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 A weak Galerkin finite element method for the second order elliptic problems with mixed boundary conditions Keywords:Galerkin finite element methods, discrete gradient, second-order elliptic problems, mixed boundary conditions. Abstract: In this paper, a weak Galerkin finite element method is proposed and analyzed for the second-order elliptic equation with mixed boundary conditions. Optimal order error estimates are established in both discrete $H^1$ norm and the standard $L^2$ norm for the corresponding WG approximations. The numerical experiments are presented to verify the efficiency of the method.