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<html>
<head>
<title>
FEM2D_POISSON_SPARSE - Finite Element Solution on Arbitrary 2D Region
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM2D_POISSON_SPARSE <br>
Finite Element Solution of Poisson's Equation <br>
on a Triangulated Region<br>
using a Sparse Matrix Solver
</h1>
<hr>
<p>
<b>FEM2D_POISSON_SPARSE</b>
is a FORTRAN90 program which
applies the finite element method to solve
a form of Poisson's equation over an arbitrary triangulated region,
using sparse matrix storage and an iterative solver.
</p>
<h3 align = "center">
Sparse Matrix Modifications:
</h3>
<p>
This program is a revised version of <b>FEM2D_POISSON</b>.
</p>
<p>
<b>FEM2D_POISSON</b> used a direct LINPACK/LAPACK banded matrix
solver. The storage requirements for the banded solver are less than
for a full storage solver, but still grow quickly as the
problem size increases.
</p>
<p>
<b>FEM2D_POISSON_SPARSE</b> reduces the storage requirements even further,
by using sparse matrix techniques. The storage format chosen is
known as <b>DSP</b> or "sparse triplet" format, which essentially
simply saves in three vectors A, IA, JA, which record the value,
row and column of every nonzero entry. To solve the linear system,
the <b>MGMRES</b> iterative solver was applied. With these
modifications, <b>FEM2D_POISSON_SPARSE</b> can handle problems much larger
than those that <b>FEM2D_POISSON</b> could. A new issue
is that the iterative solver must be monitored to ensure that
convergence is proceeding properly, and has reached the desired
tolerance.
</p>
<h3 align = "center">
The Triangulated Region:
</h3>
<p>
The computational region is unknown by the program. The user
specifies it by preparing a file containing the coordinates of
the nodes, and a file containing the indices of nodes that make
up triangles that form a triangulation of the region.
</p>
<p>
Normally, the user does not type in this information by hand, but has
a program fill in the nodes, and perhaps another program that
constructs the triangulation. However, in the simplest case,
the user might construct a very crude triangulation by hand, and
have <a href = "../../f_src/triangulation_refine/triangulation_refine.html">
TRIANGULATION_REFINE</a> refine it to something more reasonable.
</p>
<p>
For the following ridiculously small example:
<pre>
4----5
|\ |\
| \ | \
| \ | \
| \| \
1----2----3
</pre>
the node file would be:
<pre>
0.0 0.0
1.0 0.0
2.0 0.0
0.0 1.0
1.0 1.0
</pre>
and the triangle file would be
<pre>
1 2 4
5 4 2
2 3 5
</pre>
</p>
<h3 align = "center">
The Poisson Equation:
</h3>
<p>
The program is set up to handle the linear Poisson
equation with a right hand side function, and nonhomogeneous
Dirichlet boundary conditions. The state variable
U(X,Y) is then constrained by:
<pre>
- Del H(x,y) Del U(x,y) + K(x,y) * U(x,y) = F(x,y) inside the region;
U(x,y) = G(x,y) on the boundary.
</pre>
</p>
<p>
A fancier version of the program is eventually intended, which will
handle a more interesting nonlinear PDE, and include optional Neumann
boundary conditions.
</p>
<h3 align = "center">
User Interface:
</h3>
<p>
To specify the boundary condition function G(x,y),
the linear coefficients H(x,y) and K(x,y) and the right hand side function F(x,y),
the user has to modify a file containing three subroutines,
<ul>
<li>
<b>SUBROUTINE DIRICHLET_CONDITION ( NODE_NUM, NODE_XY, NODE_BC )</b>
evaluates G(X,Y).
</li>
<li>
<b>SUBROUTINE H_COEF ( NODE_NUM, NODE_XY, NODE_H )</b>
evaluates H(x,y).
</li>
<li>
<b>SUBROUTINE K_COEF ( NODE_NUM, NODE_XY, NODE_K )</b>
evaluates K(x,y).
</li>
<li>
<b>SUBROUTINE RHS ( NODE_NUM, NODE_XY, NODE_F )</b>
evaluates F(x,y).
</li>
</ul>
</p>
<p>
To run the program, the user compiles the user routines,
links them with <b>FEM2D_POISSON_SPARSE</b>, and runs the executable.
</p>
<p>
The program writes out a file containing an Encapsulated
PostScript image of the nodes and elements, with numbers.
If there are too many nodes, the plot may be too cluttered
to read. For lower values, however, it is
a valuable map of what is going on in the geometry.
</p>
<p>
The program is also able to write out a file containing the
solution value at every node. This file may be used to create
contour plots of the solution.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
The user must create an executable by compiling the user routines
and linking them with the main program, perhaps by commands like:
<pre>
gfortran -c fem2d_poisson_sparse.f90
gfortran -c user.f90
gfortran fem2d_poisson_sparse.o user.o
mv a.out fem2d_poisson_sparse
</pre>
</p>
<p>
Assuming the executable program is called "fem2d_poisson_sparse", then
the program is executed by
<blockquote>
<b>fem2d_poisson_sparse</b> <i>prefix</i>
</blockquote>
where prefix is the common filename prefix, so that:
<ul>
<li>
<i>prefix</i><b>_nodes.txt</b>, is a file containing the node coordinates;
</li>
<li>
<i>prefix</i><b>_elements.txt</b>, is a file listing the 3 nodes that
make up each element;
</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>FEM2D_POISSON_SPARSE</b> is available in
<a href = "../../cpp_src/fem2d_poisson_sparse/fem2d_poisson_sparse.html">a C++ version</a> and
<a href = "../../f_src/fem2d_poisson_sparse/fem2d_poisson_sparse.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem2d_poisson_sparse/fem2d_poisson_sparse.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Programs:
</h3>
<p>
<a href = "../../f_src/fem2d_poisson_sparse_ell/fem2d_poisson_sparse_ell.html">
FEM2D_POISSON_SPARSE_ELL</a>,
a FORTRAN90 library which
defines the geometry of an L-shaped region, as well as boundary
conditions for a given Poisson problem, and is called by FEM2D_POISSON_SPARSE
as part of a solution procedure.
</p>
<p>
<a href = "../../f_src/fem2d_poisson_sparse_lake/fem2d_poisson_sparse_lake.html">
FEM2D_POISSON_SPARSE_LAKE</a>,
a FORTRAN90 library which
defines the geometry of a lake-shaped region, as well as boundary
conditions for a given Poisson problem, and is called by FEM2D_POISSON_SPARSE
as part of a solution procedure.
</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 = "fem2d_poisson_sparse.f90">fem2d_poisson_sparse.f90</a>,
the source code;
</li>
<li>
<a href = "fem2d_poisson_sparse.sh">fem2d_poisson_sparse.sh</a>,
commands to compile the partial program;
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program of FEM2D_POISSON_SPARSE.
</li>
<li>
<b>ASSEMBLE_POISSON_DSP</b> assembles the system for the Poisson equation.
</li>
<li>
<b>AX</b> computes A * X for a sparse matrix.
</li>
<li>
<b>BASIS_ONE_T3</b> evaluates a linear basis function.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>DIRICHLET_APPLY_DSP</b> accounts for Dirichlet boundary conditions.
</li>
<li>
<b>DSP_IJ_TO_K</b> seeks the compressed index of the (I,J) entry of A.
</li>
<li>
<b>DSP_PRINT_SOME</b> prints some of a DSP matrix.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_NAME_SPECIFICATION</b> determines the names of the input files.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>I4_HUGE</b> returns a "huge" I4.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of integer division.
</li>
<li>
<b>I4_WRAP</b> forces an integer to lie between given limits by wrapping.
</li>
<li>
<b>I4COL_COMPARE</b> compares columns I and J of an I4COL.
</li>
<li>
<b>I4COL_SORT_A</b> ascending sorts an I4COL.
</li>
<li>
<b>I4COL_SWAP</b> swaps columns I and J of an I4COL.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of the transpose of an I4mat.
</li>
<li>
<b>I4VEC2_COMPARE</b> compares pairs of integers stored in two vectors.
</li>
<li>
<b>I4VEC2_SORT_A</b> ascending sorts a vector of pairs of integers.
</li>
<li>
<b>I4MAT_DATA_READ</b> reads data from an I4MAT file.
</li>
<li>
<b>I4MAT_HEADER_READ</b> reads the header from an I4MAT.
</li>
<li>
<b>MGMRES</b> applies the restarted GMRES iteration to a linear system.
</li>
<li>
<b>MULT_GIVENS</b> applies a Givens rotation to two successive entries of a vector.
</li>
<li>
<b>POINTS_PLOT</b> plots a pointset.
</li>
<li>
<b>QUAD_RULE</b> sets the quadrature rule for assembly.
</li>
<li>
<b>R8MAT_DATA_READ</b> reads data from an R8MAT file.
</li>
<li>
<b>R8MAT_HEADER_READ</b> reads the header from an R8MAT file.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>R8VEC_PRINT_SOME</b> prints "some" of an R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
</li>
<li>
<b>REFERENCE_TO_PHYSICAL_T3</b> maps reference points to physical points.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_I4VEC</b> reads an I4VEC from a string.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</li>
<li>
<b>SOLUTION_EVALUATE</b> evaluates the solution at a point in an element.
</li>
<li>
<b>SORT_HEAP_EXTERNAL</b> externally sorts a list of items into ascending order.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TRIANGLE_AREA_2D</b> computes the area of a triangle in 2D.
</li>
<li>
<b>TRIANGULATION_ORDER3_ADJ_COUNT</b> counts adjacencies in a triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_ADJ_SET2</b> sets adjacencies in a triangulation.
</li>
<li>
<b>TRIANGULATION_ORDER3_BOUNDARY_NODE</b> indicates which nodes are on the boundary.
</li>
<li>
<b>TRIANGULATION_ORDER3_NEIGHBOR_TRIANGLES</b> determines triangle neighbors.
</li>
<li>
<b>TRIANGULATION_ORDER3_PLOT</b> plots a 3-node triangulation of a pointset.
</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 02 January 2011.
</i>
<!-- John Burkardt -->
</body>
</html>