Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations

TL;DR

This paper develops a novel analytical framework combining maximal regularity and energy estimates to derive optimal error bounds for implicit time discretization methods applied to quasilinear parabolic equations, without growth assumptions.

Contribution

It introduces a new approach that merges maximal regularity with energy estimates to analyze fully implicit and linearly implicit BDF methods for quasilinear parabolic equations, achieving optimal error bounds.

Findings

Optimal-order error bounds in maximum and energy norms.

Applicable to fully implicit and linearly implicit BDF methods.

No assumptions on coefficient growth or decay.

Abstract

We analyze fully implicit and linearly implicit backward difference formula (BDF) methods for quasilinear parabolic equations, without making any assumptions on the growth or decay of the coefficient functions. We combine maximal parabolic regularity and energy estimates to derive optimal-order error bounds for the time-discrete approximation to the solution and its gradient in the maximum norm and energy norm.