Maximal regularity of evolving FEMs for parabolic equations on an evolving surface

In this paper, we prove that the spatially semi-discrete evolving finite element methods (FEMs) for parabolic equations on a given evolving hypersurface of arbitrary dimensions preserves the maximal $L^{p}$-regularity at the discrete level. We first establish the results on a stationary surface and then extend them, via a perturbation argument, to the case where the underlying surface is evolving under a prescribed velocity field. The proof combines techniques in evolving FEM, properties of Green’s functions on (discretized) closed surfaces, and local energy estimates for FEMs.