Finite difference method and its convergent error analyses for thermistor problem

The therm istor problem is an initial-boundary value problem ofcoupled nonlineardif- ferentialequations.The nonlinear PDEs consist of a heat equation w ith the Joule heating as a source and a currentconservation equation w ith tem perature-dependentelectricalconductivity. This problem has im portant applications in industry.In this paper,A new finite difference schem e is proposed on nonuniform rectangularpartition forthe therm istor problem .In thetheo- reticalanalyses,the second-order error estim ates are obtained for electricalpotentialin discrete L2 and H1 norm s,and for the tem perature in L2 norm .In order to getthese second-order error estim ates,the Joule heating source is used in a changed equivalentform .