Higher Order Galerkin Time Discretization for Nonstationary Incompressible Flow

Abstract

We present special numerical techniques for viscoelastic fluid flow uti- lizing a fully coupled monolithic multigrid finite element approach with consis- tent edge-oriented stabilization technique. The governing equations arise from the Navier–Stokes for the Oldroyd-B type of fluidwith the help of the log-conformation reformulation to allow a wide range of Weissenberg numbers. The resulting non- linear 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 triple Q2=Pdisc 1 =Q2. We utilize the discrete damped Newton method with divided dif- ferences 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 cost. The presented methodology is implemented on the open source software package FEATFLOW (www.featflow.de) and validated for several well-known benchmark problems.