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419a 28 analysis of the finite element method where

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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...
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