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The paper is concerned with Galerkin finite element solutions for parabolicequations in a convex polygon or polyhehron with a diffusion coefficient in$W^{1,N+\epsilon}$ for some $\epsilon>0$, where $N$ denotes the dimension ofthe domain. We prove the analyticity of the semigroup generated by the discreteelliptic operator, the discrete maximal $L^p$ regularity and the optimal $L^p$error estimate of the finite element solution for the parabolic equation.

Maximal $\bf L^p$ analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra