Stability and convergence of fully discrete Galerkin FEMs for the nonlinear thermistor equations in a nonconvex polygon

Abstract

In this paper, we establish the unconditional stability and optimal error estimates of a linearized backward Euler---Galerkin finite element method (FEM) for the time-dependent nonlinear thermistor equations in a two-dimensional nonconvex polygon. Due to the nonlinearity of the equations and the non-smoothness of the solution in a nonconvex polygon, the analysis is not straightforward, while most previous efforts for problems in nonconvex polygons mainly focused on linear models. Our theoretical analysis is based on an error splitting proposed in [30, 31] together with rigorous regularity analysis of the nonlinear thermistor equations and the corresponding iterated (time-discrete) elliptic system in a nonconvex polygon. With the proved regularity, we establish the stability in $$l^\infty (L^\infty )$$lź(Lź) and the convergence in $$l^\infty (L^2)$$lź(L2) for the fully discrete finite element solution without any restriction on the time-step size. The approach used in this paper may also be applied to other nonlinear parabolic systems in nonconvex polygons. Numerical results confirm our theoretical analysis and show clearly that no time-step condition is needed.