Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems

TL;DR

This paper establishes maximal regularity for backward difference time discretizations of linear and nonlinear parabolic PDEs on evolving surfaces, ensuring stability, convergence, and optimal error estimates for numerical solutions.

Contribution

It introduces a novel approach to preserve maximal $L^p$-regularity in discrete time for PDEs on evolving surfaces, extending the analysis to nonlinear problems.

Findings

Backward difference schemes preserve maximal $L^p$-regularity on evolving surfaces.

Stability and convergence of nonlinear surface PDE discretizations are proven.

Optimal error estimates are achieved in the $L^ abla(0,T;W^{1, abla})$ norm.

Abstract

Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability and convergence of time discretizations of nonlinear parabolic equations on an evolving surface, with linearly implicit backward differentiation formulae characteristic trajectories of the surface, for general locally Lipschitz nonlinearities. The discrete maximal $L^p$-regularity is used to prove the boundedness and stability of numerical solutions in the $L^\infty(0,T;W^{1,\infty})$ norm, which is used to bound the nonlinear terms in the stability analysis. Optimal-order error estimates of time discretizations in the $L^\infty(0,T;W^{1,\infty})$ norm is obtained by combining the stability analysis with the consistency estimates.