forked from johannesgerer/jburkardt-cpp
-
Notifications
You must be signed in to change notification settings - Fork 0
/
fem1d_bvp_linear.html
313 lines (275 loc) · 8.79 KB
/
fem1d_bvp_linear.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
<html>
<head>
<title>
FEM1D_BVP_LINEAR - Finite Element Method, 1D, Boundary Value Problem, Piecewise Linear Elements
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM1D_BVP_LINEAR <br> Finite Element Method, 1D, Boundary Value Problem, Piecewise Linear Elements
</h1>
<hr>
<p>
<b>FEM1D_BVP_LINEAR</b>
is a C++ program which
applies the finite element method, with piecewise linear elements,
to a two point boundary value problem in one spatial dimension,
and compares the computed and exact solutions
with the L2 and seminorm errors.
</p>
<p>
The boundary value problem (BVP) that is to be solved has the form:
<pre>
- d/dx ( a(x) * du/dx ) + c(x) * u(x) = f(x)
</pre>
in the interval 0 < x < 1. The functions a(x), c(x), and f(x) are
given.
</p>
<p>
Boundary conditions are applied at the endpoints, and in this case,
these are assumed to have the form:
<pre>
u(0.0) = 0.0;
u(1.0) = 0.0.
</pre>
</p>
<p>
To compute a finite element approximation, a set of n equally spaced
nodes is defined from 0.0 to 1.0, a set of piecewise linear basis functions
is set up, with one basis function associated with each node,
and then an integral form of the BVP is used, in which the differential
equation is multiplied by each basis function, and integration by parts is
used to simplify the integrand.
</p>
<p>
A simple two point Gauss quadrature formula is used to estimate the
resulting integrals over each interval.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>fem1d_bvp_linear</b> ( <i>n</i>, <i>a</i>, <i>c</i>, <i>f</i>,
<i>x</i>, <i>u</i> )
</blockquote>
where
<ul>
<li>
<i>n</i> is the number of equally spaced nodes.
</li>
<li>
<i>a</i> is the function which evaluates a(x);
</li>
<li>
<i>c</i> is the function which evaluates c(x);
</li>
<li>
<i>f</i> is the function which evaluates f(x).
</li>
<li>
<i>x</i> is the input vector of <i>n</i> nodes.
</li>
<li>
<i>u</i> is the output vector of <i>n</i> finite difference values.
</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>FEM1D_BVP_LINEAR</b> is available in
<a href = "../../c_src/fem1d_bvp_linear/fem1d_bvp_linear.html">a C version</a> and
<a href = "../../cpp_src/fem1d_bvp_linear/fem1d_bvp_linear.html">a C++ version</a> and
<a href = "../../f77_src/fem1d_bvp_linear/fem1d_bvp_linear.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fem1d_bvp_linear/fem1d_bvp_linear.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem1d_bvp_linear/fem1d_bvp_linear.html">a MATLAB version</a> and
<a href = "../../py_src/fem1d_bvp_linear/fem1d_bvp_linear.html">a Python version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<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/fem1d/fem1d.html">
FEM1D</a>,
a C++ program which
applies the finite element method to a linear two point boundary value problem
in a 1D region.
</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_quadratic/fem1d_bvp_quadratic.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/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_nonlinear/fem1d_nonlinear.html">
FEM1D_NONLINEAR</a>,
a C++ program which
applies the finite element method to a nonlinear two point boundary value problem
in a 1D region.
</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/fem2d_bvp_linear/fem2d_bvp_linear.html">
FEM2D_BVP_LINEAR</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>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Dianne O'Leary,<br>
Finite Differences and Finite Elements: Getting to Know You,<br>
Computing in Science and Engineering,<br>
Volume 7, Number 3, May/June 2005.
</li>
<li>
Dianne O'Leary,<br>
Scientific Computing with Case Studies,<br>
SIAM, 2008,<br>
ISBN13: 978-0-898716-66-5,<br>
LC: QA401.O44.
</li>
<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_bvp_linear.cpp">fem1d_bvp_linear.cpp</a>, the source code.
</li>
<li>
<a href = "fem1d_bvp_linear.hpp">fem1d_bvp_linear.hpp</a>, the include file.
</li>
<li>
<a href = "fem1d_bvp_linear.sh">fem1d_bvp_linear.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "fem1d_bvp_linear_prb.cpp">fem1d_bvp_linear_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "fem1d_bvp_linear_prb.sh">fem1d_bvp_linear_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "fem1d_bvp_linear_prb_output.txt">fem1d_bvp_linear_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>COMPUTE_L2_ERROR</b> estimates the L2 error norm of a finite element solution.
</li>
<li>
<b>COMPUTE_SEMINORM_ERROR</b> estimates the seminorm error of a finite element solution.
</li>
<li>
<b>FEM1D_BVP_LINEAR</b> solves a two point boundary value problem.
</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 20 August 2010.
</i>
<!-- John Burkardt -->
</body>
</html>