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<html>
<head>
<title>
FEM2D_BVP_QUADRATIC - Finite Element Method, 2D, Boundary Value Problem, Piecewise Quadratic Elements
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM2D_BVP_QUADRATIC <br> Finite Element Method, 2D, Boundary Value Problem, Piecewise Quadratic Elements
</h1>
<hr>
<p>
<b>FEM2D_BVP_QUADRATIC</b>
is a C++ program which
applies the finite element method, with piecewise quadratic elements,
to a 2D boundary value problem over a rectangle.
</p>
<p>
The boundary value problem (BVP) that is to be solved has the form:
<pre>
- d/dx ( a(x,y) * du/dx ) - d/dy ( a(x,y) * du/dy ) + c(x,y) * u(x,y) = f(x,y)
</pre>
This equation holds in the interior of some rectangle R.
The functions a(x,y), c(x,y), and f(x,y) are given.
</p>
<p>
Zero boundary conditions are imposed on the boundary of R.
</p>
<p>
To compute a finite element approximation, the X and Y extents of R
are gridded with NX and NY equally spaced values, respectively, where
NX and NY are odd.
This defines NX*NY nodes, and divides R into ((NX-1)/2)*((NY-1)/2) rectangular
elements. At the K-th node, (X(I),Y(J)), a piecewise quadratic basis
function PHI(K)(X,Y) is defined. The solution will be represented as
a linear combination of these basis functions. An integral form
of the BVP is written, in which the differential equation is
multiplied by each basis function, and integration by parts is
used to simplify the integrand.
</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_BVP_QUADRATIC</b> is available in
<a href = "../../c_src/fem2d_bvp_quadratic/fem2d_bvp_quadratic.html">a C version</a> and
<a href = "../../cpp_src/fem2d_bvp_quadratic/fem2d_bvp_quadratic.html">a C++ version</a> and
<a href = "../../f77_src/fem2d_bvp_quadratic/fem2d_bvp_quadratic.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fem2d_bvp_quadratic/fem2d_bvp_quadratic.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem2d_bvp_quadratic/fem2d_bvp_quadratic.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/fem1d_bvp_linear/fem1d_bvp_linear.html">
FEM1D_BVP_QUADRATIC</a>,
a C++ program which
applies the finite element method (FEM), with piecewise quadratic
elements, to a two point boundary value problem (BVP) in one
spatial dimension, and compares the computed and exact solutions
with the L2 and seminorm errors.
</p>
<p>
<a href = "../../cpp_src/fem2d_bvp_linear/fem2d_bvp_linear.html">
FEM2D_BVP_QUADRATIC</a>,
a C++ program which
applies the finite element method (FEM), with piecewise linear
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>
<p>
<a href = "../../cpp_src/fem2d_bvp_serene/fem2d_bvp_serene.html">
FEM2D_BVP_SERENE</a>,
a C++ program which
applies the finite element method (FEM), with serendipity 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">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "fem2d_bvp_quadratic.cpp">fem2d_bvp_quadratic.cpp</a>, the source code.
</li>
<li>
<a href = "fem2d_bvp_quadratic.hpp">fem2d_bvp_quadratic.hpp</a>, the include file.
</li>
<li>
<a href = "fem2d_bvp_quadratic.sh">fem2d_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 = "fem2d_bvp_quadratic_prb.cpp">fem2d_bvp_quadratic_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "fem2d_bvp_quadratic_prb.sh">fem2d_bvp_quadratic_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "fem2d_bvp_quadratic_prb_output.txt">fem2d_bvp_quadratic_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>FEM2D_BVP_QUADRATIC</b> solves boundary value problem on a rectangle.
</li>
<li>
<b>FEM2D_H1S_ERROR_QUADRATIC:</b> seminorm error of a finite element solution.
</li>
<li>
<b>FEM2D_L1_ERROR</b> estimates the l1 error norm of a finite element solution.
</li>
<li>
<b>FEM2D_L2_ERROR_QUADRATIC:</b> 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>R8VEC_EVEN</b> returns an R8VEC of evenly spaced values.
</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 26 June 2014.
</i>
<!-- John Burkardt -->
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