  This paper is concerned with the time-step condition of commonly-used
linearized semi-implicit schemes for nonlinear parabolic PDEs with Galerkin
finite element approximations. In particular, we study the time-dependent
nonlinear Joule heating equations. We present optimal error estimates of the
semi-implicit Euler scheme in both the $L^2$ norm and the $H^1$ norm without
any time-step restriction. Theoretical analysis is based on a new splitting of
the error and precise analysis of a corresponding time-discrete system. The
method used in this paper can be applied to more general nonlinear parabolic
systems and many other linearized (semi)-implicit time discretizations for
which previous works often require certain restriction on the time-step size
$\tau$.
