The strain smoothing technique over higher order elements and arbitrary polytopes yields less accurate solutions than other techniques such as the conventional polygonal finite element method. In this work, we propose a linear strain smoothing scheme that improves the accuracy of linear and quadratic approximations over convex polytopes. The main idea is to subdivide the polytope into simplicial subcells and use a linear smoothing function in each subcell to compute the strain. This new strain is then used in the computation of the stiffness matrix. The convergence properties and accuracy of the proposed scheme are discussed by solving a few benchmark problems. Numerical results show that the proposed linear strain smoothing scheme makes the approximation based on polytopes able to deliver the same optimal convergence rate as traditional quadrilateral and hexahedral approximations. The accuracy is also improved, and all the methods tested pass the patch test to machine precision. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.