This talk consists two parts: For the variational inequalities model obstacle problems, we propose and analyze some numerical algorithms based on an equivalent formulation. The algorithms for the related discrete Hamilton-Jacobi-Bellman equation are also discussed. The second part concerns a Quasistatic history-dependent contact model. It is a hemi-variational inequalities model. We analyze the finite element methods to approximate it, derive some error estimates and analyze some algorithms. Several models for different contact conditions in hemi-variational inequalities are mentioned.
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