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In this paper, a two-dimensional mesh-free formulation using a first-order convex approximation is presented for the analysis of electrostatic problems. The generalized mesh-free approximation method is employed to construct the first-order convex approximation which exhibits a weak Kronecker-delta property at the boundary and allows a direct enforcement of essential boundary conditions. A two by two Gaussian quadrature rule based on finite element mesh is utilized for the domain integration of mesh-free discrete equation. One numerical example is analyzed to demonstrate the accuracy of the proposed formulation and comparison is made with the analytical and finite element solutions.