  We propose a new and simpler residual based a posteriori error estimator for
finite element approximation of the elliptic obstacle problem. The results in
the article are two fold. Firstly, we address the influence of the
inhomogeneous Dirichlet boundary condition in {\em a posteriori} error control
of the elliptic obstacle problem. Secondly by rewriting the obstacle problem in
an equivalent form, we derive simpler {\em a posteriori} error bounds which are
free from min/max functions. To accomplish this, we construct a post-processed
solution $\tilde u_h$ of the discrete solution $u_h$ which satisfies the exact
boundary conditions although the discrete solution $u_h$ may not satisfy. We
propose two post processing methods and analyze them. We remark that the
results known in the literature are either for the homogeneous Dirichlet
boundary condition or that the estimator is only weakly reliable in the case of
inhomogeneous Dirichlet boundary condition.
