Decoupled Scheme for Non-Stationary Viscoelastic Fluid Flow (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, we present a decoupled finite element scheme for two-dimensionaltime-dependent viscoelastic fluid flow obeying an Oldroyd-B constitutiveequation. The key idea of our decoupled scheme is to divide the full problem intotwo subproblems, one is the constitutive equation which is stabilized by using discontinuousGalerkin (DG) approximation, and the other is the Stokes problem, can becomputed parallel. The decoupled scheme can reduce the computational cost of thenumerical simulation and implementation is easy. We compute the velocity $u$ and thepressure $p$ from the Stokes like problem, another unknown stress $σ$ from the constitutiveequation. The approximation of stress, velocity and pressure are respectively, $P_1$-discontinuous, $P_2$-continuous, and $P_1$-continuous finite elements. The well-posednessof the finite element scheme is presented and derive the stability analysis of the decoupledalgorithm. We obtain the desired error bound also demonstrate the order of theconvergence, stability and the flow behavior with the support of two numerical experimentswhich reveals that decoupled scheme is more efficient than coupled scheme.