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This paper is concerned with the time-step condition of commonly-usedlinearized semi-implicit schemes for nonlinear parabolic PDEs with Galerkinfinite element approximations. In particular, we study the time-dependentnonlinear Joule heating equations. We present optimal error estimates of thesemi-implicit Euler scheme in both the $L^2$ norm and the $H^1$ norm withoutany time-step restriction. Theoretical analysis is based on a new splitting ofthe error and precise analysis of a corresponding time-discrete system. Themethod used in this paper can be applied to more general nonlinear parabolicsystems and many other linearized (semi)-implicit time discretizations forwhich previous works often require certain restriction on the time-step size$\tau$.

Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations