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<html>
<head>
<title>
FEM1D_PROJECT - Project Data onto a 1D Finite Element Mesh
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM1D_PROJECT <br> Project Data onto a 1D Finite Element Mesh
</h1>
<hr>
<p>
<b>FEM1D_PROJECT</b>
is a FORTRAN90 program which
computes the projection of data onto a given 1D finite element mesh.
</p>
<p>
This procedure can be used to determine the least squares approximant to
data (actually, to the piecewise linear interpolant of that data) or to
determine the finite element coefficients appropriate when recomputing
a finite element solution from a fine mesh to a coarse mesh.
</p>
<p>
The sample data is given simply as two files, one containing the points,
the other, the corresponding data values.
</p>
<p>
The user must also supply two files defining the finite element mesh,
containing the points, and the pairs of points that form the elements.
</p>
<p>
The program takes the sample data, projects it onto the finite element mesh,
and determines a file of finite element coefficients that represent the
projection of the data onto the finite element space.
</p>
<p>
The current version of the program can only handle finite element
meshes which are made of piecewise linear basis functions.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>fem1d_project</b> <i>sample_prefix</i> <i>fem_prefix</i>
</blockquote>
where <i>sample_prefix</i> is the common prefix for the SAMPLE files:
<ul>
<li>
<i>sample_prefix</i><b>_nodes.txt</b>, the node coordinates.
</li>
<li>
<i>sample_prefix</i><b>_values.txt</b>, the data values;
</li>
</ul>
and <i>fem_prefix</i> is the common prefix for the FEM files:
<ul>
<li>
<i>fem_prefix</i><b>_nodes.txt</b>, the node coordinates.
</li>
<li>
<i>fem_prefix</i><b>_elements.txt</b>, the nodes that make up each element;
</li>
<li>
<i>fem_prefix</i><b>_values.txt</b>, created by the program, and contains
the projections of the data values onto the finite element space, that is,
these may be regarded as coefficients of finite element functions
representing the projections of the data.
</li>
</ul>
</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>FEM1D_PROJECT</b> is available in
<a href = "../../cpp_src/fem1d_project/fem1d_project.html">a C++ version</a> and
<a href = "../../f77_src/fem1d_project/fem1d_project.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fem1d_project/fem1d_project.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem1d_project/fem1d_project.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../data/fem1d/fem1d.html">
FEM1D</a>,
a data directory which
contains examples of 1D FEM files,
three text files that describe a 1D finite element model;
</p>
<p>
<a href = "../../f_src/fem1d/fem1d.html">
FEM1D</a>,
a FORTRAN90 program which
applies the finite element method to a 1D linear two point boundary value problem.
</p>
<p>
<a href = "../../f_src/fem1d_adaptive/fem1d_adaptive.html">
FEM1D_ADAPTIVE</a>,
a FORTRAN90 program which
applies the finite element method to a 1D linear two point boundary value problem
using adaptive refinement to improve the solution.
</p>
<p>
<a href = "../../f_src/fem1d_bvp_linear/fem1d_bvp_linear.html">
FEM1D_BVP_LINEAR</a>,
a FORTRAN90 program which
applies the finite element method, with piecewise linear elements,
to a two point boundary value problem in one spatial dimension.
</p>
<p>
<a href = "../../f_src/fem1d_heat_steady/fem1d_heat_steady.html">
FEM1D_HEAT_STEADY</a>,
a FORTRAN90 program which
uses the finite element method to solve the steady (time independent)
heat equation in 1D.
</p>
<p>
<a href = "../../f_src/fem1d_nonlinear/fem1d_nonlinear.html">
FEM1D_NONLINEAR</a>,
a FORTRAN90 program which
applies the finite element method to a 1D nonlinear two point boundary value problem.
</p>
<p>
<a href = "../../f_src/fem1d_pack/fem1d_pack.html">
FEM1D_PACK</a>,
a FORTRAN90 library which
contains utilities for 1D finite element calculations.
</p>
<p>
<a href = "../../f_src/fem1d_pmethod/fem1d_pmethod.html">
FEM1D_PMETHOD</a>,
a FORTRAN90 program which
applies the p-method version of the finite element method to a 1D linear
two point boundary value problem.
</p>
<p>
<a href = "../../f_src/fem1d_sample/fem1d_sample.html">
FEM1D_SAMPLE</a>,
a FORTRAN90 program which
samples a scalar or vector finite element function of one variable,
defined by <a href = "../../data/fem/fem.html">FEM files</a>,
returning interpolated values at the sample points.
</p>
<p>
<a href = "../../f_src/fem2d_project/fem2d_project.html">
FEM2D_PROJECT</a>,
a FORTRAN90 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 = "../../f_src/fem2d_project_function/fem2d_project_function.html">
FEM2D_PROJECT_FUNCTION</a>,
a FORTRAN90 program which
projects a function F(X,Y), given as a formula, into a given finite element space
of piecewise linear triangular elements.
</p>
<p>
<a href = "../../f_src/fem2d_sample/fem2d_sample.html">
FEM2D_SAMPLE</a>,
a FORTRAN90 program which
evaluates a finite element function defined on an order 3 or order 6 triangulation.
</p>
<p>
<a href = "../../f_src/piecewise_linear_product_integral/piecewise_linear_product_integral.html">
PIECEWISE_LINEAR_PRODUCT_INTEGRAL</a>,
a FORTRAN90 library which
calculates the exact value of the integral of the product of two
piecewise linear functions F(X) and G(X).
</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 = "fem1d_project.f90">fem1d_project.f90</a>, the source code.
</li>
<li>
<a href = "fem1d_project.sh">fem1d_project.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>POWERS</b> starts with sample data for the vector function f(x)=[ 1, x, x^2], on a grid
of 11 equally spaced nodes from 0 to 10, and projects this onto a piecewise linear
grid of 6 equally spaced nodes from 0 to 10.
<ul>
<li>
<a href = "sample_powers_nodes.txt">sample_powers_nodes.txt</a>,
the sample nodes.
</li>
<li>
<a href = "sample_powers_values.txt">sample_powers_values.txt</a>,
the sample nodal values.
</li>
<li>
<a href = "fem_powers_nodes.txt">fem_powers_nodes.txt</a>,
the FEM nodes.
</li>
<li>
<a href = "fem_powers_elements.txt">fem_powers_elements.txt</a>,
the FEM elements.
</li>
<li>
<a href = "fem_powers_values.txt">fem_powers_values.txt</a>,
the projected FEM nodal values.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for FEM1D_PROJECT.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>DTABLE_DATA_READ</b> reads data from a DTABLE file.
</li>
<li>
<b>DTABLE_HEADER_READ</b> reads the header from a DTABLE file.
</li>
<li>
<b>DTABLE_WRITE0</b> writes a DTABLE file with no headers.
</li>
<li>
<b>FEM1D_APPROXIMATE</b> approximates data at sample points with an FEM function.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>ITABLE_DATA_READ</b> reads data from an ITABLE file.
</li>
<li>
<b>ITABLE_HEADER_READ</b> reads the header from an integer table file.
</li>
<li>
<b>PIECEWISE_LINEAR_PRODUCT_QUAD:</b> piecewise linear product integral.
</li>
<li>
<b>R83_NP_FSS</b> factors and solves multiple R83 systems.
</li>
<li>
<b>R8VEC_BRACKET3</b> finds the interval containing or nearest a given value.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_I4VEC</b> reads an I4VEC from a string.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</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 13 June 2009.
</i>
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