Numerical analysis of time filter method for the stabilized incompressible diffusive Peterlin viscoelastic fluid model

Abstract

The Diffusion Peterlin Viscoelastic Fluid (DPVF) model describes the movement of specific incompressible polymeric fluids. In this paper, we introduce and evaluate a new low-complexity linear-time filter finite element (FE) method for the DPVF model. In order to avoid the value at time $t=-\Delta t$, the proposed time filter method consists of three steps, including a post-processing step. Firstly, a first-order Euler backward nonlinear fully discrete mixed FE scheme is employed to compute the numerical solutions at time $t_1=\Delta t$. For $n\ge 1$, we obtain the intermediate values $({\tilde{\text{u}}}_h^{n+1},{\tilde{p}}_h^{n+1},{\tilde{\text{d}}}_h^{n+1})$ in Step II using a fully implicit backward Euler scheme. At the same time level, we proceed with these intermediate values $({\tilde{\text{u}}}_h^{n+1},{\tilde{p}}_h^{n+1},{\tilde{\text{d}}}_h^{n+1})$ using the linear time filters. The linear time filters step does not significantly increase computational complexity. However, it can enhance temporal convergence accuracy from first order to second order for backward Euler time filter (BE time filter), and from second order to three order for BDF2 time filter. We demonstrate the almost unconditional stability of the scheme. Error estimates for the time filter method are derived and presented. Several numerical experiments are conducted to validate the theoretical findings and showcase the efficiency of the proposed method.