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<html>
<head>
<title>
FEM2D_POISSON_RECTANGLE_LINEAR - 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_LINEAR <br> Finite Element Solution of the 2D Poisson Equation
</h1>
<hr>
<p>
<b>FEM2D_POISSON_RECTANGLE_LINEAR</b>
is a FORTRAN90 program which
solves the 2D Poisson equation using the finite element method
over a rectangular region using a uniform mesh of linear triangular elements.
</p>
<p>
The computational region is a rectangle, with 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 region
U(x,y) = G(x,y) on the region 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".
</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. 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>
<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_LINEAR</b> is available in
<a href = "../../c_src/fem2d_poisson/fem2d_poisson.html">a C version</a> and
<a href = "../../cpp_src/fem2d_poisson_rectangle_linear/fem2d_poisson_rectangle_linear.html">a C++ version</a> and
<a href = "../../f77_src/fem2d_poisson_rectangle_linear/fem2d_poisson_rectangle_linear.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fem2d_poisson_rectangle_linear/fem2d_poisson_rectangle_linear.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem2d_poisson_rectangle_linear/fem2d_poisson_rectangle_linear.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../data/fem2d/fem2d.html">
FEM2D</a>,
a data directory which
contains examples of 2D FEM files,
text files that describe a 2D finite element geometry
and associated nodal values;
</p>
<p>
<a href = "../../f_src/fem2d_heat_rectangle/fem2d_heat_rectangle.html">
FEM2D_HEAT_RECTANGLE</a>,
a FORTRAN90 program which
solves the 2D time dependent heat equation on the unit square.
</p>
<p>
<a href = "../../f_src/fem2d_poisson/fem2d_poisson.html">
FEM2D_POISSON</a>,
a FORTRAN90 program which
solves the 2D Poisson equation on an arbitrary triangulated region,
using the finite element method,
and piecewise quadratic triangular elements.
</p>
<p>
<a href = "../../f_src/fem2d_poisson_rectangle/fem2d_poisson_rectangle.html">
FEM2D_POISSON_RECTANGLE</a>,
a FORTRAN90 program which
solves the 2D Poisson equation on a rectangle, using the finite element method,
and piecewise quadratic 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_linear.f90">fem2d_poisson_rectangle_linear.f90</a>, the source code;
</li>
<li>
<a href = "fem2d_poisson_rectangle_linear.sh">fem2d_poisson_rectangle_linear.sh</a>,
commands to compile and run the program;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "fem2d_poisson_rectangle_linear_output.txt">fem2d_poisson_rectangle_linear_output.txt</a>,
the output file;
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main routine of the finite element program FEM2D_POISSON_RECTANGLE_LINEAR.
</li>
<li>
<b>R8GE_FS</b> factors and solves a R8GE system.
</li>
<li>
<b>EXACT</b> calculates the exact solution and its first derivatives.
</li>
</ul>
<p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 24 December 2010.
</i>
<!-- John Burkardt -->
</body>
</html>