42a vfdxdy 742b prv ru qvudxdy 742c and

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Unformatted text preview: seek an error estimation technique that only requires local (element level) mesh computations, so let's construct a local Galerkin problem on element e by integrating (7.4.1a) over e and applying the divergence theorem to obtain: determine u 2 H 1( e) such that Ae(v u) = (v f )e+ < v pun >e where (v f )e = ZZ 8v 2 H 1 ( e ) (7.4.4a) vfdxdy (7.4.4b) prv ru + qvu]dxdy (7.4.4c) e Ae(v u) = ZZ e and < v u >e= Z @e vuds: (7.4.4d) As usual, e is the domain of element e, s is a coordinate along @ e , and n is a unit outward normal to @ e . Let u=U +e (7.4.5) where e(x y) is the discretization error of the nite element solution, and substitute (7.4.5) into (7.4.4a) to obtain Ae(v e) = (v f )e ; Ae(v U )+ < v pun >e 8v 2 H 1 ( e ): (7.4.6) Equation (7.4.6), of course, cannot be solved because (i) v, u, and e are elements of an in nite-dimensional space and (ii) the ux pun is unknown on @ e . We could obtain a nite element solution of (7.4.6) by approximating e and v by E and V in a n...
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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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