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Unformatted text preview: ite di erence techniques, derivatives in (1.2.1a)
are approximated by nite di erences with respect to a mesh introduced on 0 1] 12].
With the nite element method, the method of weighted residuals (MWR) is used to
construct an integral formulation of (1.2.1) called a variational problem. To this end, let
us multiply (1.2.1a) by a test or weight function v and integrate over (0 1) to obtain
(v L u] ; f ) = 0:
We have introduced the L2 inner product
(v u) := Z 1
0 vudx (1.2.2b) to represent the integral of a product of two functions.
The solution of (1.2.1) is also a solution of (1.2.2a) for all functions v for which the
inner product exists. We'll express this requirement by writing v 2 L2(0 1). All functions
of class L2(0 1) are \square integrable" on (0 1) thus, (v v) exists. With this viewpoint
and notation, we write (1.2.2a) more precisely as
(v L u] ; f ) = 0
8v 2 L2 (0 1):
(1.2.2c) 4 Introduction Equation (1.2.2c) is referred to as a variational form of problem (1.2.1). The reaso...
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- Spring '14