Mathematical analysis of a penalisation strategy for incompressible elastodynamics

Article type: Research Article

Authors: Caforio, F.^{a; b; *} | Imperiale, S.^{a; b; **}

Affiliations: [a] Inria, Université Paris-Saclay, France. E-mails: [email protected], [email protected] | [b] LMS, Ecole Polytechnique, CNRS, Université Paris-Saclay, France

Correspondence: [**] Corresponding author. E-mail: [email protected].

Note: [*] Current affiliation: E-mail: [email protected]. Gottfried Schatz Research Center: Division of Biophysics, Medical University of Graz, Austria.

Abstract: This work addresses the mathematical analysis – by means of asymptotic analysis – of a penalisation strategy for the full discretisation of elastic wave propagation problems in quasi-incompressible media that has been recently developed by the authors. We provide a convergence analysis of the solution to the continuous version of the penalised problem towards its formal limit when the penalisation parameter tends to infinity. Moreover, as a fundamental intermediate step we provide an asymptotic analysis of the convergence of solutions to quasi-incompressible problems towards solutions to purely incompressible problems when the incompressibility parameter tends to infinity. Finally, we further detail the regularity assumptions required to guarantee that the mentioned convergence holds.

Keywords: Elastodynamics, incompressibility, asymptotic analysis

DOI: 10.3233/ASY-201617

Journal: Asymptotic Analysis, vol. 122, no. 3-4, pp. 225-255, 2021