In this paper, we present a novel error analysis for a linearized backward Euler finite element scheme applied to the Landau-Lifshitz equation under low regularity conditions. A key ingredient of our approach is the introduction of a new projection operator based on the temporal average of the magnetization. This operator allows us to fully exploit the intrinsic regularity property \(\varvec{m} \in L^2(0, T; (H^3(\varOmega ))^3)\) , thereby avoiding higher regularity assumptions. Our error estimates are developed for two distinct scenarios. Firstly, under the assumption that the initial data satisfies \(\varvec{m}^0 \in (H^2(\varOmega ))^3 \cap (W^{1,\infty }(\varOmega ))^3\) and \(\varvec{m}_t^0 \in (H^2(\varOmega ))^3\) , we establish a conditionally optimal error estimate whose convergence optimality adapts to the spatial regularity of the solution at current time step. Secondly, by strengthening the regularity to \(\varvec{m} \in L^\infty (0,T; (W^{1,\infty }(\varOmega ))^3)\) while relaxing the requirement on the initial time derivative to \(\varvec{m}_t^0 \in (H^1(\varOmega ))^3\) , we derive the optimal error estimate of order \(O(\tau + h^2)\) . Additionally, the analysis is extended to smoother solutions, and the error in preserving the unit length constraint of the Landau–Lifshitz equation is rigorously bounded for all cases considered.
