Analysis of Stabilized Crank-Nicolson Time-Stepping Scheme for the Evolutionary Peterlin Viscoelastic Model

Abstract

The Peterlin viscoelastic model describes the motion of certain incompressible polymeric fluids. It employs a nonlinear dumbbell model with a nonlinear spring force law making it more nonlinear than other viscoelastic models. In this paper, we propose and study a fully implicit stabilized Crank-Nicolson time stepping scheme for finite element spatial discretization of the non-stationary Peterlin viscoelastic fluid model with non-homogeneous boundary conditions. The proposed scheme adds a suitable stabilizing term to improve the structural and stability properties of the scheme. We prove that the scheme is almost unconditionally stable, i.e., stable when the time step is less than or equal to a constant. Further, with the help of the a priori error bounds of the Stokes and Ritz projections, optimal error estimates for the velocity, the conformation tensor and the pressure are presented in suitable norms. Numerical examples are presented that illustrate the accuracy and stability of the scheme.

Keywords:

• Error estimates
• finite element method
• non-homogeneous
• Peterlin viscoelastic model
• stability
• stabilized Crank-Nicolson scheme

AMS SUBJECT CLASSIFICATIONS:

• 65N30
• 76M10
• 65M12
• 65N15
• 76A10