# 32bc exist and that the trivial boundary conditions

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

This is the end of the preview. Sign up to access the rest of the document.

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)...
View Full Document

## This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

Ask a homework question - tutors are online