The authors propose a new mixed formulation for the Poisson equation on Lipschitz domains and establish the wellposedness of the new formulation. Unlike the traditional mixed formulation which solves

$p$ in

${L}^{2}\left({\Omega}\right)$ and

$u=\nabla p$ in

$H(\text{div},{\Omega})$, the new formulation solves

$p$ in

${H}^{1}\left({\Omega}\right)$ and

$u$ in

${L}^{2}\left({\Omega}\right)$. This lowers the regularity restriction for

$u$. Section 2 presents the new mixed formulation and proves its wellposedness. Section 3 proposes the

${\left({P}_{0}\right)}^{2}$-

${P}_{1}$ finite element approximation to the new formulation, that is,

$u$ is solved with piecewise constant polynomials and

$p$ is solved with piecewise linear continuous polynomials. Optimal error estimates are also proved. Section 4 proposes the

${\left({P}_{1}\right)}^{2}$-

${P}_{1}$ finite element approximation to the new formulation and proves the optimal error estimates. The last section presents the numerical experiments for verifying the theories.