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Integrals like 124 or 126 exist for some pretty wild

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Unformatted text preview: 6) exist for some pretty \wild" choices of V . Valid methods exist when V is a Dirac delta function (although such functions are not elements of L2 ) and when V is a piecewise constant function (cf. Problems 1 and 2 at the end of this section). There are many reasons to prefer a more symmetric variational form of (1.2.1) than (1.2.2), e.g., problem (1.2.1) is symmetric (self-adjoint) and the variational form should re ect this. Additionally, we might want to choose the same trial and test spaces, as with Galerkin's method, but ask for less continuity on the trial space S N . This is typically the case. As we shall see, it will be di cult to construct continuously di erentiable approximations of nite element type in two and three dimensions. We can construct the symmetric variational form that we need by integrating the second derivative terms in (1.2.2a) by parts thus, using (1.2.1a) Z1 v ;(pu ) + qu ; f ]dx = 0 0 Z 1 0 (v pu + vqu ; vf )dx ; vpu j1 = 0 0 0 0 0 0 (1.2.8) where ( ) = d( )=dx. The treatmen...
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