*Submitted Paper*

**Inserted:** 17 dec 2017

**Last Updated:** 21 jun 2018

**Year:** 2017

**Abstract:**

We perform a convergence analysis of a discrete-in-time minimization scheme approximating a finite dimensional singularly perturbed gradient flow. We allow for different scalings between the viscosity parameter $\varepsilon$ and the time scale $\tau$. When the ratio $\frac{\varepsilon}{\tau}$ diverges, we rigorously prove the convergence of this scheme to a (discontinuous) Balanced Viscosity solution of the quasistatic evolution problem obtained as formal limit, when $\varepsilon\to 0$, of the gradient flow. We also characterize the limit evolution corresponding to an asymptotically finite ratio between the scales, which is of a different kind. In this case, a discrete interfacial energy is optimized at jump times.

**Keywords:**
singular perturbations, Gradient Flow, Variational methods, rate-independent systems, minimizing movement, Balanced Viscosity solutions, crease energy

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