Numerical Analysis of the Quasistatic Thermoviscoelastic

In this work, the quasistatic thermoviscoelastic thermistor problem is considered. The thermistor model describes the combination of the effects due to the heat, electrical current conduction and Joule’s heat generation. The variational formulation leads to a coupled system of nonlinear variational equations for which the existence of a weak solution is recalled. Then, a fully discrete algorithm is introduced based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity assumptions, the linear convergence of the scheme is deduced. Finally, some numerical simulations are performed in order to show the behaviour of the algorithm. Mathematics Subject Classification. 65N15, 65N30, 74D10, 74S05, 74S20. Received: June 13, 2005. Introduction During the last twenty years, the thermistor problem has been studied by many authors (see, e.g., [1–4, 8, 10, 14, 15, 21, 22, 25]), dealing with different issues such as the existence and uniqueness of weak solutions or the degenerate cases, the so-called capacity solutions. Recently, in [5, 16–19, 26] the numerical solution of the thermistor problem has been considered, under different mechanical conditions, using finite differences or the finite element method. The thermistor model describes the combination of the effects due to the heat, electrical current conduction and Joule’s heat generation. The electrical conductivity is assumed increasing (see [20] for details concerning the physical setting). The novelty of this model is that thermoviscoelastic effects are taken into account. The existence of a unique weak solution for the dynamical problem was established in [20], and its proof is based on the regularization, time-retarding and a fixed point argument. This work continues [20] and it is parallel to [12], where a finite element algorithm is presented and numerical simulations are shown. Here, our aim is to provide the numerical analysis for the quasistatic thermoviscoelastic thermistor problem. The paper is structured as follows. In Section 2, the mechanical problem and its variational formulation are