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<html>
<head>
<title>
FEM1D_BVP_QUADRATIC - Finite Element Method, 1D, Boundary Value Problem, Piecewise Quadratic Elements
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM1D_BVP_QUADRATIC <br> Finite Element Method, 1D, Boundary Value Problem, Piecewise Quadratic Elements
</h1>
<hr>
<p>
<b>FEM1D_BVP_QUADRATIC</b>
is a C++ program which
applies the finite element method, with piecewise quadratic 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 quadratoc 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 three point Gauss quadrature formula is used to estimate the
resulting integrals over each interval.
</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_QUADRATIC</b> is available in
<a href = "../../c_src/fem1d_bvp_quadratic/fem1d_bvp_quadratic.html">a C version</a> and
<a href = "../../cpp_src/fem1d_bvp_quadratic/fem1d_bvp_quadratic.html">a C++ version</a> and
<a href = "../../f77_src/fem1d_bvp_quadratic/fem1d_bvp_quadratic.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fem1d_bvp_quadratic/fem1d_bvp_quadratic.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem1d_bvp_quadratic/fem1d_bvp_quadratic.html">a MATLAB version</a> and
<a href = "../../py_src/fem1d_bvp_quadratic/fem1d_bvp_quadratic.html">a Python version</a>..
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/fd1d_bvp/fd1d_bvp.html">
FD1D_BVP</a>,
a C++ program which
applies the finite difference method
to a two point boundary value problem in one spatial dimension.
</p>
<p>
<a href = "../../cpp_src/fem1d/fem1d.html">
FEM1D</a>,
a C++ program which
applies the finite element method to a linear two point boundary value problem
in a 1D region.
</p>
<p>
<a href = "../../cpp_src/fem1d_adaptive/fem1d_adaptive.html">
FEM1D_ADAPTIVE</a>,
a C++ 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 = "../../cpp_src/fem1d_bvp_linear/fem1d_bvp_linear.html">
FEM1D_BVP_LINEAR</a>,
a C++ 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>
<a href = "../../cpp_src/fem1d_heat_steady/fem1d_heat_steady.html">
FEM1D_HEAT_STEADY</a>,
a C++ program which
uses the finite element method to solve the steady (time independent)
heat equation in 1D.
</p>
<p>
<a href = "../../cpp_src/fem1d_nonlinear/fem1d_nonlinear.html">
FEM1D_NONLINEAR</a>,
a C++ program which
applies the finite element method to a nonlinear two point boundary value problem
in a 1D region.
</p>
<p>
<a href = "../../cpp_src/fem1d_pmethod/fem1d_pmethod.html">
FEM1D_PMETHOD</a>,
a C++ 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 = "../../cpp_src/fem2d_bvp_quadratic/fem2d_bvp_quadratic.html">
FEM2D_BVP_QUADRATIC</a>,
a C++ program which
applies the finite element method (FEM), with piecewise quadratic 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 = "fem1d_bvp_quadratic.cpp">fem1d_bvp_quadratic.cpp</a>, the source code.
</li>
<li>
<a href = "fem1d_bvp_quadratic.hpp">fem1d_bvp_quadratic.hpp</a>, the include file.
</li>
<li>
<a href = "fem1d_bvp_quadratic.sh">fem1d_bvp_quadratic.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_quadratic_prb.cpp">fem1d_bvp_quadratic_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "fem1d_bvp_quadratic_prb.sh">fem1d_bvp_quadratic_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "fem1d_bvp_quadratic_prb_output.txt">fem1d_bvp_quadratic_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>FEM1D_BVP_QUADRATIC</b> solves a two point boundary value problem.
</li>
<li>
<b>H1S_ERROR_QUADRATIC</b> estimates the seminorm error of a finite element solution.
</li>
<li>
<b>I4VEC_ZERO_NEW</b> creates and zeroes an I4VEC.
</li>
<li>
<b>L1_ERROR</b> estimates the l1 error norm of a finite element solution.
</li>
<li>
<b>L2_ERROR_QUADRATIC</b> estimates the L2 error norm of a finite element solution.
</li>
<li>
<b>R8MAT_SOLVE2</b> computes the solution of an N by N linear system.
</li>
<li>
<b>R8MAT_ZERO_NEW</b> returns a new zeroed R8MAT.
</li>
<li>
<b>R8VEC_EVEN_NEW</b> returns N real values, evenly spaced between ALO and AHI.
</li>
<li>
<b>R8VEC_ZERO_NEW</b> creates and zeroes an R8VEC.
</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 = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 18 June 2014.
</i>
<!-- John Burkardt -->
</body>
</html>