Maximum-norm stability and maximal $$L^{p}$$ L p regularity of FEMs for parabolic equations with Lipschitz continuous coefficients

Abstract

In this paper, we study the semi-discrete Galerkin finite element method for parabolic equations with Lipschitz continuous coefficients. We prove the maximum-norm 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 $$L^\infty ( Q_T)$$L?(QT) and $$L^p((0,T);L^q(\Omega ))$$Lp((0,T)?Lq(Ω)), $$1