On energy-dissipative finite element approximations for rate-type viscoelastic fluids with stress diffusion

TL;DR

This paper develops a fully discrete finite element method for simulating unsteady viscoelastic fluid flows with stress diffusion, ensuring energy stability, positivity, and convergence to weak solutions.

Contribution

It introduces an energy-preserving discretization for a complex viscoelastic model, guaranteeing stability, positivity, and convergence in finite element approximations.

Findings

Unconditional solvability and stability of the discretized system.

Discrete solutions converge to a global-in-time weak solution.

Numerical tests confirm convergence and stability.

Abstract

We study a fully discrete finite element approximation of a model for unsteady flows of rate-type viscoelastic fluids with stress diffusion in two and three dimensions. The model consists of the incompressible Navier--Stokes equation for the velocity, coupled with a diffusive variant of a combination of the Oldroyd-B and the Giesekus model for the left Cauchy--Green tensor. The discretization of the model is chosen such that an energy inequality is preserved at the fully discrete level. Thus, unconditional solvability and stability for the discrete system are guaranteed and the discrete Cauchy--Green tensor is positive definite. Moreover, subsequences of discrete solutions converge to a global-in-time weak solution, as the discretization parameters tend to zero. In the end, we present numerical convergence tests.