The (differential) operator $G$ can be either linear or nonlinear.
\subsection{Example}
Let $\Omega \subset \mathbb{R}^d$ denote a polygonal bounded domain, $\mathcal{T}_H = \{ T_1, ..., T_N \}$ a corresponding regular triangulation, $\mathcal{N}_H = \{ x_1, ...., x_{\tilde{N}}\}$ the set of nodes and $\{ \Phi_1, ...., \Phi_{\tilde{N}}\}$ the associated Lagrange basis of order $1$. The (discrete) linear space of solutions is given by
\begin{align*}
V := \{ v_H \in C^0(\Omega)| \enspace (v_H)_{|T}\in \mathbb{P}^1(T) \forall T \in \mathcal{T}_H \}
\end{align*}
and the linear subspace by $\mathring{V} := V \cap \mathring{H}^1(\Omega)$. Now, let us consider the following discrete problem:
\begin{problem}
For $g \in V \subset C^0(\Omega)$ find $u \in V$ with $u=g$ on $\partial \Omega$ and
\begin{align*}
\int_{\Omega} \nabla u \cdot \nabla v = \int_{\Omega} f v \quad \forall v \in \mathring{V}.
\end{align*}
(Note: $\mathring{V}$ can not be replaced by $V$).
\end{problem}
Putting this into the general framework above, we idenitfy the (linear) functional $F:\mathring{V}\rightarrow \mathbb{R}$ by
\begin{align*}
F[v] := \int_{\Omega} f v \enspace \mbox{for} \enspace v \in \mathring{V}.
\end{align*}
$\mathring{V}$ is a constraint subspace with the constraint $C[i][v] := v(x_i^b) = 0$ for any boundary node $x_i^b$ (i.e. $\mathcal{N}_H \cap \partial \Omega = \{ x_1^b, ...., x_{\bar{N}}^b\}$). We can therefore identify
\begin{align*}
\mathring{V} = \{ v \in V| \hspace{3pt} C[i][v] = 0 \enspace \forall 1\le i\le \bar{N} \} =: V_C.
\end{align*}
The affine space is given by:
\begin{align*}
V_{g_H} := \{ v + g_H | \hspace{3pt} v \in V \enspace \mbox{and} \enspace g_H := \sum_{i=1}^{\tilde{N}} g(x_i) \Phi_i \}.
\end{align*}
and the (differential) operator $G : V_{g_H} \rightarrow V_C^{\prime}$ by:
\begin{align*}
G(u)[v] := \int_{\Omega} \nabla u \cdot \nabla v.
\end{align*}
With these notations the problem reads:
\begin{align*}
\mbox{Find} \enspace u \in V_{g_H} \enspace \mbox{with} \enspace G(u) = F \enspace \mbox{on} \enspace V_C.
\end{align*}
\section{Programming concept}
In this section we describe the general programming concept. Details on the implementation of the various classes are given later. We assume that we use a {\tt namespace Functionals} in order to avoid conflicts with other existing \dunefem-classes.
\subsection{Required classes}
First of all we give an overview on the various classes that are required in our concept. In particuler we comment on the functionality of each class.\\
\\
{\tt typedef Functional < DiscreteFunctionSpace > FunctionalType;}
\begin{itemize}
\item[$\circ$] from the the general type {\tt Functional} we can derive various realizations of functionals
\item[$\circ$] at first, we restrict ourselves to linear functionals
\item[$\circ$] we might distinguish the types of functionals according to the codim: CodimFunctionals, for example {\tt FunctionalType::CodimFunctional<codim>}; combinations of functionals for different codims must be possible (for instance $F(\Phi) = \int_{\Omega} f \Phi + \int_{\partial \Omega} g \Phi$)
\item[$\circ$] required methods:
\item[$\cdot$] method: {\tt apply( function )} $\leftrightarrow$ $F[ v ]$, $v$ analytical function
\item[$\cdot$] method: {\tt apply( discreteFunction )} $\leftrightarrow$ $F[v_H]$, $v_H$ discrete function
\item[$\cdot$] method: {\tt applyLocal( localBasefunctionSet )} $\rightarrow$ an abstract method depending on the specific realisation of a functional; we can say it returns a vector of local contributions for a specific grid element; it is required for assembling the right hand side in our system of equations; details are given later
\end{itemize}
{\tt typedef Constraint < FunctionalType > ConstraintType;}
\begin{itemize}
\item[$\circ$] various realizations of constraints $C$ are possible (boundary conditions, periodicity, zero-average, ...); they are derived from the general {\tt Constraint} class
......
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Please register or to comment