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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
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
(x) = > xx +1 xx if xj x < xj+1 :
; ; j ; j; j
j ; j φ j (x)
x0 xj-1 xj xj+1 xN Figure 1.3.1: One-dimensional nite element mesh and piecewise linear hat function
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
j (xk ) = j k := 0 otherwise :
Using this basis with (1.2.3)...
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- Spring '14