Unconditional Optimal Error Estimates of BDF–Galerkin FEMs for Nonlinear Thermistor Equations

In this paper we study linearized backward differential formula (BDF) type schemes with Galerkin finite element approximations for the time-dependent nonlinear thermistor equations. Optimal $$L^2$$L2 error estimates for the proposed schemes are proved unconditionally. The proof consists of two steps. First, the boundedness of the numerical solution in certain strong norms is obtained by a temporal-spatial error splitting argument. Second, a traditional approach is used to provide an optimal $$L^2$$L2 error estimate for $$r$$r-th order FEMs $$(r \ge 1)$$(r≥1). Numerical experiments in both two and three dimensional spaces are conducted to confirm our theoretical analysis and show the high order accuracy and unconditional stability (convergence) of the linearized BDF–Galerkin FEMs.

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