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# 46 equation 746 of course cannot be solved because i

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Unformatted text preview: ite~ dimensional subspace S N ( e) of H 1( e ). Thus, ~ Ae(V E ) = (V f )e ; Ae(V U )+ < V pun >e 8V 2 S N ( e ): (7.4.7) ~ We will discuss selection of S N momentarily. Let us rst prescribe the ux pun appearing in the last term of (7.4.7). The simplest possibility is to use an average ux obtained from pUn across the element boundary, i.e., + ; ~ Ae(V E ) = (V f )e ; Ae(V U )+ < V (pUn) + (pUn ) >e 8V 2 S N ( e ) (7.4.8) 2 7.4. A Posteriori Error Estimation 21 where superscripts + and ;, respectively, denote values of pUn on the exterior and interior of @ e . Equation (7.4.8) is a local Neumann problem for determining the error approximation E on each element. No assembly and global solution is involved. Some investigators prefer to apply the divergence theorem to the second term on the right to obtain + ; Ae(V E ) = (V r)e; < V (pUn); >e + < V (pUn ) + (pUn) >e 2 or + ; Ae(V E ) = (V r)e+ < V (pUn ) ; (pUn) >e (7.4.9a) 2 where r(x y) = f + r prU ; qU (7.4.9b) is the residual. This form involves jumps in the ux across element boundaries. ~ ~ Now let us select the error approximation space S N . Choosing S N = S N does not ~ work since th...
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