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<html>
<head>
<title>
FEM2D_POISSON_RECTANGLE - Finite Element Solution of the 2D Poisson Equation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM2D_POISSON_RECTANGLE <br> Poisson Equation in 2D <br> Finite Element Solution
</h1>
<hr>
<p>
<b>FEM2D_POISSON</b>
is a MATLAB program which
solves the 2D Poisson equation using the finite element method,
and quadratic basis functions.
</p>
<p>
The computational region is a rectangle, with homogenous Dirichlet
boundary conditions applied along the boundary. The state variable
U(X,Y) is then constrained by:
<pre>
- ( Uxx + Uyy ) = F(x,y) in the box
U(x,y) = G(x,y) on the box boundary
</pre>
</p>
<p>
The computational region is first covered with an NX by NY
rectangular array of points, creating (NX-1)*(NY-1) subrectangles.
Each subrectangle is divided into two triangles, creating a total
of 2*(NX-1)*(NY-1) geometric "elements". Because quadratic basis
functions are to be used, each triangle will be associated not only
with the three corner nodes that defined it, but with three extra
midside nodes. If we include these additional nodes, there are
now a total of (2*NX-1)*(2*NY-1) nodes in the region.
</p>
<p>
We now assume that the unknown function U(x,y) can be represented
as a linear combination of the basis functions associated with each
node. The value of U at the boundary nodes is obvious, so we
concentrate on the NUNK interior nodes where U(x,y) is unknown.
For each node I, we determine a basis function PHI(I)(x,y), and
evaluate the following finite element integral:
<pre>
Integral ( Ux(x,y) * PHIx(I)(x,y) + Uy(x,y) * PHIy(I)(x,y) ) =
Integral ( F(x,y) * PHI(I)(x,y)
</pre>
The set of all such equations yields a linear system for the
coefficients of the representation of U.
</p>
<p>
The program allows the user to supply two routines:
<ul>
<li>
<b>FUNCTION VALUE = RHS ( X, Y )</b> returns the right hand side
F(x,y) of the Poisson equation.
</li>
<li>
<b>FUNCTION [U, DUDX, DUDY ] = EXACT ( X, Y )</b> returns
the exact solution of the Poisson equation (assuming this is
known.) This routine is necessary so that error analysis
can be performed, reporting the L2 and H1 seminorm errors
between the true and computed solutions.
</li>
</ul>
</p>
<p>
There are a few variables that are easy to manipulate. In particular,
the user can change the variables NX and NY in the main program,
to change the number of nodes and elements. The variables (XL,YB)
and (XR,YT) define the location of the lower left and upper right
corners of the rectangular region, and these can also be changed
in a single place in the main program.
</p>
<p>
The program writes out a file containing an Encapsulated
PostScript image of the nodes and elements, with numbers.
Unfortunately, for values of NX and NY over 10, the plot is
too cluttered to read. For lower values, however, it is
a valuable map of what is going on in the geometry.
</p>
<p>
The program is also able to write out a file containing the
solution value at every node. This file may be used to create
contour plots of the solution.
</p>
<p>
The original version of this code comes from Professor Janet Peterson.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>FEM2D_POISSON_RECTANGLE</b> is available in
<a href = "../../c_src/fem2d_poisson_rectangle/fem2d_poisson_rectangle.html">a C version</a> and
<a href = "../../cpp_src/fem2d_poisson_rectangle/fem2d_poisson_rectangle.html">a C++ version</a> and
<a href = "../../f77_src/fem2d_poisson_rectangle/fem2d_poisson_rectangle.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fem2d_poisson_rectangle/fem2d_poisson_rectangle.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem2d_poisson_rectangle/fem2d_poisson_rectangle.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/fem2d_poisson_rectangle_linear/fem2d_poisson_rectangle_linear.html">
FEM2D_POISSON_RECTANGLE_LINEAR</a>,
a MATLAB program which
solves the 2D Poisson equation on a rectangle,
using the finite element method,
and piecewise linear triangular elements.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Hans Rudolf Schwarz,<br>
Finite Element Methods,<br>
Academic Press, 1988,<br>
ISBN: 0126330107,<br>
LC: TA347.F5.S3313.
</li>
<li>
Gilbert Strang, George Fix,<br>
An Analysis of the Finite Element Method,<br>
Cambridge, 1973,<br>
ISBN: 096140888X,<br>
LC: TA335.S77.
</li>
<li>
Olgierd Zienkiewicz,<br>
The Finite Element Method,<br>
Sixth Edition,<br>
Butterworth-Heinemann, 2005,<br>
ISBN: 0750663200,<br>
LC: TA640.2.Z54
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "fem2d_poisson_rectangle.m">fem2d_poisson_rectangle.m</a>,
the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "rectangle_output.txt">rectangle_output.txt</a>,
the output file.
</li>
<li>
<a href = "rectangle_nodes.png">rectangle_nodes.png</a>,
a PNG image of
the nodes, for NX = NY = 7 (the picture is
hard to read for larger values of NX and NY);
</li>
<li>
<a href = "rectangle_nodes.txt">rectangle_nodes.txt</a>,
a text file containing a list, for each node, of its X and Y
coordinates;
</li>
<li>
<a href = "rectangle_elements.png">rectangle_elements.png</a>,
a PNG image of
the elements, for NX = NY = 7 (the picture is
hard to read for larger values of NX and NY);
</li>
<li>
<a href = "rectangle_elements.txt">rectangle_elements.txt</a>,
a text file containing a list, for each element, of the six
nodes that compose it;
</li>
<li>
<a href = "rectangle_solution.txt">rectangle_solution.txt</a>,
a text file containing a list, for each node, of the value
of the solution;
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 16 May 2009.
</i>
<!-- John Burkardt -->
</body>
</html>