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<html>
<head>
<title>
FEM1D_BVP_LINEAR - Finite Element Method, 1D, Boundary Value Problem, Piecewise Linear Elements
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM1D_BVP_LINEAR <br> Finite Element Method, 1D, Boundary Value Problem, Piecewise Linear Elements
</h1>
<hr>
<p>
<b>FEM1D_BVP_LINEAR</b>
is a FORTRAN90 program which
applies the finite element method, with piecewise linear elements,
to a two point boundary value problem in one spatial dimension,
and compares the computed and exact solutions
with the L2 and seminorm errors.
</p>
<p>
The boundary value problem (BVP) that is to be solved has the form:
<pre>
- d/dx ( a(x) * du/dx ) + c(x) * u(x) = f(x)
</pre>
in the interval 0 < x < 1. The functions a(x), c(x), and f(x) are
given functions, and a formula for a'(x) is also available.
</p>
<p>
Boundary conditions are applied at the endpoints, and in this case,
these are assumed to have the form:
<pre>
u(0.0) = 0.0;
u(1.0) = 0.0.
</pre>
</p>
<p>
To compute a finite element approximation, a set of n equally spaced
nodes is defined from 0.0 to 1.0, a set of piecewise linear basis functions
is set up, with one basis function associated with each node,
and then an integral form of the BVP is used, in which the differential
equation is multiplied by each basis function, and integration by parts is
used to simplify the integrand.
</p>
<p>
A simple two point Gauss quadrature formula is used to estimate the
resulting integrals over each interval.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
call <b>fem1d_bvp_linear</b> ( <i>n</i>, <i>a</i>, <i>c</i>, <i>f</i>,
<i>x</i>, <i>u</i> )
</blockquote>
where
<ul>
<li>
<i>n</i> is the number of equally spaced nodes.
</li>
<li>
<i>a</i> is the function which evaluates a(x);
</li>
<li>
<i>c</i> is the function which evaluates c(x);
</li>
<li>
<i>f</i> is the function which evaluates f(x).
</li>
<li>
<i>x</i> is the input vector of <i>n</i> nodes.
</li>
<li>
<i>u</i> is the output vector of <i>n</i> finite difference values.
</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_BVP_LINEAR</b> is available in
<a href = "../../c_src/fem1d_bvp_linear/fem1d_bvp_linear.html">a C version</a> and
<a href = "../../cpp_src/fem1d_bvp_linear/fem1d_bvp_linear.html">a C++ version</a> and
<a href = "../../f77_src/fem1d_bvp_linear/fem1d_bvp_linear.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fem1d_bvp_linear/fem1d_bvp_linear.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem1d_bvp_linear/fem1d_bvp_linear.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/fd1d_bvp/fd1d_bvp.html">
FD1D_BVP</a>,
a FORTRAN90 program which
applies the finite difference method
to a two point boundary value problem in one spatial dimension.
</p>
<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 linear two point boundary value problem
in a 1D region.
</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 linear two point boundary value problem
in a 1D region, using adaptive refinement to improve the solution.
</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 nonlinear two point boundary value problem
in a 1D region.
</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 linear
two point boundary value problem in a 1D region.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Dianne O'Leary,<br>
Finite Differences and Finite Elements: Getting to Know You,<br>
Computing in Science and Engineering,<br>
Volume 7, Number 3, May/June 2005.
</li>
<li>
Dianne O'Leary,<br>
Scientific Computing with Case Studies,<br>
SIAM, 2008,<br>
ISBN13: 978-0-898716-66-5,<br>
LC: QA401.O44.
</li>
<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_bvp_linear.f90">fem1d_bvp_linear.f90</a>, the source code.
</li>
<li>
<a href = "fem1d_bvp_linear.sh">fem1d_bvp_linear.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "fem1d_bvp_linear_prb.f90">fem1d_bvp_linear_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "fem1d_bvp_linear_prb.sh">fem1d_bvp_linear_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "fem1d_bvp_linear_prb_output.txt">fem1d_bvp_linear_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>COMPUTE_L2_ERROR</b> estimates the L2 error norm of a finite element solution.
</li>
<li>
<b>COMPUTE_SEMINORM_ERROR</b> estimates the seminorm error of a finite element solution.
</li>
<li>
<b>FEM1D_BVP_LINEAR</b> solves a two point boundary value problem.
</li>
<li>
<b>R8MAT_SOLVE2</b> computes the solution of an N by N linear system.
</li>
<li>
<b>R8VEC_EVEN</b> returns an R8VEC of evenly spaced values.
</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 19 August 2010.
</i>
<!-- John Burkardt -->
</body>
</html>