  We study fully discrete linearized Galerkin finite element approximations to
a nonlinear gradient flow, applications of which can be found in many areas.
Due to the strong nonlinearity of the equation, existing analyses for implicit
schemes require certain restrictions on the time step and no analysis has been
explored for linearized schemes. This paper focuses on the unconditionally
optimal $L^2$ error estimate of a linearized scheme. The key to our analysis is
an iterated sequence of time-discrete elliptic equations and a rigorous
analysis of its solution. We prove the $W^{1,\infty}$ boundedness of the
solution of the time-discrete system and the corresponding FE solution, based
on a more precise estimate of elliptic PDEs in $W^{2,2+\epsilon}$ and a
physical feature of the gradient-dependent diffusion coefficient. Numerical
examples are provided to support our theoretical analysis.
