We study an induction hardening model described by Maxwell's equations coupled with a heat equation. The magnetic induction field is assumed a nonlinear constitutional relation and the electric conductivity is temperature‐dependent. The T‐ψ method is to transform Maxwell's equations to the vector–scalar potential formulations and to solve the potentials by means of the finite element method. In this article, we present a fully discrete T‐ψ finite element scheme for this nonlinear coupled problem and discuss its solvability. We prove that the discrete solution converges to a weak solution of the continuous problem. Finally, we conclude with two numerical experiments for the coupled system.

中文翻译：

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完全离散的T-ψ有限元方法来解决非线性感应淬火问题

我们研究了由麦克斯韦方程和热方程描述的感应淬火模型。假定磁场是非线性的本构关系，并且电导率与温度有关。所述Ť - ψ方法是变换麦克斯韦方程的矢量标量势的制剂和通过有限元方法的手段来解决的电位。在本文中，我们针对此非线性耦合问题提出了一种完全离散的T - ψ有限元方案，并讨论了其可解性。我们证明了离散解收敛于连续问题的弱解。最后，我们以耦合系统的两个数值实验作为结束。