419a 28 analysis of the finite element method where

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: at smooth solutions of the weak problem (7.4.19) satisfy the Neumann problem ;r prE + qE = r pEn = R (x y) 2 e (x y) 2 @ e : (7.4.20a) (7.4.20b) Solutions of (7.4.20) only exist when the data R and r satisfy the equilibrium condition ZZ r(x y)dxdy + e Z @e R(s)ds = 0: (7.4.20c) This condition will most likely not be satis ed by the choice of = 1=2. Ainsworth and Oden 5] describe a relatively simple procedure that requires the solution of the Poisson problem ; !e = r @!e = R @n (x y ) 2 (x y) 2 @ e ; @ !e = 0 (7.4.21a) e E (x y) 2 @ E : (7.4.21b) (7.4.21c) The error estimate is kE kA = 2 N X e=1 Ae(!e !e): (7.4.21d) The function is approximated by a piecewise-linear polynomial in a coordinate s on @ e and may be determined explicitly prior to solving (7.4.21). Let us illustrate the e ect of this equilibrated error estimate. Example 7.4.4. Oden 16] considers a \cracked panel" as shown in Figure 7.4.4 and determines u as the solution of A(v u) = ZZ (vxux + vy uy )dxdy = 0: 7.4. A Posteriori Error Estimation 29 y u = r1/2 cos θ/2 r ΩL u=0 ΩR x θ u y= 0 Figure 7.4.4: Cracked panel used for Example 7.4.4...
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