Numerical analysis of a dual‐phase‐lag model involving two temperatures. (27th December 2019)

• Record Type: Journal Article
• Authors: Bazarra, Noelia; Fernández, José R.; Magaña, Antonio; Quintanilla, Ramón
• Is Part Of: Mathematical methods in the applied sciences. Volume 43:Number 5(2020)
• Page Start: 2759
• Page End: 2771
• Publication Date: 2019-12-27
• Subjects: a priori estimates -- finite elements -- heat conduction -- numerical simulations -- phase‐lag with two temperatures; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.6082
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this paper, we numerically analyse a phase‐lag model with two temperatures which arises in the heat conduction theory. The model is written as a linear partial differential equation of third order in time. The variational formulation, written in terms of the thermal acceleration, leads to a linear variational equation, for which we recall an existence and uniqueness result and an energy decay property. Then, using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives, fully discrete approximations are introduced. A discrete stability property is proved, and a priori error estimates are obtained, from which the linear convergence of the approximation is derived. Finally, some one‐dimensional numerical simulations are described to demonstrate the accuracy of the approximation and the behaviour of the solution.