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Convergence analysis of time-discretisation schemes for rate-independent systems. (English) Zbl 1437.49009

Summary: It is well known that rate-independent systems involving nonconvex energy functionals in general do not allow for time-continuous solutions even if the given data are smooth. In the last years, several solution concepts were proposed that include discontinuities in the notion of solution, among them the class of global energetic solutions and the class of BV-solutions. In general, these solution concepts are not equivalent and numerical schemes are needed that reliably approximate that type of solutions one is interested in. In this paper, we analyse the convergence of solutions of three time-discretisation schemes, namely an approach based on local minimisation, a relaxed version of it and an alternate minimisation scheme. For all three cases, we show that under suitable conditions on the discretisation parameters discrete solutions converge to limit functions that belong to the class of BV-solutions. The proofs rely on a reparametrisation argument. We illustrate the different schemes with a toy example.

MSC:

49J27 Existence theories for problems in abstract spaces
49J40 Variational inequalities
35Q74 PDEs in connection with mechanics of deformable solids
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
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