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Unformatted text preview: tions (of one variable) belonging to H 1 must
necessarily be continuous. Accepting this for the moment, let us establish the goal of
nding the simplest continuous piecewise polynomial approximations of u and v. This
would be a piecewise linear polynomial with respect to a mesh
0 = x0 < x1 < : : : < xN = 1
(1.3.3)
introduced on 0 1]. Each subinterval (xj 1 xj ), j = 1 2 : : : N , is called a nite element.
The basis is created from the \hat function"
8 x x ;1
> x x ;1 if xj 1 x < xj
<
+1
(x) = > xx +1 xx if xj x < xj+1 :
(1.3.4a)
j
:0
otherwise
; ; j ; j; j
;
j
j ; j φ j (x)
1 x
x0 xj1 xj xj+1 xN Figure 1.3.1: Onedimensional nite element mesh and piecewise linear hat function
j (x).
As shown in Figure 1.3.1, j (x) is nonzero only on the two elements containing the
node xj . It rises and descends linearly on these two elements and has a maximal unit
value at x = xj . Indeed, it vanishes at all nodes but xj , i.e.,
1 if xk = xj
(1.3.4b)
j (xk ) = j k := 0 otherwise :
Using this basis with (1.2.3)...
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 Spring '14
 JosephE.Flaherty

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