Geodesic
A contact problem in thermoviscoelastic diffusion theory with second sound
Aouadi, Moncef1 ; Copetti, Maria I.M.2 ; Fernández, José R.3
1 École Nationale d’Ingénieurs de Bizerte, Université de Carthage, BP66, Campus Universitaire, 7035 Menzel Abderrahman, Tunisia
2 Departamento de Matemática, Universidade Federal de Santa Maria, 97105-900 Santa Maria, RS Brasil.
3 Departamento de Matemática Aplicada I, Universidade de Vigo, ETSI Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 759-796

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We consider a contact problem in thermoviscoelastic diffusion theory in one space dimension with second sound. The contact is modeled by the Signorini’s condition and the stress-strain constitutive equation is of Kelvin−Voigt type. The thermal and diffusion disturbances are modeled by Cattaneo’s law for heat and diffusion equations to remove the physical paradox of infinite propagation speed in the classical theory within Fourier’s law. The system of equations is a coupling of a hyperbolic equation with four parabolic equations. It poses some new mathematical difficulties due to the nonlinear boundary conditions and the lack of regularity. We prove that the viscoelastic term provides additional regularity leading to the existence of weak solutions. Then, fully discrete approximations to a penalized problem are considered by using the finite element method. A stability property is shown, which leads to a discrete version of the energy decay property. A priori error analysis is then provided, from which the linear convergence of the algorithm is derived. Finally, we give some computational results.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016039
Classification : 65N30, 65N15
Keywords: Thermoviscoelastic, diffusion, contact, existence, exponential stability, numerical analysis

Aouadi, Moncef 1 ; Copetti, Maria I.M. 2 ; Fernández, José R. 3

1 École Nationale d’Ingénieurs de Bizerte, Université de Carthage, BP66, Campus Universitaire, 7035 Menzel Abderrahman, Tunisia
2 Departamento de Matemática, Universidade Federal de Santa Maria, 97105-900 Santa Maria, RS Brasil.
3 Departamento de Matemática Aplicada I, Universidade de Vigo, ETSI Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain 
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     title = {A contact problem in thermoviscoelastic diffusion theory with second sound},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {759--796},
     publisher = {EDP-Sciences},
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Aouadi, Moncef; Copetti, Maria I.M.; Fernández, José R. A contact problem in thermoviscoelastic diffusion theory with second sound. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 759-796. doi: 10.1051/m2an/2016039

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