This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Simpson's rule to yield
(
Finally, ( j j Ut ) = hj (_j;1 + 2_j ) + hj6+1 (2_j + c_j+1):
c
c
6c
f) Zx j xj ;1 j f (x)dx + Zx
xj j +1 j f (x)dx: Although integration of order one would do, we'll, once again, use Simpson's rule to
obtain
( j f ) hj (2fj;1=2 + fj ) + hj6+1 (fj + 2fj+1=2):
6
We could replace fj;1=2 by the average of fj;1 and fj to obtain a similar formula to the
one obtained for ( j Ut ) thus,
( j f ) hj (fj;1 + 2fj ) + hj6+1 (2fj + fj+1):
6 Combining these results yields the discrete nite element system hj (_
c
6 j ;1 p
p
+ 2_j ) + hj+1 (2_j + cj+1) + j;1=2 (cj ; cj;1) ; j+1=2 (cj+1 ; cj )
c
c_
6
hj
hj + 1=2 6 Parabolic Problems = hj (fj;1 + 2fj ) + hj+1 (2fj + fj+1)
j = 1 2 : : : N ; 1:
6
6
(We have dropped the and written the equation as an equality.)
If p is constant and the mesh spacing h is uniform, we obtain h (_
c
6 j ;1 + 4_j + cj+1) ; p (cj;1 ; 2cj + cj+1) = h (fj;1 + 4fj + fj+1)
c_
h
6
j = 1 2 : : : N ; 1: The discrete systems may be written in matrix form and, for simplicity, we'll do so for
the constant coe cient, uniform mesh case to obtain
_
Mc + Kc = l
where 24
61
h6
M= 66
6
6
4 (9.2.4a) 3 1
7
41
7
... ... ... 7
7
7
1 4 15
14 2 2 ;1
6 ;1 2 ;1
p6
K= h6
6 ... ...
6
4
;1 3
7
7
7
7
5
;1 7 ...
2
;1 2 2f
6f
l= h6
66
4 + 4f1 + f2
1 + 4f2 + f3
...
fN ;2 + 4fN ;1 + fN
0 c = c1 c2 : : : cN ;1]T : 3
7
7
7
5 (9.2.4b) (9.2.4c) (9.2.4d)
(9.2.4e) The matrices M, K, and l are the global mass matrix, the global sti ness matrix, and
the global load vector. Actually, M has little to do with mass and should more correctly
be called a global dissipation matrix however, we'll stay with our prior terminology.
In practical problems, elementbyelement assembly should be used to construct global
matrices and vectors and not the nodal approach used here.
The discrete nite element system (9.2.4) is an implicit system of ordinary di erential
_
equations for c. The mass matrix M can be \lumped" by a variety of tricks to yield an 9.2. SemiDiscrete Galerkin Problems 7 explicit ordinary di erential system. One such trick is to approximate (
the rightrectangular rule on each element to obtain
( j Ut) = Zx j j xj ;1 (_j;1 j;1 + cj j )dx +
c
_ Zx j +1 xj j (_j j + cj+1
c
_ j +1 j Ut) by using )dx hcj : The resulting nite element system would be
_
hIc + Kc = l:
Recall (cf. Section 6.3), that a onepoint quadrature rule is satisfactory for the convergence of a piecewiselinear polynomial nite element solution.
N
With the initial data determined by L2 projection onto SE , we have
( j j = 1 2 : : : N ; 1: U ( 0)) = ( j u0) Numerical integration will typically be needed to evaluate ( j u0) and we'll approximate
it in the manner used for the loading term ( j f ). Thus, with uniform spacing, we have 2u
6u
Mc(0) = u = h 6
66
4 3 + 4u0 + u0
1
2
+ 4u0 + u0 7
2
3
7:
7
...
5
0
0
0
uN ;2 + 4uN ;1 + uN
0
0
0
1 0 (9.2.4f) If the initial data is consistent with the trivial Dirichlet boundary data, i.e., if u0 2 H01
then the above system reduces to cj (0) = u0(xj ) j =...
View
Full
Document
This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty
 The Land

Click to edit the document details