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

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.