# Finite element method - Dual functional Error estimate

I have an equation -$u''=1,~~x\in(0,1)$. I solved it numerically by Finite element method. and find an approximate solution. $u_{h}$. As you know to do this I defined bilinear and linear functionals for weak formulation s.t

a(u,v)=l(v) for every v in $H_{0}^{1}$ where

a(u,v)=$\int_{0}^{1}u'v'dx$ and l(v)=$\int_{0}^{1}vdx$.

But to find a functinal estimates, |J($u-u_{h}$ )|, I defined a functinal J(v)=$\int_{0}^{1}vdx$ and try to find z in dual problem a(v,z)=J(v) for every v in $H_{0}^{1}$

and I solved the dual problem then find approximation for z.

But J($u-u_{h}$)=a($u-u_{h}$,z)=a(u,z)-a($u_{h}$,z) =l(z)-a($u_{h}$,z)$~~~~~~~$ (**)

(**) is called the error indicator and my questions is that how can I find it ?

You compute $z$ approximately as you described. Then plug this $z$ into the formula $J(u-u_h)=\ell(z)-a(u_h,z)$.
• Actually you can do it analytically. The coefficients of the expansion of $z$ in terms of the hat functions are given. So you just need to analytically compute a few integrals involving hat functions. – timur Aug 15 '12 at 18:57