21 with nite di erence techniques derivatives in 121a

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online