Unformatted text preview: etermining U by solving (7.2.8). The approximate strain energy A (V U )
or L2 inner product (V f ) re ect the numerical integration that has been used. For
example, consider the loading
(V f ) = N
X
e=1 (V f )e (V f )e = ZZ V (x y)f (x y)dxdy e where e is the domain occupied by element e in a mesh of N elements. Using an
npoint quadrature rule (cf. (6.1.2a)) on element e, we would approximate (V f ) by
(V f ) =
where
(V f )e = n
X
k=1 N
X
e=1 (V f )e Wk V (xk yk )f (xk yk): (7.3.1b) (7.3.1c) The e ects of transformations to a canonical element have not been shown for simplicity
and a similar formula applies for A (V U ).
Deriving an estimate for the perturbation introduced by (7.3.1a) is relatively simple
if A(V U ) and A (V U ) are continuous and coercive. Theorem 7.3.1. Suppose that A(v u) and A (V U ) are bilinear forms with A being
continuous and A being coercive in H 1 thus, there exists constants and such that
jA(u v )j kuk1kv k1 1
8u v 2 H0 (7.3.2a) and A (U U )
Then
ku ; U k1 kU k2
1 N
8U 2 S0 : );A
C fku ; V k1 + supN jA(V W kW k (V W )j +
1
W 2S0
);(
N
sup j(W f kW kW f ) j g
8V 2 S0 :
N
1
W 2S0 (7.3.2b) (7.3.3) 12 Analysis of...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, Finite Element Method, KU

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