A finite element method for a non-Newtonian dilute polymer fluid

We study the discretisation of a uniaxial (rank-one) reduction of the Oldroyd-B model for dilute polymer solutions, in which the conformation tensor is represented as $\sig = \vec b \otimes \vec b$. Building on structural analogies with MHD, we formulate a finite element framework compatible with the de Rham complex, so that the discrete velocity is exactly divergence-free. The spatial discretisation combines an interior-penalty treatment of viscosity with upwind transport to control stress layers and we prove inf-sup conditions on the mixed pairs. For time-stepping, we design an IMEX scheme that is linear at each step and show well-posedness of the fully discrete problem together with a discrete energy law mirroring the continuum dissipation. Numerical experiments on canonical benchmarks (lid-driven cavity, pipe-with-cavity and $4{:}1$ planar contraction) demonstrate accuracy and robustness for moderate-to-high Weissenberg numbers, capturing sharp stress gradients and corner singularities while retaining the efficiency gains of the uniaxial model. The results indicate that de Rham-compatible discretisations coupled with energy-stable IMEX time integration provide a reliable pathway for viscoelastic computations at elevated elasticity.