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The paper is concerned with Galerkin finite element solutions of parabolic equations in a convex polygon or polyhedron with a diffusion coefficient in W1, N+α for some α > 0, where N denotes the dimension of the domain. We prove the analyticity of the semigroup generated by the discrete elliptic operator, the discrete maximal Lp regularity and the optimal Lp error estimate of the finite element solution for the parabolic equation. © 2016 American Mathematical Society.