Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions
نویسندگان
چکیده
In this paper we present a finite element discretization of the Joule-heating problem. We prove existence of solution to the discrete formulation and strong convergence of the finite element solution to the weak solution, up to a sub-sequence. We also present numerical examples in three spatial dimensions. The first example demonstrates the convergence of the method in the second example we consider an engineering application.
منابع مشابه
Optimal Error Analysis of Galerkin FEMs for Nonlinear Joule Heating Equations
We study in this paper two linearized backward Euler schemes with Galerkin finite element approximations for the time-dependent nonlinear Joule heating equations. By introducing a time-discrete (elliptic) system as proposed in Li and Sun (Int J Numer Anal Model 10:622–633, 2013; SIAM J Numer Anal (to appear)), we split the error function as the temporal error function plus the spatial error fun...
متن کاملFinite Element Approximation of a Stefan Problem with Degenerate Joule Heating
We consider a fully practical finite element approximation of the following degenerate system ∂ ∂t ρ(u) −∇.(α(u)∇u) σ(u) |∇φ|, ∇.(σ(u)∇φ) = 0 subject to an initial condition on the temperature, u, and boundary conditions on both u and the electric potential, φ. In the above ρ(u) is the enthalpy incorporating the latent heat of melting, α(u) > 0 is the temperature dependent heat conductivity, an...
متن کاملAnalysis of High-order Approximations by Spectral Interpolation Applied to One- and Two-dimensional Finite Element Method
The implementation of high-order (spectral) approximations associated with FEM is an approach to overcome the difficulties encountered in the numerical analysis of complex problems. This paper proposes the use of the spectral finite element method, originally developed for computational fluid dynamics problems, to achieve improved solutions for these types of problems. Here, the interpolation n...
متن کاملA Finite Element Model for the Time-dependent Joule Heating Problem
We study a spatially semidiscrete and a completely discrete finite element model for a nonlinear system consisting of an elliptic and a parabolic partial differential equation describing the electric heating of a conducting body. We prove error bounds of optimal order under minimal regularity assumptions when the number of spatial variables d < 3. We establish the existence of solutions with th...
متن کاملError Analysis of Linearized Semi-implicit Galerkin Finite Element Methods for Nonlinear Parabolic Equations
This paper is concerned with the time-step condition of commonly-used linearized semi-implicit schemes for nonlinear parabolic PDEs with Galerkin finite element approximations. In particular, we study the time-dependent nonlinear Joule heating equations. We present optimal error estimates of the semi-implicit Euler scheme in both the L norm and the H norm without any time-step restriction. Theo...
متن کامل