  In this article, we introduce a new method for discretizing micro-macro
models of dilute polymeric fluids by integrating a finite element
discretization for the macroscopic fluid dynamic equation with a deterministic
variational particle scheme for the microscopic Fokker-Planck equation. To
address challenges arising from micro-macro coupling, we employ a discrete
energetic variational approach to derive a coarse-grained micro-macro model
with a particle approximation first and then develop a particle-FEM
discretization for the coarse-grained model. The accuracy of the proposed
method is evaluated for a Hookean dumbbell model in a Couette flow by comparing
the computed velocity field with existing analytical solutions. We also use our
method to study nonlinear FENE dumbbell models in different scenarios, such as
extension flow, pure shear flow, and lid-driven cavity flow. Numerical examples
demonstrate that the proposed deterministic particle approach can accurately
capture the various key rheological phenomena in the original FENE model,
including hysteresis and {\delta}-function-like spike behavior in extension
flows, velocity overshoot phenomenon in pure shear flow, symmetries breaking,
vortex center shifting and vortices weakening in the lid-driven cavity flow,
with a small number of particles.
