LAUSR

Maximal regularity is a fundamental concept in the theory of partial differential equations. In this paper, we establish a fully discrete version of maximal regularity for parabolic equations on a polygonal or polyhedral domain $$\varOmega $$Ω. We derive various stability results in the discrete $$L^p(0,T;L^q(\varOmega ))$$Lp(0,T;Lq(Ω)) norms for the finite element approximation with the mass-lumping to the linear heat equation. Our method of analysis is an operator theoretical one using pure imaginary powers of operators and might be a discrete version of the result of Dore and Venni. As an application, optimal order error estimates in those norms are proved. Furthermore, we study the finite element approximation for semilinear heat equations with locally Lipschitz continuous nonlinear terms and offer a new method for deriving optimal order error estimates. Some interesting auxiliary results including discrete Gagliardo–Nirenberg and Sobolev inequalities are also presented.