Error estimates of the form \[ (1)\quad \| \nabla u-(\nabla u)_ h\| \leq ch^{r+1}| \log h|^{\mu_ 1},\quad \| P_ mu- u_ h\| \leq ch^{r+2}| \log h|^{\mu_ 2} \] are characteristic of mixed finite element approximation to 2nd order elliptic boundary value problem, where \(u_ h\) and \((\nabla u)_ h\) are the approximations of u and \(\nabla u\), respectively, \(P_ m\) is an operator defined on each element that is invariant on polynomials of degree \(m\leq r\) there, and the norms \(\|.\|\) are usually \(L_ p\)- norms \(\|.\|_{O,p}\) with \(1\leq p\leq \infty\). In the post- processing, the authors construct an improved approximation \(u^*_ h\) of u by means of the formula \(u^*_ h=u_ h+\vec d_{r,m}\cdot (\nabla u)_ h\) such that the estimate \[ (2)\quad \| u-u^*_ h\|_{O,p}\leq c\{\| P_ mu-u_ h\|_{O,p}+h\| \nabla u- (\nabla u)_ h\|_{O,p}+h^{r+2}| u|_{r+2,p}\} \] holds for \(u\in W_ p^{r+2}(\Omega)\), where \(\vec d_{r,m}is\) a known vector differential operator. Applications to various mixed finite element schemes are given. However, estimate (2) is a pure approximation- theoretic result based on estimates of the form (1) only, and hence it is not restricted to mixed finite element schemes.