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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 $L^2(\Omega)$ and $u=\nabla p$ in $H(\text{div},\Omega)$, the new formulation solves $p$ in $H^1(\Omega)$ and $u$ in $L^2(\Omega)$. This lowers the regularity restriction for $u$. Section 2 presents the new mixed formulation and proves its wellposedness. Section 3 proposes the $(P_0)^2$-$P_1$ 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 $(P_1)^2$-$P_1$ finite element approximation to the new formulation and proves the optimal error estimates. The last section presents the numerical experiments for verifying the theories.

65N12Stability and convergence of numerical methods (BVP of PDE)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
65N15Error bounds (BVP of PDE)
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