21 with nite di erence techniques derivatives in 121a

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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: (1.2.2a) 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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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

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