Finite element discretization and Newton-multigrid solution techniques for the log-conformation reformulation (LCR) of viscoelastic flow problems ⋆

We present special numerical techniques for viscoelastic fluid flow utilizing a fully coupled monolithic multigrid finite element approach with consistent edge-oriented stabilization technique. The governing equations arise from the Navier-Stokes equations for the Oldroyd-B type of fluid with the help of the log-conformation reformulation (”Kupferman’s trick”) to allow a wide range of Weissenberg numbers. The resulting nonlinear system consists of 6 variables for velocity, pressure and the logarithm of the conformation stress tensor in 2D. The system is discretized in time by using a fully implicit second order accurate time integrator. In each time step, we have to solve a discretized system in space employing the high order finite element pair Q2/P1. We utilize the discrete damped Newton method with divided differences for handling the Jacobian, and apply a geometrical multigrid solver with a special Vanka smoother to handle the linear subproblems. Local refinement can be assigned at regions of interest to reduce the computational efforts. The presented methodology is implemented and validated for several well-known benchmark problems.