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<html>
<head>
<title>
MULTIGRID_POISSON_1D - Multigrid Solver for 1D Poisson Problem
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
MULTIGRID_POISSON_1D <br> Multigrid Solver for 1D Poisson Problem
</h1>
<hr>
<p>
<b>MULTIGRID_POISSON_1D</b>
is a C++ library which
applies a multigrid method to solve the linear system associated
with a discretized version of the 1D Poisson equation.
</p>
<p>
The 1D Poisson equation is assumed to have the form
<pre>
-u''(x) = f(x), for a < x < b
u(a) = ua, u(b) = ub
</pre>
</p>
<p>
Let K be a small positive integer called the mesh index, and let
N = 2^K be the corresponding number of uniform subintervals into which
[A,B] is divided. Assigning a value U(I) to each of the N+1 equally
spaced nodes with coordinate X(I), we approximate the equation by
<pre>
-U(i-1) + 2 U(i) - U(i+1)
------------------------- = f( X(i) ), 1 < I < N+1
h^2
U(1) = ua, U(N+1) = ub.
</pre>
</p>
<p>
It remains to solve the linear system for the desired values of U.
This could be done directly, or iteratively. An iterative method
such as Jacobi, Gauss-Seidel or SOR might be suitable, but experience
shows that the convergence rate of these iterative methods decreases
drastically as the value of K is increased - that is, as a more
refined and accurate answer is sought.
</p>
<p>
The multigrid method defines a nested set of grids, and corresponding
solutions, to the problem, and applies an iterative linear solver.
By transfering information from one grid to a finer or coarser one,
a more rapid convergence behavior can be encouraged.
</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>MULTIGRID_POISSON_1D</b> is available in
<a href = "../../c_src/multigrid_poisson_1d/multigrid_poisson_1d.html">a C version</a> and
<a href = "../../cpp_src/multigrid_poisson_1d/multigrid_poisson_1d.html">a C++ version</a> and
<a href = "../../f77_src/multigrid_poisson_1d/multigrid_poisson_1d.html">a FORTRAN77 version</a> and
<a href = "../../f_src/multigrid_poisson_1d/multigrid_poisson_1d.html">a FORTRAN90 version</a> and
<a href = "../../m_src/multigrid_poisson_1d/multigrid_poisson_1d.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/cyclic_reduction/cyclic_reduction.html">
CYCLIC_REDUCTION</a>,
a C++ library which
solves a tridiagonal linear system using cyclic reduction.
</p>
<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/mgmres/mgmres.html">
MGMRES</a>,
a C++ library which
applies the restarted GMRES algorithm to solve a sparse linear system,
by Lili Ju.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
William Briggs, Van Emden Henson, Steve McCormick,<br>
A Multigrid Tutorial,<br>
SIAM, 2000,<br>
ISBN13: 978-0-898714-62-3,<br>
LC: QA377.B75.
</li>
<li>
William Hager,<br>
Applied Numerical Linear Algebra,<br>
Prentice-Hall, 1988,<br>
ISBN13: 978-0130412942,<br>
LC: QA184.H33.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "multigrid_poisson_1d.cpp">multigrid_poisson_1d.cpp</a>, the source code.
</li>
<li>
<a href = "multigrid_poisson_1d.hpp">multigrid_poisson_1d.hpp</a>, the include file
</li>
<li>
<a href = "multigrid_poisson_1d.sh">multigrid_poisson_1d.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "multigrid_poisson_1d_prb.cpp">multigrid_poisson_1d_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "multigrid_poisson_1d_prb.sh">multigrid_poisson_1d_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "multigrid_poisson_1d_prb_output.txt">multigrid_poisson_1d_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CTOF</b> transfers data from a coarse to a finer grid.
</li>
<li>
<b>FTOC</b> transfers data from a fine grid to a coarser grid.
</li>
<li>
<b>GAUSS_SEIDEL</b> carries out one step of a Gauss-Seidel iteration.
</li>
<li>
<b>I4_LOG_2</b> returns the integer part of the logarithm base 2 of an I4.
</li>
<li>
<b>I4_POWER</b> returns the value of I^J.
</li>
<li>
<b>MONOGRID_POISSON_1D</b> solves a 1D PDE, using the Gauss-Seidel method.
</li>
<li>
<b>MULTIGRID_POISSON_1D</b> solves a 1D PDE using the multigrid method.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_MAX</b> returns the maximum of two R8's.
</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 07 December 2011.
</i>
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