  This paper focuses on unconditionally optimal error analysis of an uncoupled
and linearized Crank--Nicolson Galerkin finite element method for the
time-dependent nonlinear thermistor equations in $d$-dimensional space,
$d=2,3$. We split the error function into two parts, one from the spatial
discretization and one from the temporal discretization, by introducing a
corresponding time-discrete (elliptic) system. We present a rigorous analysis
for the regularity of the solution of the time-discrete system and error
estimates of the time discretization. With these estimates and the proved
regularity, optimal error estimates of the fully discrete Crank--Nicolson
Galerkin method are obtained unconditionally. Numerical results confirm our
analysis and show the efficiency of the method.
