32bc exist and that the trivial boundary conditions

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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 xj-1 xj xj+1 xN Figure 1.3.1: One-dimensional 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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