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<html>
<head>
<title>
FEM2D_HEAT_SPARSE_SQUARE - A Problem in a Square Region for FEM2D_HEAT_SPARSE
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM2D_HEAT_SPARSE_SQUARE <br> A Problem in a Square Region for FEM2D_HEAT_SPARSE
</h1>
<hr>
<p>
<b>FEM2D_HEAT_SPARSE_SQUARE</b> is a MATLAB library which
defines the geometry and other data of the square problem,
so that a solution can be computed by <b>fem2d_heat_sparse</b>.
</p>
<p>
This mesh uses 81 nodes and forms 32 quadratic triangular elements.
</p>
<p>
We assume that the equation to be solved is
<blockquote><b>
dUdT - Laplacian U + K * U = F
</b></blockquote>
with
<blockquote><b>
K = 0,
</b></blockquote>
and
<blockquote><b>
F = (2*pi*pi-1)*sin(pi*x)*sin(pi*y)*exp(-t).
</b></blockquote>
The exact solution is:
<blockquote><b>
U = sin(pi*x) * sin(pi*y) * exp(-t).
</b></blockquote>
which is the formula we use to define our functions <b>G</b>
(for the boundary conditions) and <b>H</b> (for the initial
conditions).
</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_HEAT_SPARSE_SQUARE</b> is available in
<a href = "../../m_src/fem2d_heat_sparse_square/fem2d_heat_sparse_square.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/contour_sequence4/contour_sequence4.html">
CONTOUR_SEQUENCE4</a>,
a MATLAB program which
makes a sequence of contour plots based on a single XY coordinate file,
and a sequence of Z(X,Y) data files.
</p>
<p>
<a href = "../../m_src/fem2d_heat_sparse/fem2d_heat_sparse.html">
FEM2D_HEAT_SPARSE</a>,
a MATLAB program which
solves the time dependent heat equation
in an arbitrary triangulated 2D region,
using MATLAB's sparse matrix storage format and solver.
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "dirichlet_condition.m">dirichlet_condition.m</a>,
the user-supplied routine to evaluate the boundary conditions;
</li>
<li>
<a href = "initial_condition.m">initial_condition.m</a>,
the user-supplied routines to evaluate the initial conditions;
</li>
<li>
<a href = "k_coef.m">k_coef.m</a>,
the user-supplied routines to evaluate the linear coefficient.
</li>
<li>
<a href = "rhs.m">rhs.m</a>,
the user-supplied routines to evaluate the right hand side;
</li>
<li>
<a href = "square_output.txt">square_output.txt</a>,
output from a run of the program;
</li>
<li>
<a href = "square_nodes.png">square_nodes.png</a>,
a PNG image of
the 49 nodes;
</li>
<li>
<a href = "square_nodes.txt">square_nodes.txt</a>,
a text file containing a list, for each node, of its X and Y
coordinates;
</li>
<li>
<a href = "square_elements.png">square_elements.png</a>,
a PNG image of
the 32 elements;
</li>
<li>
<a href = "square_elements.txt">square_elements.txt</a>,
a text file containing a list, for each element, of the six
nodes that compose it;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
The following solution data was computed over 10 time steps:
<ul>
<li>
<a href = "square_time.txt">square_time.txt</a>,
a text file containing the solution times;
</li>
<li>
<a href = "u0000.txt">u0000.txt</a>,
the solution U at time step 0;
</li>
<li>
<a href = "u0001.txt">u0001.txt</a>,
the solution U at time step 1;
</li>
<li>
<a href = "u0002.txt">u0002.txt</a>,
the solution U at time step 2;
</li>
<li>
<a href = "u0003.txt">u0003.txt</a>,
the solution U at time step 3;
</li>
<li>
<a href = "u0004.txt">u0004.txt</a>,
the solution U at time step 4;
</li>
<li>
<a href = "u0005.txt">u0005.txt</a>,
the solution U at time step 5;
</li>
<li>
<a href = "u0006.txt">u0006.txt</a>,
the solution U at time step 6;
</li>
<li>
<a href = "u0007.txt">u0007.txt</a>,
the solution U at time step 7;
</li>
<li>
<a href = "u0008.txt">u0008.txt</a>,
the solution U at time step 8;
</li>
<li>
<a href = "u0009.txt">u0009.txt</a>,
the solution U at time step 9;
</li>
<li>
<a href = "u0010.txt">u0010.txt</a>,
the solution U at time step 10;
</li>
</ul>
</p>
<p>
The MATLAB program <b>CONTOUR_SEQUENCE4</b> can make contour
plots from the sequence of solutions:
<ul>
<li>
<a href = "u0000.png">u0000.png</a>,
the solution U at time step 0;
</li>
<li>
<a href = "u0001.png">u0001.png</a>,
the solution U at time step 1;
</li>
<li>
<a href = "u0002.png">u0002.png</a>,
the solution U at time step 2;
</li>
<li>
<a href = "u0003.png">u0003.png</a>,
the solution U at time step 3;
</li>
<li>
<a href = "u0004.png">u0004.png</a>,
the solution U at time step 4;
</li>
<li>
<a href = "u0005.png">u0005.png</a>,
the solution U at time step 5;
</li>
<li>
<a href = "u0006.png">u0006.png</a>,
the solution U at time step 6;
</li>
<li>
<a href = "u0007.png">u0007.png</a>,
the solution U at time step 7;
</li>
<li>
<a href = "u0008.png">u0008.png</a>,
the solution U at time step 8;
</li>
<li>
<a href = "u0009.png">u0009.png</a>,
the solution U at time step 9;
</li>
<li>
<a href = "u0010.png">u0010.png</a>,
the solution U at time step 10;
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source code page</a>.
</p>
<hr>
<i>
Last revised on 06 July 2011.
</i>
<!-- John Burkardt -->
</body>
</html>