A penalized version of the local minimization scheme for rate-independent systems. (May 2021)

• Record Type: Journal Article
• Authors: Knees, Dorothee; Shcherbakov, Viktor
• Is Part Of: Applied mathematics letters. Volume 115(2021)
• Publication Date: 2021-05
• Subjects: Rate-independent system -- Local minimization scheme -- Moreau–Yosida approximation -- Parametrized BV solution; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2020.106954
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: The letter presents a penalized version of the time-discretization local minimization scheme first proposed by Efendiev and Mielke in 2006 to resolve time discontinuities in rate-independent systems with nonconvex energies. In order to penalize inequality constrains enforcing the local minimality, the Moreau–Yosida approximation is employed. We prove the convergence of time-discrete solutions to functions that are parametrized BV solutions of the time-continuous problem (in an abstract infinite-dimensional setting), provided that the discretization and approximation parameters are chosen appropriately. We test our scheme on a one-dimensional example and find a notable improvement compared with the original version.