Maximal regularity of multistep fully discrete finite element methods for parabolic equations

TL;DR

This paper proves maximal regularity and error estimates for multistep fully discrete finite element methods applied to parabolic equations with general diffusion coefficients, extending previous semidiscrete results.

Contribution

It extends semidiscrete maximal $L^p$-regularity results to multistep fully discrete methods for parabolic equations with less regular diffusion coefficients.

Findings

Maximal $L^p$-regularity established for multistep finite element methods.

Optimal $ ext{ell}^p(L^q)$ error estimates derived.

Maximal angles of $R$-boundedness characterized for associated operators.

Abstract

This article extends the semidiscrete maximal $L^p$-regularity results in [27] to multistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in $W^{1,d+\beta}$, where $d$ is the dimension of space and $\beta>0$. The maximal angles of $R$-boundedness are characterized for the analytic semigroup $e^{zA_h}$ and the resolvent operator $z(z-A_h)^{-1}$, respectively, associated to an elliptic finite element operator $A_h$. Maximal $L^p$-regularity, optimal $\ell^p(L^q)$ error estimate, and $\ell^p(W^{1,q})$ estimate are established for fully discrete finite element methods with multistep backward differentiation formula.