Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers (Q2842472)

This work studies a model within the kinetic theory of dilute solutions of polymeric liquids. In particular, it studies dimers which do not interact among themselves. The model considered falls in the class of bead-spring chain type models, and the spring potential is assumed to be anharmonic. The reason is that the simpler harmonic potential allows arbitrarily large extensions of the spring. The nonlinear elastic force chosen in this work prohibits this type of behavior. The dynamics of the solvent is modeled by the incompressible Navier-Stokes equations provided with an extra-stress tensor. This extra-stress tensor couples the Navier-Stokes system with a Fokker-Planck equation that describes the random motion of the polymer chains. The main objective of this work is the rigorous numerical analysis of this model; in particular, a fully discrete finite element approximation to this system of partial differential equations is constructed. Moreover, the sequence of finite element approximations is shown to converge (up to passing to a subsequence), when the spatial discretization parameter in the first place and the temporal discretization parameter afterwards tend to zero, to a weak solution of the system of partial differential equations. The result is proven with minimal assumptions on the regularity of the data.