Existence for the thermoviscoelastic thermistor problem

Abstract

The existence of a weak solution to a dynamic model for a thermistor, which takes into account the thermoelastic properties of the device, is established. The model consists of a coupled system of the equations of dynamic thermoviscoelasticity, the heat equation with the Joule heating term, and the quasistatic charge conservation equation. The system is strongly nonlinear since the electrical conductivity is assumed to be temperature dependent, and the Joule heating term is quadratic in the gradient of the electric potential. The existence of a solution is obtained by considering a sequence of approximate time-retarded problems. After obtaining the necessary a priori estimates, a solution of the problem is found by passing to the approximation limit. The uniqueness of the solution remains an open problem. c 2008 Foundation for Scientific Research and Technological Innovation(FSRTI). All rights reserved.MSC: 74D05, 74F05, 74H20, 74H30

Numerical analysis of the quasistatic thermoviscoelastic thermistor problem

“…In this section, we follow [12,20] and we refer the reader there for details. Let S d denote the space of second order symmetric tensors defined on R d , and let ":" and | · | represent the inner product and the norm on S d , respectively.…”

“…An existence result for a dynamic version of problem (13)- (16) was given in [20]. It is based on time-retarded approximate problems, a priori estimates and compactness arguments.…”

“…Remark 2.1. Notice that the discrete problem (20) can be seen as a linear system which is solved using Cholesky's method and then, displacements are updated from (21). Besides, discretized problem (22)- (23) is a nonlinear system due to the temperature dependence on the physical parameters, as the electrical and thermal conductivities, and the Joule's source term in (23).…”

“…Secondly, we rewrite the variational equation (15) at time t = t n for all w = w h ∈ E h ⊂ E and we substract it to (20) to obtain…”

“…(24) hold. Let {u, θ, φ} denote the solution to problem (13)- (16) and {u h , θ h , φ h } the solution to discrete problem (20)- (23). Then, the following error estimates are obtained for all {z…”