Maximum-norm stability and maximal Lp regularity of FEMs for parabolic equations with Lipschitz continuous coefficients

In this paper, we study the semi-discrete Galerkin finite element method for parabolic equations with Lipschitz continuous coefficients. We prove the maximumnorm stability of the semigroup generated by the corresponding elliptic finite element operator, and prove the space-time stability of the parabolic projection onto the finite element space in L∞(QT ) and L p((0, T ); Lq ( )), 1 < p, q < ∞. The maximal L p regularity of the parabolic finite element equation is also established. Mathematics Subject Classification 65M30 · 65M12 · 35K20