4.1 Dynamics of Systems with Friction or Elastoplastic Terms

Properties of numerical schemes for discrete (and even more general) systems including friction or elastoplastic terms have been studied in [Bastien (2000)], [Bastien and Schatzman (1999)] and [Bastien and Schatzman (2000)]. Let us recall the main results, adapted from these two last references. Let H be a separable Hilbert space with the scalar product ( , ) and norm . Let us consider the problem:

(4.1)

with

(4.2)

where B is a Lipschitz-continuous operator and, H is an elliptic one, i.e.:

(4.3)

and

(4.4)

A is a maximal monotone operator on H with bounded section A ⁰, and

(4.5)

is Lipschitz-continuous on H, i.e.

(4.6)

and

(4.7)

The map

(4.8)

is bounded in L ²([ t ₀, t ₀+ T], H) on every compact subset of H. Then, existence and uniqueness results have been obtained in [Brezis (1973)] and the following numerical scheme [Bastien and Schatzman (2000)] with constant time step h>0 can be used in order to approximate the exact solution X(t) of (4.1):

(4.9)

It is proved in [Bastien (2000)] or [Bastien and Schatzman (1999)] (and an idea of proof is given in [Bastien and Schatzman (2000)]) that this numerical scheme is of order 1/2, i.e., it verifies:

(4.10)

where is the function defined by linear interpolation of X _nh.

In many physical situations as we already explained, A derives...