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 A Weak Galerkin Method for Second Order Elliptic Problems with Polynomial Reduction Nolisa Malluwawadu,Saqib Hussain Keywords:Galerkin finite element methods, second-order elliptic problems, discrete gradient, mixed finite element methods Abstract: The second order elliptic equation, which is also know as the diffusion-convection equation, is of great interest in many branches of physics and industry. In this paper, we use the weak Galerkin finite element method to study the general elliptic equation. A weak Galerkin finite element method is proposed and analyzed. This scheme features piecewise polynomials of degree $k\geq 1$ on each element and piecewise polynomials of degree $k-1\geq 0$ on each edge or face of the element. Error estimates of optimal order are established in both discrete $H^1$ and standard $L^2$ norm. The paper also presents some numerical experiments to verify the efficiency of the method.