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revision 100 by jgs, Wed Dec 15 03:48:48 2004 UTC revision 102 by jgs, Wed Dec 15 07:08:39 2004 UTC
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1  % $Id$  % $Id$
2    
 \chapter{The First Steps}  
3    
4  \section{Introduction}  \chapter{Introduction}
5    \label{INTRO}
6    
7  \subsection{Getting the software}  \subsection{Getting the software}
8    
 \escript, \ESyS, all freely available.  Where do people get \finley from?  
9    
10  \begin{enumerate}  \escript, \ESyS, all freely available.  Where do people get \finley from?
 \item how to get the software  
 \item a few words about the general structure  
 \item installation  
 \end{enumerate}  
   
 \section{How to solve a linear PDE}  
11    
 \begin{figure}  
 \centerline{\includegraphics[width=\figwidth]{FirstStepDomain}}  
 \caption{Domain $\Omega$ with outer normal field $n$.}  
 \label{fig:FirstSteps.1}  
 \end{figure}  
   
 \begin{figure}  
 \centerline{\includegraphics[width=\figwidth]{FirstStepMesh}}  
 \caption{Mesh of $4 \time 4$ elements on a rectangular domain.  Here  
 each element is a quadrilateral and described by four nodes, namely  
 the corner points. The solution is interpolated by a bi-linear  
 polynomial.}  
 \label{fig:FirstSteps.2}  
 \end{figure}  
   
 We want to solve the \index{partial differential equation}(\index{PDE})  
 \begin{equation}  
 -\Delta u + \alpha u =f  
 \label{eq:FirstSteps.1}  
 \end{equation}  
 for the solution $u$ on the domain $\Omega$.  Here we assume that the  
 domain is the rectangle of length $1$ and height $2$, see  
 \fig{fig:FirstSteps.1}.  $\Delta$ denotes the \index{Laplace  
 operator} which is defined by  
 \begin{equation}  
 \Delta u = (u\hackscore {,1})\hackscore{,1}+(u\hackscore{,2})\hackscore{,2}  
 \label{eq:FirstSteps.1.1}  
 \end{equation}  
 where for any function $w$ and any direction $i$ $u\hackscore{,i}$  
 denotes the derivative of $w$ with respect to $i$.  $\alpha$ is a  
 known constant (we will set $\alpha=10$) and $f$ is a given function  
 which may depend on the location in the domain.  On the boundary of  
 the domain $\Omega$ the solution $u$ shall fulfill the so-called  
 homogeneous \index{Neumann boundary condition}  
 \begin{equation}  
 \frac{\partial u}{\partial n} = 0  
 \label{eq:FirstSteps.2}  
 \end{equation}  
 where $n=(n\hackscore1,n\hackscore2)$ denotes the outer normal field  
 of the domain, see Figure~\ref{fig:FirstSteps.1} and  
 \begin{equation}  
 \frac{\partial u}{\partial n} = n\hackscore1 u\hackscore{,1} +  
 n\hackscore2 u\hackscore{,2}  
 \label{eq:FirstSteps.2.1}  
 \end{equation}  
 denotes the normal derivative on the boundary.  The partial  
 differential \eqn{eq:FirstSteps.1}) together with the  
 boundary condition~(\eqn{eq:FirstSteps.2}) forms a boundary value  
 problem (\index{BVP}) for unknown function $u$.  
   
 In most cases, the BVP cannot be solved analytically and numerical  
 methods have to be used construct an approximation of the solution  
 $u$. Here we will use the \index{finite element method}  
 (\index{FEM}). The basic idea is to fill the domain with a set of  
 points, so called nodes. The solution is approximated by its values on  
 the nodes. Moreover, the domain is subdivide into small subdomain,  
 so-called elements. On each element the solution is represented by a  
 polynomial of a certain degree through its values at the nodes located  
 in the element. The nodes and its connection through elements is  
 called a \index{mesh}.  \fig{fig:FirstSteps.2} shows an example  
 of a FEM mesh with four elements in the $x_0$ and for elements in the  
 $x_1$ direction over a rectangular domain. On more details we referring  
 to the literature, for instance \cite{X1,X2,X3}.  
   
 \escript provides the class \linearPDE to define a  
 general linear, steady differential partial differential equation of  
 second order. We will discuss the most general form that can be  
 defined through \class{linearPDE} later. The components which are  
 relevant for us here is as follows:  
 \begin{equation}  
 -\sum\hackscore{i,j=0}^k (A\hackscore{ij}  
  u\hackscore{,j})\hackscore{,i} + D u = Y  
 \label{eq:FirstSteps.3}  
 \end{equation}  
 In this form $D$ and $Y$ are scalars and $A$ is a $k \times k$ matrix  
 where $k$ denotes the spatial dimension (in our example we have  
 $k=2$).  By comparing the template~(\ref{eq:FirstSteps.3}) with the  
 differential equation~(\ref{eq:FirstSteps.1}) we want to solve we can  
 immediately identify the appropriate values for $A$, $D$ and $Y$:  
 \begin{equation}  
 \begin{array}{lcccc}  
 A\hackscore{ij} & = & \delta\hackscore{ij}&  =&  
                  \left[  
                                   \begin{array}{cc}  
                                   1 & 0\\  
                                   0 & 1  
                                   \end{array}  
                  \right] \\  
 D & =& \alpha \\  
 Y & = & f \\  
 \end{array}  
 \label{eq:FirstSteps.3.1}  
 \end{equation}  
 When the PDE is defined via~(\eqn{eq:FirstSteps.3}), \class{linearPDE}  
 makes a particular assumptions about the boundary conditions:  
 \begin{equation}  
 -\sum\hackscore{i,j=0}^k n\hackscore{i} A\hackscore{ij}  
  u\hackscore{,j} = 0  
 \label{eq:FirstSteps.4}  
 \end{equation}  
 Note that this boundary condition does not require extra information  
 as it only refers to the coefficient $A$ which already appears in the  
 PDE and so natural for it. Therefore this boundary condition is called  
 a \index{natural boundary condition}.  
   
 We will set $\alpha=10$ and $f=10$ such that $u=1$ becomes the exact  
 solution of the boundary value problem.  We make this very simple  
 choice to be able to test our program as we can compare the result  
 with the exact solution. Later we will set $\alpha$ and $f$ to  
 functions of their locations in the domain in which case we will not  
 be able to give an analytic solution. However, after testing our  
 program on this very simple case, we can be confident that it working  
 correctly before we apply it is a more complicated situation.  
   
 This is the program to solve the boundary value problem: (Remember that  
 lines starting with '\#' are commend lines in Python)  
 %\verbatiminput{examples/FirstSteps1.py}  
 \begin{python}  
 # import ESyS and finley  
 from ESyS import *  
 import finley  
 # set a value for alpha:  
 alpha=10  
 # generate mesh:  
 mydomain=finley.Rectangle(n0=40,n1=20,l0=2.,l1=1.)  
 # generate a system:  
 mypde=linearPDE(A=[[1,0],[0,1]],D=alpha,Y=10,domain=mydomain)  
 # generate a test solution:  
 u=mypde.getSolution()  
 # calculate the error of the solution  
 error=u-1.  
 print "norm of the approximation error is ",Lsup(error)  
 \end{python}  
 Line 2 import \escript and a few other tools for \ESyS. In Line 3 is  
 importing \finley which is used to solve the partial differential  
 equation. In line 7, a rectangular domain of length $l\hackscore 0=2$  
 and height $l\hackscore 1=1$ is generated and subdivided in  
 $n\hackscore 0=40$ and $n\hackscore 1=20$ elements in $x\hackscore 0$  
 and $x\hackscore 1$ direction, respectively.  
12    
 We are using a function of \finley. This determines later in the code  
 which solver for the PDE is actually being used to solve the PDE.  
13    
 The solution is done in three steps:  
14  \begin{enumerate}  \begin{enumerate}
15  \item generate a finite element mesh subdividing the domain into elements.   \item general structure
16  \item assemble the system of linear equations $Mu=b$ from the BVP   \item how to get the software
17  \item solve the linear system to get $u$   \item a few words about the general structure
18    \item installation
19  \end{enumerate}  \end{enumerate}
 The returned $u$ is given an approximation of the solution of the BVP  
 at the nodes of the finite element mesh. The quality of the  
 approximation depends on the size of the elements of the finite  
 element mesh: As smaller the element size the better the  
 approximation.  In our example we know the solution of the BVP so we  
 can compare the returned approximation with the true solution.  In  
 fact, as the true solution is simple, we can expect that the  
 approximation is exact.  
   
 The first step imports the package \ESyS which includes among  
 others the module \finley.  
   
 \section{A Time Dependent Problem}  
   
 % \verbatiminput{exam/finley\hackscoretime.py}  
   
 \section{With Dirichlet Conditions}  
   
 % \verbatiminput{exam/finley\hackscoredirichlet.py}  
   
 \section{Systems of PDEs}  
   
 % \verbatiminput{exam/system\hackscoretime.py}  
20    
 \section{Explicit Schemes}  
 % \verbatiminput{exam/explicit\hackscoretime.py}  

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