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<html>
<head>
<title>
FEM2D_PROJECT_FUNCTION - Piecewise Linear Finite Element Approximation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM2D_PROJECT_FUNCTION <br> Piecewise Linear Finite Element Approximation
</h1>
<hr>
<p>
<b>FEM2D_PROJECT_FUNCTION</b>
is a C++ program which
projects a function W(X,Y), given as a formula, into a given finite element space
of piecewise linear triangular elements.
</p>
<p>
The result is that the function W(x,y) is approximated by a finite element
function U(x,y) formed using piecewise linear triangular elements.
</p>
<p>
The computational region is a rectangle, which is divided up into a mesh
of triangles using a grid of NX by NY points.
For node K at grid point (I,J)
in the interior,
the associated basis function Vk(x,y) is used to pose the equation:
<pre>
Integral U(x,y) Vk(x,y) dx = Integral W(x,y) Vk(x,y) dx
</pre>
while, for node K at grid point (I,J) on the boundary, the associated degree of freedom
is determined by the boundary condition
<pre>
U(x,y) = W(x,y),
</pre>
</p>
<p>
The conditions define a linear system for the coefficients Uk in the finite
element expansion of U(x,y):
<pre>
U(x,y) = sum ( 1 <= K <= M*N ) Uk * Vk(x,y)
</pre>
</p>
<p>
The program computes these coefficients, compares U and W pointwise at
the nodes, and computes the L2 norms of U, W and U-W.
</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_PROJECT_FUNCTION</b> is available in
<a href = "../../cpp_src/fem2d_project_function/fem2d_project_function.html">a C++ version</a> and
<a href = "../../f_src/fem2d_project_function/fem2d_project_function.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem2d_project_function/fem2d_project_function.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/fem1d_project/fem1d_project.html">
FEM1D_PROJECT</a>,
a C++ program which
projects data into a finite element space, including the least squares
approximation of data, or the projection of a finite element solution
from one mesh to another.
</p>
<p>
<a href = "../../data/fem2d/fem2d.html">
FEM2D</a>,
a data directory which
contains examples of 2D FEM files,
three text files that describe a 2D finite element geometry;
</p>
<p>
<a href = "../../cpp_src/fem2d_heat/fem2d_heat.html">
FEM2D_HEAT</a>,
a C++ program which
solves the time dependent heat equation in the unit square.
</p>
<p>
<a href = "../../cpp_src/fem2d_pack/fem2d_pack.html">
FEM2D_PACK</a>,
a C++ library which
includes utitlies for 2D finite element calculations.
</p>
<p>
<a href = "../../cpp_src/fem2d_poisson/fem2d_poisson.html">
FEM2D_POISSON</a>,
a C++ program which
solves the 2D Poisson equation on a rectangle, using the finite element method,
and piecewise quadratic triangular elements.
</p>
<p>
<a href = "../../cpp_src/fem2d_project/fem2d_project.html">
FEM2D_PROJECT</a>,
a C++ program which
projects a function F(X,Y), given as a data, into a given finite element space
of piecewise linear triangular elements.
</p>
<p>
<a href = "../../cpp_src/fem2d_sample/fem2d_sample.html">
FEM2D_SAMPLE</a>,
a C++ library which
evaluates a finite element function defined on an order 3 or order 6 triangulation.
</p>
<p>
<a href = "../../m_src/hot_pipe/hot_pipe.html">
HOT_PIPE</a>,
a MATLAB program which
uses FEM_50_HEAT to solve the heat equation for a pipe model.
</p>
<p>
<a href = "../../m_src/hot_point/hot_point.html">
HOT_POINT</a>,
a MATLAB program which
uses <b>FEM_50_HEAT</b> to solve the heat equation for a 2D region with a hot point.
</p>
<p>
<a href = "../../data/triangulation_order3/triangulation_order3.html">
TRIANGULATION_ORDER3</a>,
a data directory which
contains a description and
examples of how an order 3 (linear) triangulation is defined by a node
and element file.
</p>
<p>
<a href = "../../m_src/triangulation_order3_contour/triangulation_order3_contour.html">
TRIANGULATION_ORDER3_CONTOUR</a>,
a MATLAB program which
makes contour plot of data defined on an order 3 triangulation.
</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_project_function.cpp">fem2d_project_function.cpp</a>, the source code;
</li>
<li>
<a href = "fem2d_project_function.sh">fem2d_project_function.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_project_function_output.txt">fem2d_project_function_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_PROJECT_FUNCTION.
</li>
<li>
<b>EXACT</b> calculates the exact solution.
</li>
<li>
<b>R8GE_FS</b> factors and solves a R8GE system.
</li>
</ul>
<p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 02 June 2009.
</i>
<!-- John Burkardt -->
</body>
</html>