Numerical analysis of the quasistatic thermoviscoelastic thermistor problem

Abstract

Abstract. 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.

Finite element convergence analysis for the thermoviscoelastic Joule heating problem

“…Lemma 3.1. Let Assumptions 3.1-3.4 be satisfied and let (θ, φ, u) and (Θ n , Φ n , U n ) be solutions to the equations (4)-( 6) and (10)- (12), respectively. Then we have…”

“…Many authors have analyzed methods for similar problems. For example, [12] considers the quasi-static version where the ü-term is ignored, [1], [11] and [22] considers the non-deformable case, [13,14] treat the purely thermoviscoelastic case (no φ) with nonlinear constituent law, etc. Additionally, in the deformable case a common theme seems to be suboptimal convergence orders, i.e.…”

“…Let Assumptions 3.1-3.4 be satisfied and (Θ n , Φ n , U n ) and (Θ n h , Φ n h , U n h ) be solutions to equations (10)-(12) and(13)-(15), respectively. Then there are positive constants k 0 and h 0 such that if k < k 0 and h < h 0 then for n = 1, .…”

Invariance of flows in doubly-connected domains with the same modulus

“…By the conservation of charge and energy we obtain the so-called thermistor problem, which has been thoroughly studied by many authors, [3,4,10] and recently by [6,7] among others. In particular in [9] the invariance by conformal mappings of the problem is noted.…”