-
Notifications
You must be signed in to change notification settings - Fork 64
Expand file tree
/
Copy pathfem2d_bvp_serene.html
More file actions
221 lines (195 loc) · 7.32 KB
/
Copy pathfem2d_bvp_serene.html
File metadata and controls
221 lines (195 loc) · 7.32 KB
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
<html>
<head>
<title>
FEM2D_BVP_SERENE - Finite Element Method, 2D, Boundary Value Problem, Serendipity Elements
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM2D_BVP_SERENE <br> Finite Element Method, 2D, Boundary Value Problem, Serendipity Elements
</h1>
<hr>
<p>
<b>FEM2D_BVP_SERENE</b>
is a MATLAB program which
applies the finite element method, with serendipity 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>
The MATLAB "gallery()" command can produce a copy of the Wathen matrix,
which arises as a weighted sum of elementary mass matrices associated
with a grid of serendipity elements.
</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_SERENE</b> is available in
<a href = "../../c_src/fem2d_bvp_serene/fem2d_bvp_serene.html">a C version</a> and
<a href = "../../cpp_src/fem2d_bvp_serene/fem2d_bvp_serene.html">a C++ version</a> and
<a href = "../../f77_src/fem2d_bvp_serene/fem2d_bvp_serene.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fem2d_bvp_serene/fem2d_bvp_serene.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem2d_bvp_serene/fem2d_bvp_serene.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/fem2d_bvp_linear/fem2d_bvp_linear.html">
FEM2D_BVP_LINEAR</a>,
a MATLAB program which
applies the finite element method (FEM), with piecewise linear
elements, to a 2D boundary value problem (BVP) over a rectangle,
and compares the computed and exact solutions
with the L2 and seminorm errors.
</p>
<p>
<a href = "../../m_src/fem2d_bvp_quadratic/fem2d_bvp_quadratic.html">
FEM2D_BVP_QUADRATIC</a>,
a MATLAB program which
applies the finite element method (FEM), with piecewise quadratic
elements, to a 2D boundary value problem (BVP) over a rectangle,
and compares the computed and exact solutions
with the L2 and seminorm errors.
</p>
<p>
<a href = "../../m_src/wathen/wathen.html">
WATHEN</a>,
a MATLAB library which
compares storage schemes (full, banded, sparse triplet, sparse) and
solution strategies (A\x, Linpack, conjugate gradient (CG))
for linear systems involving the Wathen matrix,
which can arise when solving a problem using the
finite element method (FEM).
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "basis_serene.m">basis_serene.m</a>,
evaluates the basis functions for the serendipity element.
</li>
<li>
<a href = "basisd_serene.m">basisd_serene.m</a>,
evaluates the X and Y derivatives of the basis functions
for the serendipity element.
</li>
<li>
<a href = "fem2d_bvp_serene.m">fem2d_bvp_serene.m</a>,
sets up and solves the finite element problem.
</li>
<li>
<a href = "fem2d_bvp_serene_extend.m">fem2d_bvp_serene_extend.m</a>,
extends a solution from the serendipity grid to a regular grid.
</li>
<li>
<a href = "fem2d_bvp_serene_node_num.m">fem2d_bvp_serene_node_num.m</a>,
returns the number of nodes in a grid made of serendipity elements.
</li>
<li>
<a href = "fem2d_h1s_error_serene.m">fem2d_h1s_error_serene.m</a>,
evaluates the H1S seminorm error for a serendipity finite element solution.
</li>
<li>
<a href = "fem2d_l1_error_serene.m">fem2d_l1_error_serene.m</a>,
evaluates the l1 error for a serendipity finite element solution.
</li>
<li>
<a href = "fem2d_l2_error_serene.m">fem2d_l2_error_serene.m</a>,
evaluates the L2 error for a serendipity finite element solution.
</li>
<li>
<a href = "not1.m">not1.m</a>,
evaluates a one-term factor for serendipity basis functions.
</li>
<li>
<a href = "not1d.m">not1d.m</a>,
differentiates a one-term factor for serendipity basis functions.
</li>
<li>
<a href = "not2.m">not2.m</a>,
evaluates a two-term factor for serendipity basis functions.
</li>
<li>
<a href = "not2dx.m">not2dx.m</a>,
differentiates a two-term factor for serendipity basis functions with respect to X.
</li>
<li>
<a href = "not2dy.m">not2dy.m</a>,
differentiates a two-term factor for serendipity basis functions with respect to Y.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "fem2d_bvp_serene_test.m">fem2d_bvp_serene_test.m</a>,
a program which calls fem_bvp_serene with some test cases.
</li>
<li>
<a href = "fem2d_bvp_serene_test_output.txt">fem2d_bvp_serene_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "fem2d_bvp_serene_test01.m">fem2d_bvp_serene_test01.m</a>,
a simple test in which the exact solution is a quadratic function.
</li>
<li>
<a href = "fem2d_bvp_serene_test02.m">fem2d_bvp_serene_test02.m</a>,
makes some checks on the serendipity basis functions.
</li>
<li>
<a href = "fem2d_bvp_serene_test03.m">fem2d_bvp_serene_test03.m</a>,
sets up a contrived problem so that the system matrix is formed using
the Wathen matrix. It prints out one copy of the (scaled) element
mass matrix, and compares it to a (scaled and permuted) copy of
the Wathen matrix returned by Matlab's gallery('wathen',1,1) command.
</li>
<li>
<a href = "fem2d_bvp_serene_test04.m">fem2d_bvp_serene_test04.m</a>,
repeats the calculation in fem2d_bvp_serene_test01(), but now uses
the fem2d_bvp_serene_extend() function to extend the solution from
the irregular serendipity grid to a matrix grid, which can then be
plotted with the surf() command.
</li>
<li>
<a href = "fem2d_bvp_serene_test04.png">fem2d_bvp_serene_test04.png</a>,
an image of the solution after extension.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 30 June 2014.
</i>
<!-- John Burkardt -->
</body>
</html>