A penalized version of the local minimization scheme for rate-independent systems

• Published in: Applied mathematics letters Vol. 115; p. 106954
• Main Authors: Knees, Dorothee, Shcherbakov, Viktor
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.05.2021
• Subjects: Local minimization scheme; Moreau–Yosida approximation; Parametrized BV solution; Rate-independent system; Rate-independent system; Parametrized BV solution; Local minimization scheme; Moreau–Yosida approximation
• ISSN: 0893-9659; 1873-5452
• DOI: 10.1016/j.aml.2020.106954

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.