Numerical approximation of the Oldroyd‐B model by the weighted least‐squares/discontinuous Galerkin method

Abstract

We consider a numerical method for the Oldroyd‐B model of viscoelastic fluid flows by a combination of the weighted least‐squares (WLS) method and the discontinuous Galerkin (DG) finite element method. The constitutive equation is decoupled from the momentum and continuity equations, and the approximate solution is computed iteratively by solving the Stokes problem and a linearized constitutive equation using WLS and DG, respectively. An a priori error estimate for the WLS/DG method is derived and numerical results supporting the estimate are presented. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Decoupled algorithm for solving Phan-Thien–Tanner viscoelastic fluid by finite element method

“…for all r ∈ R. Thus, by applying the above priori estimate to (3.11), we have that 4 , we obtain the following error estimate using Theorem 4 in [7] ∥ǔ…”

“…Suppose that the PTT model (2.4) admits a smooth and small enough solution (u, p, 4 , we then have a priori error estimate for the WLS/SUPG method. Let…”

“…Therefore, it is natural to introduce a decoupled algorithm to divide the system into two simple problems [2][3][4][5][6]. Najib and Sandri in [2] presented a decoupled algorithm of a fixed point for the Oldroyd-B model.…”

“…In [7], Bochev and Gunzburger gave a weighted least-squares finite element method for an equivalent first order Stokes system. The least-squares finite element method has also been applied to the Oldroyd-B, Carreau, Giesekus and upperconvected Maxwell models for numerical simulations, see e.g., [4,[11][12][13]. The least-squares finite element method is based on the minimization of a quadratic functional including the residuals of each equation multiplied by a proper weight.…”

“…The convergence of the decoupled iterative method was shown and the error bounds were given. In [4], the momentum equation was decoupled from the constitutive equation and each subproblem was solved by different algorithms.…”