Stability and convergence analysis of stabilized finite element method for the Kelvin-Voigt viscoelastic fluid flow mode
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Stability and convergence analysis of stabilized finite element method for the Kelvin-Voigt viscoelastic fluid flow model Tong Zhang1 · Mengmeng Duan2 Received: 20 February 2019 / Accepted: 17 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we consider the Galerkin finite element method (FEM) for the KelvinVoigt viscoelastic fluid flow model with the lowest equal-order pairs. In order to overcome the restriction of the so-called inf-sup conditions, a pressure projection method based on the differences of two local Gauss integrations is introduced. Under some suitable assumptions on the initial data and forcing function, we firstly present some stability and convergence results of numerical solutions in spatial discrete scheme. By constructing the dual linearized Kelvin-Voigt model, stability and optimal error estimates of numerical solutions in various norms are established. Secondly, a fully discrete stabilized FEM is introduced, the backward Euler scheme is adopted to treat the time derivative terms, the implicit scheme is used to deal with the linear terms and semi-implicit scheme is applied to treat the nonlinear term, unconditional stability and convergence results are also presented. Finally, some numerical examples are presented to verify the developed theoretical analysis and show the performances of the considered numerical schemes. Keywords Stabilized method · Kelvin-Voigt viscoelastic fluid flow model · The lowest equal-order mixed elements · The L’Hospital rule · Negative norm technique Mathematics Subject Classification (2010) 65N15 · 65N30 · 76D05
Tong Zhang
[email protected] 1
School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, China
2
School of Mathematics & Information Science, Henan Polytechnic University, Jiaozuo, 454003, China
Numerical Algorithms
1 Introduction As an important component of the non-Newton fluids, the viscoelastic fluid flow model has been widely used in food products, molten plastic, biologic fluid, etc. In recent years, some important viscoelastic models have been researched not only from the viewpoint of theoretical analysis but also from the numerical simulations point. For example, we can refer to [1–3] and the reference therein. In this paper, we consider the following Kelvin-Voigt viscoelastic flow model ⎧ ∂u ⎪ ⎪ (x, t) ∈ × R + , ⎪ ∂t − κut − νu + u · ∇u + ∇p = f, ⎪ ⎪ ⎨ div u = 0, (x, t) ∈ × R + , (1) ⎪ + ⎪ ⎪ u = 0, (x, t) ∈ ∂ × R , ⎪ ⎪ ⎩ x ∈ , u(x, 0) = u0 , where ∈ R d (d = 2, 3) be a bounded convex domain or the boundary ∂ ∈ C 2 , u and p are the fluid velocity and pressure, the positive parameters ν and κ are the kinematic coefficient viscosity and the retardation time or the time of relaxation of deformations, respectively, f is the prescribed body force, and u0 is the initial velocity. The Kelvin-Voigt model was first introduced by Pavlovskii [4], which can be used to describe the motion of weakly concentrated water-polymer solution. It was named the Kelvin-
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