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A new stabilized mixed finite-element method for Poisson equation based on two local Gauss integrations for linear element pair. (English) Zbl 1241.65091
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 $H\left(\text{div},{\Omega }\right)$, 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.
##### MSC:
 65N12 Stability and convergence of numerical methods (BVP of PDE) 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation 65N15 Error bounds (BVP of PDE)