Unconditionally optimal error analysis of fully discrete Galerkin methods for general nonlinear parabolic equations

TL;DR

This paper establishes unconditionally optimal error bounds for fully discrete Galerkin methods applied to general nonlinear parabolic equations, ensuring accuracy without restrictions on discretization parameters.

Contribution

It provides a novel unconditionally optimal error analysis framework for fully discrete Galerkin methods on nonlinear parabolic systems, including error splitting and boundedness proofs.

Findings

Error estimates are independent of time step size.

Numerical examples confirm theoretical error bounds.

Boundedness of numerical solutions in key norms is proven.

Abstract

The paper focuses on unconditionally optimal error analysis of the fully discrete Galerkin finite element methods for a general nonlinear parabolic system in $\R^d$ with $d=2,3$. In terms of a corresponding time-discrete system of PDEs as proposed in \cite{LS1}, we split the error function into two parts, one from the temporal discretization and one the spatial discretization. We prove that the latter is $\tau$-independent and the numerical solution is bounded in the $L^{\infty}$ and $W^{1,\infty}$ norms by the inverse inequalities. With the boundedness of the numerical solution, optimal error estimates can be obtained unconditionally in a routine way. Several numerical examples in two and three dimensional spaces are given to support our theoretical analysis.