Unformatted text preview: .3. Finite Element Methods in Time 13 Di erentiating and setting t = tn+1
_
_
_
N0 (tn+1) = 2 3 t
N1 (tn+1) = ; 2t
N2(tn+1 ) = 2 1 t :
Thus,
n+1
n
n;1
_
Y(tn+1) = 3y ;24yt + y
and the secondorder BDF is
n+1
n
n;1
f (tn+1 yn+1 3y ;24yt + y ) = 0:
Applying this method to (9.2.4a) yields
3cn+1 ; 4cn + cn;1 + Kn+1cn+1 = ln+1:
M
2t
Thus, computation of cn+1 requires inversion of M + K:
2t Backward di erence formulas through order six are available 2, 3, 6, 7, 8]. 9.3 Finite Element Methods in Time
It is, of course, possible to use the nite element method in time. This can be done
on spacetime triangular or quadrilateral elements for problems in one space dimension
on hexahedra, tetrahedra, and prisms in two space dimensions and on fourdimensional
parallelepipeds and prisms in three space dimensions. However, for simplicity, we'll focus
on the time aspects of the spacetime nite element method by assuming that the spatial
discretization has already been performed. Thus, we'll consider an ODE system in the
form (9.2.4a) and construct a Galerkin problem in time by multiplying it by a test
function w 2 L2 and integrating on (tn tn+1] to obtain
_
(w Mc)n + (w Kc)n = (w l)n
where the L2 inner product in time is
(w c)n = Zt n+1 tn 8w 2 L2 (tn tn+1] wT cdt: (9.3.1a)
(9.3.1b) Only rst derivatives are involved in (9.2.4a) thus, neither the trial space for c nor the
test space for w have to be continuous. For our initial method, let us assume that c(t)
is continuous at tn. By assumption, c(tn) is known in this case and, hence, w(tn) = 0. 14 Parabolic Problems Example 9.3.1. Let us examine the method that results when c(t) and w(t) are linear
on (tn tn+1]. We represent c(t) in the manner used for a spatial basis as c( ) cnNn( ) + cn+1Nn+1( )
where Nn( ) = 1 ;
2
are hat functions in time and Nn+1( ) = 1 +
2 = 2t ; tn ; tn+1 (9.3.2a)
(9.3.2b)
(9.3.2c) t de nes the canonical element in time. The test function w = Nn+1 ( ) 1 1 : : : 1]T (9.3.2d) vanishes at tn ( = ;1) and is linear on (tn tn+1).
Transforming the integrals in (9.3.1a) to (;1 1) using (9.3.2c) and using (9.3.2a,b,d)
yields
t Z 1 1 + M cn+1 ; cn d
_
(w Mc)n = 2
2
t tZ ;1 1 + K cn 1 ; + cn+1 1 + ]d :
2 ;1 2
2
2
(Again, we have written equality instead of for simplicity.) Assuming that M and K
are independent of time, we have
(w Kc)n = 1 _
(w Mc)n = M c 2; c
n+1 n (w Kc)n = 6t K(cn + 2cn+1):
Substituting these into (9.3.1a) cn+1 ; cn + t K(cn + 2cn+1) = t Z 1 1 + l( )d
M2
6
2
2 or, if l is approximated like c, ;1 (9.3.3a) n+1
n
M c 2; c + 6t K(cn + 2cn+1) = 6t (ln + 2ln+1): (9.3.3b) 1
M + 2 tK]cn+1 = M ; 3 tK]cn + 1 t ln + 2ln+1]
3
3 (9.3.3c) Regrouping terms 9.3. Finite Element Methods in Time 15 we see that the piecewiselinear Galerkin method in time is a weighted average scheme
(9.2.12c) with = 2=3. Thus, at least to this low order, there is not much di erence between nite di erence and nite element methods. Other similarities appear in Problem
1 at the end of this section.
Loworder schemes such as (9.2.12) are popular in nite element packages. Our preference is for BD...
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 Spring '14
 JosephE.Flaherty
 The Land, Tn, Boundary value problem, Numerical ordinary differential equations, nite element, parabolic problems

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