Mimetic finite differences for nonlinear and control problems

In this paper we review some recent applications of the mimetic finite difference method to nonlinear problems (variational inequalities and quasilinear elliptic equations) and optimal control problems governed by linear elliptic partial differential equations. Several numerical examples show the effectiveness of mimetic finite differences in building accurate numerical approximations. Finally, driven by a real-world industrial application (the numerical simulation of the extrusion process) we explore possible further applications of the mimetic finite difference method to nonlinear Stokes equations and shape optimization/free-boundary problems.