Abstarct: In this thesis we focus on estimating A posteriori error using Dual Weighted Residual of partial diffrential equations.This end we solve the equations by Galerkin finite element method baxse on continuous piecwise constant in space and discontinuous piecwise constant functions in time. We compute the A posteriori error estimate for this equations using Dual Weighted Residual method. Finally we have some numerical results for the adpativity baxsed on this a posteriori error estimate.