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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 MATLAB 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.
</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>
<i>u</i> = <b>fem1d_bvp_linear</b> ( <i>n</i>, <i>@a</i>, <i>@c</i>,
<i>@f</i>, <i>x</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> values at the nodes,
which can also be regarded as the finite element coefficients.
</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> and
<a href = "../../py_src/fem1d_bvp_linear/fem1d_bvp_linear.html">a Python version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/bvp4c/bvp4c.html">
BVP4C</a>,
MATLAB programs which
illustrate how to use the MATLAB command <b>bvp4c()</b>, which can solve
boundary value problems (BVP's) in one spatial dimension.
</p>
<p>
<a href = "../../m_src/fd1d_bvp/fd1d_bvp.html">
FD1D_BVP</a>,
a MATLAB program which
applies the finite difference method
to a two point boundary value problem in one spatial dimension.
</p>
<p>
<a href = "../../m_src/fem_neumann/fem_neumann.html">
FEM_NEUMANN</a>,
a MATLAB program which
sets up a time-dependent reaction-diffusion equation in 1D,
with Neumann boundary conditions,
discretized using the finite element method.
</p>
<p>
<a href = "../../m_src/fem1d/fem1d.html">
FEM1D</a>,
a MATLAB program which
applies the finite element method to a linear two point boundary value problem
in a 1D region.
</p>
<p>
<a href = "../../m_src/fem1d_adaptive/fem1d_adaptive.html">
FEM1D_ADAPTIVE</a>,
a MATLAB 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 = "../../m_src/fem1d_bvp_quadratic/fem1d_bvp_quadratic.html">
FEM1D_BVP_QUADRATIC</a>,
a MATLAB program which
applies the finite element method (FEM), with piecewise quadratic
elements, to a two point boundary value problem (BVP) in one
spatial dimension, and compares the computed and exact solutions
with the L2 and seminorm errors.
</p>
<p>
<a href = "../../m_src/fem1d_display/fem1d_display.html">
FEM1D_DISPLAY</a>,
a MATLAB program which
reads three files defining a 1D arbitrary degree finite element function,
and displays a plot.
</p>
<p>
<a href = "../../m_src/fem1d_function_10_display/fem1d_function_10_display.html">
FEM1D_FUNCTION_10_DISPLAY</a>,
a MATLAB program which
reads a prefix defining three finite element data files,
reads the data, samples the finite element function, and displays
a plot.
</p>
<p>
<a href = "../../m_src/fem1d_lagrange/fem1d_lagrange.html">
FEM1D_LAGRANGE</a>,
a MATLAB library which
sets up the matrices and vectors associated with the finite element
method (FEM) solution of a boundary value problem (BVP) -u''+u=f(x),
using Lagrange basis polynomials.
</p>
<p>
<a href = "../../m_src/fem1d_nonlinear/fem1d_nonlinear.html">
FEM1D_NONLINEAR</a>,
a MATLAB program which
applies the finite element method to a nonlinear two point boundary value problem
in a 1D region.
</p>
<p>
<a href = "../../m_src/fem1d_pmethod/fem1d_pmethod.html">
FEM1D_PMETHOD</a>,
a MATLAB 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>
<p>
<a href = "../../m_src/fem2d_bvp_linear/fem2d_bvp_linear.html">
FEM2D_BVP_LINEAR</a>,
a MATLAB program which
applies the finite element method (FEM), with piecewise linear elements,
to a 2D boundary value problem (BVP) in a rectangle,
and compares the computed and exact solutions
with the L2 and seminorm errors.
</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 = "l1_error.m">l1_error.m</a>,
estimates the little l1 norm of the error, given the solution of
the finite element problem, and a function that evaluates
the exact solution.
</li>
<li>
<a href = "l2_error_linear.m">l2_error_linear.m</a>,
estimates the L2 norm of the error, given the
piecewise linear solution of
the finite element problem, and a function that evaluates
the exact solution.
</li>
<li>
<a href = "h1s_error_linear.m">h1s_error_linear.m</a>,
estimates the seminorm of the error, given the
piecewise linear solution of
the finite element problem, and a function that evaluates
the derivative of the exact solution.
</li>
<li>
<a href = "fem1d_bvp_linear.m">fem1d_bvp_linear.m</a>,
sets up and solves the finite element problem.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>FEM1D_BVP_TEST</b> runs example problems described
by Dianne O'Leary, and several others.
<ul>
<li>
<a href = "fem1d_bvp_linear_test.m">fem1d_bvp_linear_test.m</a>,
a program which calls fem_bvp_linear with some test cases.
</li>
<li>
<a href = "fem1d_bvp_linear_test_output.txt">fem1d_bvp_linear_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "fem1d_bvp_linear_test01.m">fem1d_bvp_linear_test01.m</a>
</li>
<li>
<a href = "fem1d_bvp_linear_test02.m">fem1d_bvp_linear_test02.m</a>
</li>
<li>
<a href = "fem1d_bvp_linear_test03.m">fem1d_bvp_linear_test03.m</a>
</li>
<li>
<a href = "fem1d_bvp_linear_test04.m">fem1d_bvp_linear_test04.m</a>
</li>
<li>
<a href = "fem1d_bvp_linear_test05.m">fem1d_bvp_linear_test05.m</a>
</li>
<li>
<a href = "fem1d_bvp_linear_test06.m">fem1d_bvp_linear_test06.m</a>
</li>
<li>
<a href = "fem1d_bvp_linear_test07.m">fem1d_bvp_linear_test07.m</a>
</li>
<li>
<a href = "fem1d_bvp_linear_test08.m">fem1d_bvp_linear_test08.m</a>
</li>
<li>
<a href = "fem1d_bvp_linear_test09.m">fem1d_bvp_linear_test09.m</a>
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 16 June 2014.
</i>
<!-- John Burkardt -->
</body>
</html>