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- Title
Finite element discretization of local minimization schemes for rate-independent evolutions.
- Authors
Meyer, Christian; Sievers, Michael
- Abstract
This paper is concerned with a space-time discretization of a rate-independent evolution governed by a non-smooth dissipation and a non-convex energy functional. For the time discretization, we apply the local minimization scheme introduced in Efendiev and Mielke (J Convex Anal 13(1):151-167, 2006), which is known to resolve time discontinuities, which may show up due to the non-convex energy. The spatial discretization is performed by classical linear finite elements. We show that accumulation points of the sequence of discrete solutions for mesh size tending to zero exist and are so-called parametrized solutions of the continuous problem. The discrete problems are solved by means of a mass lumping scheme for the non-smooth dissipation functional in combination with a semi-smooth Newton method. A numerical test indicates the efficiency of this approach. In addition, we compared the local minimization scheme with a time stepping scheme for global energetic solutions, which shows that both schemes yield different solutions with differing time discontinuities.
- Subjects
DISCRETIZATION methods; FINITE element method; SPACE-time mathematical models; NUMERICAL analysis; CONTINUOUS probability theory
- Publication
Calcolo, 2019, Vol 56, Issue 1, p1
- ISSN
0008-0624
- Publication type
Article
- DOI
10.1007/s10092-018-0301-4