1 
jgs 
102 
% $Id$ 
2 



3 


\chapter{The First Steps} 
4 


\label{FirstSteps} 
5 



6 


\begin{figure} 
7 


\centerline{\includegraphics[width=\figwidth]{FirstStepDomain}} 
8 


\caption{Domain $\Omega=[0,1]^2$ with outer normal field $n$.} 
9 


\label{fig:FirstSteps.1} 
10 


\end{figure} 
11 



12 


In this chapter we will show the basics of how to use \escript to solve 
13 


a partial differential equation \index{partial differential equation} (PDE \index{partial differential equation!PDE}). The reader should be familiar with 
14 


the basics of Python. The knowledge presented the Python tutorial at \url{http://docs.python.org/tut/tut.html} 
15 


is sufficient. It is helpful if the reader has some basic knowledge on PDEs \index{PDE}. 
16 



17 


The \index{PDE} we want to solve is the Poisson equation \index{Poisson equation} 
18 


\begin{equation} 
19 


\Delta u =f 
20 


\label{eq:FirstSteps.1} 
21 


\end{equation} 
22 


for the solution $u$. The domain of interest which we denote by $\Omega$ 
23 


is the unit square 
24 


\begin{equation} 
25 


\Omega=[0,1]^2=\{ (x\hackscore 0;x\hackscore 1)  0\le x\hackscore{0} \le 1 \mbox{ and } 0\le x\hackscore{1} \le 1 \} 
26 


\label{eq:FirstSteps.1b} 
27 


\end{equation} 
28 


The domain is shown in Figure~\fig{fig:FirstSteps.1}. 
29 



30 


$\Delta$ denotes the Laplace operator\index{Laplace operator} which is defined by 
31 


\begin{equation} 
32 


\Delta u = (u\hackscore {,0})\hackscore{,0}+(u\hackscore{,1})\hackscore{,1} 
33 


\label{eq:FirstSteps.1.1} 
34 


\end{equation} 
35 


where for any function $w$ and any direction $i$ $u\hackscore{,i}$ 
36 


denotes the partial derivative \index{partial derivative} of $w$ with respect to $i$. 
37 


\footnote{Some readers 
38 


may be more familiar with the Laplace operator\index{Laplace operator} being written 
39 


as $\nabla^2$, and written in the form 
40 


\begin{equation*} 
41 


\nabla^2 u = \frac{\partial^2 u}{\partial x\hackscore 0^2} 
42 


+ \frac{\partial^2 u}{\partial x\hackscore 1^2} 
43 


\end{equation*} 
44 


and \eqn{eq:FirstSteps.1} as 
45 


\begin{equation*} 
46 


\nabla^2 u = f 
47 


\end{equation*} 
48 


} 
49 


Basically in the subindex of a function any index left to the comma denotes a spatial derivative with respect 
50 


to the index. To get a more compact form we will write $w\hackscore{,ij}=(w\hackscore {,i})\hackscore{,j}$ 
51 


which leads to 
52 


\begin{equation} 
53 


\Delta u = u\hackscore{,00}+u\hackscore{,11}=\sum\hackscore{i=0}^2 u\hackscore{,ii} 
54 


\label{eq:FirstSteps.1.1b} 
55 


\end{equation} 
56 


In some cases, and we will see examples for this in the next chapter, 
57 


the usage of the nested $\sum$ symbols blows up the formulas and therefore 
58 


it is convenient to use Einstein summation convention \index{summation convention} which 
59 


says that $\sum$ sign is dropped and a summation over a repeated index is performed 
60 


("repeated index means summation"). For instance we write 
61 


\begin{eqnarray} 
62 


x\hackscore{i}y\hackscore{i}=\sum\hackscore{i=0}^2 x\hackscore{i}y\hackscore{i} \\ 
63 


x\hackscore{i}u\hackscore{,i}=\sum\hackscore{i=0}^2 x\hackscore{i}u\hackscore{,i} \\ 
64 


u\hackscore{,ii}=\sum\hackscore{i=0}^2 u\hackscore{,ii} \\ 
65 


\label{eq:FirstSteps.1.1c} 
66 


\end{eqnarray} 
67 


With the summation convention we can write the Poisson equation \index{Poisson equation} as 
68 


\begin{equation} 
69 


 u\hackscore{,ii} =1 
70 


\label{eq:FirstSteps.1.sum} 
71 


\end{equation} 
72 


On the boundary of the domain $\Omega$ the normal derivative $n\hackscore{i} u\hackscore{,i}$ 
73 


of the solution $u$ shall be zero, ie. $u$ shall fulfill 
74 


the homogeneous Neumann boundary condition\index{Neumann 
75 


boundary condition!homogeneous} 
76 


\begin{equation} 
77 


n\hackscore{i} u\hackscore{,i}= 0 \;. 
78 


\label{eq:FirstSteps.2} 
79 


\end{equation} 
80 


$n=(n\hackscore{i})$ denotes the outer normal field 
81 


of the domain, see \fig{fig:FirstSteps.1}. Remember that we 
82 


are applying the Einstein summation convention \index{summation convention}, i.e 
83 


$n\hackscore{i} u\hackscore{,i}= n\hackscore1 u\hackscore{,1} + 
84 


n\hackscore2 u\hackscore{,2}$. 
85 


\footnote{Some readers may familiar with the notation 
86 


\begin{equation*} 
87 


\frac{\partial u}{\partial n} = n\hackscore{i} u\hackscore{,i} 
88 


\end{equation*} 
89 


for the normal derivative.} 
90 


The Neumann boundary condition of \eqn{eq:FirstSteps.2} should be fulfilled on the 
91 


set $\Gamma^N$ which is the top and right edge of the domain: 
92 


\begin{equation} 
93 


\Gamma^N=\{(x\hackscore 0;x\hackscore 1) \in \Omega  x\hackscore{0}=1 \mbox{ or } x\hackscore{1}=1 \} 
94 


\label{eq:FirstSteps.2b} 
95 


\end{equation} 
96 


On the bottom and the left edge of the domain which defined 
97 


as 
98 


\begin{equation} 
99 


\Gamma^D=\{(x\hackscore 0;x\hackscore 1) \in \Omega  x\hackscore{0}=0 \mbox{ or } x\hackscore{1}=0 \} 
100 


\label{eq:FirstSteps.2c} 
101 


\end{equation} 
102 


the solution shall be identically zero: 
103 


\begin{equation} 
104 


u=0 \; . 
105 


\label{eq:FirstSteps.2d} 
106 


\end{equation} 
107 


The kind of boundary condition is called a homogeneous Dirichlet boundary condition 
108 


\index{Dirichlet boundary condition!homogeneous}. The partial differential equation in \eqn{eq:FirstSteps.1.sum} together 
109 


with the Neumann boundary condition \eqn{eq:FirstSteps.2} and 
110 


Dirichlet boundary condition in \eqn{eq:FirstSteps.2d} form a so 
111 


called boundary value 
112 


problem\index{boundary value problem} (BVP\index{boundary value problem!BVP}) for unknown 
113 


function $u$. 
114 



115 



116 


\begin{figure} 
117 


\centerline{\includegraphics[width=\figwidth]{FirstStepMesh}} 
118 


\caption{Mesh of $4 \time 4$ elements on a rectangular domain. Here 
119 


each element is a quadrilateral and described by four nodes, namely 
120 


the corner points. The solution is interpolated by a bilinear 
121 


polynomial.} 
122 


\label{fig:FirstSteps.2} 
123 


\end{figure} 
124 



125 


In general the BVP\index{boundary value problem!BVP} cannot be solved analytically and numerical 
126 


methods have to be used construct an approximation of the solution 
127 


$u$. Here we will use the finite element method\index{finite element 
128 


method} (FEM\index{finite element 
129 


method!FEM}). The basic idea is to fill the domain with a 
130 


set of points, so called nodes. The solution is approximated by its 
131 


values on the nodes\index{finite element 
132 


method!nodes}. Moreover, the domain is subdivide into small 
133 


subdomain, socalled elements \index{finite element 
134 


method!element}. On each element the solution is 
135 


represented by a polynomial of a certain degree through its values at 
136 


the nodes located in the element. The nodes and its connection through 
137 


elements is called a mesh\index{finite element 
138 


method!mesh}. Figure~\fig{fig:FirstSteps.2} shows an 
139 


example of a FEM mesh with four elements in the $x_0$ and four elements 
140 


in the $x_1$ direction over the unit square. 
141 


For more details we refer the reader to the literature, for instance 
142 


\Ref{Zienc,NumHand}. 
143 



144 


\escript provides the class \Poisson to define a Poisson equation \index{Poisson equation}. 
145 


(We will discuss a more general form of a PDE \index{partial differential equation!PDE} 
146 


that can be defined through the \LinearPDE class later). The instantiation of 
147 


a \Poisson class object requires the specification of the domain $\Omega$. In \escript 
148 


the \Domain class objects are used to describe the geometry of a domain but it also 
149 


contains information about the discretization methods and the actual solver which is used 
150 


to solve the PDE. Here we are using the FEM library \finley \index{finite element 
151 


method}. The following statements create the \Domain object \var{mydomain} from the 
152 


\finley method \method{Rectangle} 
153 


\begin{python} 
154 


import finley 
155 


mydomain = finley.Rectangle(l0=1.,l1=1.,n0=40, n1=20) 
156 


\end{python} 
157 


In this case the domain is a rectangle with the lower, left corner at point $(0,0)$ and 
158 


the right, upper corner at $(\var{l0},\var{l1})=(1,1)$. 
159 


The arguments \var{l0} and \var{l1} define the number of elements in $x\hackscore{0}$ and 
160 


$x\hackscore{1}$direction respectively. For more details on \method{Rectangle} and 
161 


other \Domain generators within the \finley module, 
162 


see \Chap{CHAPTER ON FINLEY}. 
163 



164 


The following statements define the \Poisson object \var{mypde} with domain var{mydomain} and 
165 


the right hand side $f$ of the PDE to constant $1$: 
166 


\begin{python} 
167 


import escript 
168 


mypde = escript.Poisson(domain=mydomain,f=1) 
169 


\end{python} 
170 


We have not specified any boundary condition but the 
171 


\Poisson class implicitly assumes homogeneous Neuman boundary conditions \index{Neumann 
172 


boundary condition!homogeneous} defined by \eqn{eq:FirstSteps.2}. With this boundary 
173 


condition the BVP\index{boundary value problem!BVP} we have defined has no unique solution. In fact, with any solution $u$ 
174 


and any constant $C$ the function $u+C$ becomes a solution as well. We have to add 
175 


a Dirichlet boundary condition \index{Dirichlet boundary condition}. This is done 
176 


by defines a characteristic function \index{characteristic function} 
177 


which has a positive values at locations $(x_0,x_1)$ where Dirichlet boundary condition is set 
178 


and $0$ elsewhere. In our case of $\Gamma^D$ defined by \eqn{eq:FirstSteps.2c}, 
179 


we need a function which is positive for the cases $x_0=0$ or $x_1=0$: 
180 


\begin{python} 
181 


x=mydomain.getX() 
182 


gammaD=x[0].whereZero()+x[1].whereZero() 
183 


\end{python} 
184 


In first statement returns, the method \method{getX} of the \Domain \var{mydomain} access to the locations 
185 


in the domain defined by \var{mydomain}. The object \var{x} is actually an \Data object 
186 


which we will learn more about later. \code{x[0]} returns the $x_0$ coordinates of the locations and 
187 


\code{x[0].whereZero()} creates function which equals $1$ where \code{x[0]} is (nearly) equal to zero 
188 


and $0$ elsewhere. The sum of the results of \code{x[0].whereZero()} and \code{x[1].whereZero()} gives a function on the domain \var{mydomain} which is exactly positive where $x_0$ or $x_1$ is equal to zero. 
189 



190 


The additional parameter \var{q} of the \Poisson object creater defines the 
191 


characteristic function \index{characteristic function} of the locations 
192 


of the domain where homogeneous Dirichlet boundary condition \index{Dirichlet boundary condition!homogeneous} 
193 


are set. The complete definition of our example is now: 
194 


\begin{python} 
195 


from linearPDEs import Poisson 
196 


x = mydomain.getX() 
197 


gammaD = x[0].whereZero()+x[1].whereZero() 
198 


mypde = Poisson(domain=mydomain,f=1,q=gammaD) 
199 


\end{python} 
200 


The first statement imports the \Poisson class definition form the \linearPDEsPack module which is part of the \escript module. 
201 


To get the solution of the Poisson equation defines by \var{mypde} we just have to call its 
202 


\method{getSolution}. 
203 



204 


Now we can write the script to solve our test problem (Remember that 
205 


lines starting with '\#' are commend lines in Python) (available as \file{mypoisson.py} 
206 


in the \ExampleDirectory): 
207 


\begin{python} 
208 


import esys.finley 
209 


from esys.linearPDEs import Poisson 
210 


# generate domain: 
211 


mydomain = esys.finley.Rectangle(l0=1.,l1=1.,n0=40, n1=20) 
212 


# define characteristic function of Gamma^D 
213 


x = mydomain.getX() 
214 


gammaD = x[0].whereZero()+x[1].whereZero() 
215 


# define PDE and get its solution u 
216 


mypde = Poisson(domain=mydomain,f=1,q=gammaD) 
217 


u = mypde.getSolution() 
218 


# write u to an external file 
219 


u.saveDX("u.dx") 
220 


\end{python} 
221 


The last statement writes the solution to the external file \file{u.dx} in 
222 


\OpenDX file format. \OpenDX is a software package 
223 


for the visualization of scientific, engineering and analytical data and is freely available 
224 


from \url{http://www.opendx.org}. 
225 



226 


\begin{figure} 
227 


\centerline{\includegraphics[width=\figwidth]{FirstStepResult.eps}} 
228 


\caption{\OpenDX visualization of the Possion equation soluition for $f=1$} 
229 


\label{fig:FirstSteps.3} 
230 


\end{figure} 
231 



232 


You can edit this script using your favourite text editor (or the Integrated DeveLopment Environment IDLE 
233 


for Python). If the script file has the name \file{mypoisson.py} \index{scripts!\file{mypoisson.py}} you can run the 
234 


script from any shell using the command: 
235 


\begin{verbatim} 
236 


python mypoisson.py 
237 


\end{verbatim} 
238 


After the script has (hopefully successfully) been completed you will find the file \file{u.dx} in the current 
239 


directory. An easy way to visualize the results is the command 
240 


\begin{verbatim} 
241 


dx prompter 
242 


\end{verbatim} 
243 


to start the generic data visualization interface of \OpenDX. \fig{fig:FirstSteps.3} shows the result. 
244 



245 



246 



247 


