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<html>
<head>
<title>
FEM1D_NONLINEAR - Finite Element Method for 1D Nonlinear Problem
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM1D_NONLINEAR <br> Finite Element Method for 1D Nonlinear Problem.
</h1>
<hr>
<p>
<b>FEM1D_NONLINEAR</b>
is a C++ program which
applies the finite element method to a simple nonlinear
boundary value problem in one spatial dimension.
</p>
<p>
The nonlinear boundary problem involves an unknown function <b>u</b> on
an interval <b>[a,b]</b>. Typically, a single boundary condition, either
<b>u(a)</b> or <b>u'(a)</b>, is given at the left endpoint, and
<b>u(b)</b> or <b>u'(b)</b> at the right endpoint. The associated differential
equation, which must hold in the interior of <b>[a,b]</b>, happens to have
the form:
<pre>
-d/dx ( p(x) du/dx ) + q(x) * u + u * du/dx = f(x)
</pre>
where <b>p(x)</b>, <b>q(x)</b> and <b>f(x)</b> are given functions.
</p>
<p>
The nonlinearity arises because of the term <b>u*du/dx</b>; if this term
were not present, standard finite element techniques would allow the system
to be set up and solved almost as easily as a linear system of algebraic
equations is solved.
</p>
<p>
Newton's method, which works well for scalar nonlinear equations, can also
be applied to this problem, as long as we are willing to extend our notions
of a function and derivative. We need to imagine our differential equation
as a function <b>F(u)</b>:
<pre>
F(u) = -d/dx ( p(x) du/dx ) + q(x) * u + u * du/dx - f(x)
</pre>
(with the boundary conditions wrapped in here somewhere as well!)
If we differentiate this function, we get a Jacobian operator, which is
evaluated at <b>u</b>, and applied to any small increment <b>v</b>. This equation
implicitly describes the tangent plane of solutions near to a given
solution <b>u</b>.
<pre>
J(u,v) = -d/dx ( p(x) dv/dx ) + q(x) * v + u * dv/dx + v * du/dx
</pre>
Now if we apply the finite element formulation to represent <b>u</b> and
<b>v</b> in terms of sums of basis functions, we can set up a linear
system, to be evaluated at <b>u</b> and solved for the Newton increment
<b>delta_u</b>:
<pre>
J(u,delta_u) = - F(u)
</pre>
By using the Newton increment to update <b>u</b> and repeating the process
as needed, we can expect to get a good finite element solution of our original
nonlinear boundary value problem.
<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_NONLINEAR</b> is available in
<a href = "../../c_src/fem1d_nonlinear/fem1d_nonlinear.html">a C version</a> and
<a href = "../../cpp_src/fem1d_nonlinear/fem1d_nonlinear.html">a C++ version</a> and
<a href = "../../f77_src/fem1d_nonlinear/fem1d_nonlinear.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fem1d_nonlinear/fem1d_nonlinear.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem1d_nonlinear/fem1d_nonlinear.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 = "../../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 1D.
</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.
</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_pack/fem1d_pack.html">
FEM1D_PACK</a>,
a C++ library which
contains utilities for 1D finite element calculations.
</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/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 = "../../cpp_src/fem1d_sample/fem1d_sample.html">
FEM1D_SAMPLE</a>,
a C++ program which
samples a scalar or vector finite element function of one variable,
defined by FEM files,
returning interpolated values at the sample points.
</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_nonlinear.cpp">fem1d_nonlinear.cpp</a>, the source code.
</li>
<li>
<a href = "fem1d_nonlinear.sh">fem1d_nonlinear.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "problem1_output.txt">problem1_output.txt</a>,
the output from solving problem 1.
</li>
<li>
<a href = "problem2_output.txt">problem2_output.txt</a>,
the output from solving problem 2.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for FEM1D_NONLINEAR.
</li>
<li>
<b>ASSEMBLE_NEWTON</b> assembles the Newton linear system.
</li>
<li>
<b>ASSEMBLE_PICARD</b> assembles the Picard linear system.
</li>
<li>
<b>COMPARE</b> compares the computed and exact solutions.
</li>
<li>
<b>FF</b> returns the right hand side of the differential equation.
</li>
<li>
<b>GEOMETRY</b> sets up the geometry for the interval [XL,XR].
</li>
<li>
<b>INIT</b> initializes variables that define the problem.
</li>
<li>
<b>OUTPUT</b> prints out the computed solution at the nodes.
</li>
<li>
<b>PHI</b> evaluates a linear basis function and its derivative.
</li>
<li>
<b>PP</b> evaluates the function P in the differential equation.
</li>
<li>
<b>PRSYS</b> prints out the tridiagonal linear system.
</li>
<li>
<b>QQ</b> returns the value of the coefficient function Q(X).
</li>
<li>
<b>SOLVE</b> solves a tridiagonal matrix system of the form A*x = b.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>U_EXACT</b> returns the value of the exact solution at a point X.
</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 29 April 2007.
</i>
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