Linearly Implicit Finite Element Methods for the Time-Dependent Joule Heating Problem

Abstract

Completely discrete numerical methods for a nonlinear elliptic-parabolic system, the time-dependent Joule heating problem, are introduced and analyzed. The equations are discretized in space by a standard finite element method, and in time by combinations of rational implicit and explicit multistep schemes. The schemes are linearly implicit in the sense that they require, at each time level, the solution of linear systems of equations. Optimal order error estimates are proved under the assumption of sufficiently regular solutions. (2000): 65M30, 65M15, 35K60. AMS subject classification

Unconditional Optimal Error Estimates of BDF–Galerkin FEMs for Nonlinear Thermistor Equations

“…Theoretical analyses for the nonlinear thermistor system have been done extensively, see, e.g., [3,5,9,19,32,33]. Numerical methods and analysis for the time-dependent nonlinear thermistor system (1.1)-(1.4) can be found in [2,4,12,31,34,35]. For the two dimensional problem, optimal L 2 error estimates of linearized semi-implicit schemes with Galerkin and mixed FEMs were obtained in [31] and [34] under a weak time step condition, respectively.…”

Superconvergent estimates of conforming finite element method for nonlinear time‐dependent Joule heating equations

“…A lot of studies of theoretical and numerical analysis have been devoted to system (1.1)-(1.4) (see e.g., [1][2][3][4][5][6][7][8][9][10][11][12] and references therein). More precisely, the existence of solutions was showed in [2].…”

Uniform Bmo Estimate of Parabolic Equations and Global Well-Posedness of the Thermistor Problem

“…The electric current J is related to the electric field via J = σ (u)E, where σ (u) is the electric conductivity of the conductor, dependent on the temperature u. The heat produced (per unit volume) by the electric current is given by Joule's law: E · J = σ (u)|∇φ| 2 , and the conservation of charge is described by ∇ · J = 0.…”

“…For interested readers, we refer to [2,4,10,12,19,32] for numerical methods and numerical analysis of the thermistor problem based on the well-posedness assumption. Analyses of the related optimal control problems can be found in [15,16], and the existence of solutions for a related thermoviscoelastic thermistor problem was presented in [18].…”