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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
</h1>
<hr>
<p>
<b>FEM2D_POISSON_SPARSE</b>
is a C++ 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 right hand side function F(x,y), the linear
coefficients H(x,y) and K(x,y) and the boundary condition function G(x,y),
the user has to modify a file containing three routines,
<ul>
<li>
<b>void rhs ( int node_num, double node_xy[], double node_rhs[] )</b>
evaluates the right hand side of function F(x,y) at a list of
nodes.
</li>
<li>
<b>void h_coef ( int node_num, double node_xy[], double node_h[] )</b>
evaluates the coefficient function H(x,y) at a list of nodes.
</li>
<li>
<b>void k_coef ( int node_num, double node_xy[], double node_k[] )</b>
evaluates the coefficient function K(x,y) at a list of nodes.
</li>
<li>
<b>void dirichlet_condition ( int node_num, double node_xy[], double node_g[] )</b>
evaluates the Dirichlet boundary condition G(X,Y) at a list of nodes.
</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>
Assuming the executable program is called "my_problem", then
the program is executed by
<blockquote>
<b>my_problem</b> <i>prefix</i>
</blockquote>
where prefix is the common filename prefix, so that:
<ul>
<li>
<i>prefix</i><b>_nodes.txt</b> contains the X, Y coordinates of nodes;
</li>
<li>
<i>prefix</i><b>_elements.txt</b> contains triples of node indices that form triangles.
</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 Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/fem2d_poisson/fem2d_poisson.html">
FEM2D_POISSON</a>,
a C++ program which
solves Poisson's equation on a triangulated region,
using the finite element method and a banded solver.
In order to run, it requires
user-supplied routines that define problem data.
</p>
<p>
<a href = "../../cpp_src/fem2d_poisson_cg/fem2d_poisson_cg.html">
FEM2D_POISSON_CG</a>,
a C++ program which
solves Poisson's equation on a triangulated region,
using the finite element method, sparse storage, and a conjugate gradient solver.
</p>
<p>
<a href = "../../cpp_src/fem2d_poisson_sparse_baffle/fem2d_poisson_sparse_baffle.html">
FEM2D_POISSON_SPARSE_BAFFLE</a>,
a C++ library which
defines the geometry of a rectangle channel containing 13 hexagonal baffles,
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 = "../../cpp_src/fem2d_poisson_sparse_ell/fem2d_poisson_sparse_ell.html">
FEM2D_POISSON_SPARSE_ELL</a>,
a C++ library which
defines the geometry of an L-shaped region, as well as boundary
conditions for a given Poisson problem, and is called by
<b>fem2d_poisson_sparse</b> as part of a solution procedure.
</p>
<p>
<a href = "../../cpp_src/fem2d_poisson_sparse_lake/fem2d_poisson_sparse_lake.html">
FEM2D_POISSON_SPARSE_LAKE</a>,
a C++ library which
defines the geometry of a lake-shaped region, as well as boundary
conditions for a given Poisson problem, and is called by
<b>fem2d_poisson_sparse</b> as part of a solution procedure.
</p>
<p>
<a href = "../../cpp_src/mgmres/mgmres.html">
MGMRES</a>,
a C++ library which
applies the restarted Generalized Minimum Residual (GMRES) algorithm
to solve a sparse linear system, by Lili Ju.
</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.cpp">fem2d_poisson_sparse.cpp</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 for 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 basis functions for a linear triangular element.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is true if two characters are equal, disregarding case.
</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>DTABLE_DATA_READ</b> reads the data from a real TABLE file.
</li>
<li>
<b>DTABLE_HEADER_READ</b> reads the header from a real TABLE file.
</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>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two I4's.
</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 the columns of an I4COL.
</li>
<li>
<b>I4COL_SWAP</b> swaps two columns of an I4COL.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of an I4MAt, transposed.
</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>ITABLE_DATA_READ</b> reads data from an integer TABLE file.
</li>
<li>
<b>ITABLE_HEADER_READ</b> reads the header from an integer TABLE file.
</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>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8_NINT</b> returns the nearest integer to an R8.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8VEC_AMAX</b> returns the maximum absolute value in an R8VEC.
</li>
<li>
<b>R8VEC_DOT</b> computes the dot product of a pair of R8VEC's.
</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_LEN_TRIM</b> returns the length of a string to the last nonblank.
</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 a triangle.
</li>
<li>
<b>SOLUTION_WRITE</b> writes the solution to a file.
</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>TIMESTRING</b> returns the current YMDHMS date as a string.
</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 nodes on the boundary.
</li>
<li>
<b>TRIANGULATION_ORDER3_NEIGHBOR_TRIANGLES</b> determines triangle neighbors.
</li>
<li>
<b>TRIANGULATION_ORDER3_PLOT</b> plots a triangulation of a set of nodes.
</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 15 July 2007.
</i>
<!-- John Burkardt -->
</body>
</html>