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Two-Level Finite Element Approximation for Oseen Viscoelastic Fluid Flow

Author

Listed:
  • Nasrin Jahan Nasu

    (School of Mathematical Sciences, East China Normal University, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Shanghai 200241, China)

  • Md. Abdullah Al Mahbub

    (School of Mathematical Sciences, East China Normal University, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Shanghai 200241, China
    Department of Mathematics, Comilla University, Comilla 3506, Bangladesh)

  • Shahid Hussain

    (School of Mathematical Sciences, East China Normal University, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Shanghai 200241, China)

  • Haibiao Zheng

    (School of Mathematical Sciences, East China Normal University, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Shanghai 200241, China)

Abstract

In this paper, a two-level finite element method for Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law is presented. With the newly proposed algorithm, solving a large system of the constitutive equations will not be much more complex than the solution of one linearized equation. The viscoelastic fluid flow constitutive equation consists of nonlinear terms, which are linearized by taking a known velocity b ( x ) , and transforms into the Oseen viscoelastic fluid flow model. Since Oseen viscoelastic fluid flow is already linear, we use a two-level method with a new technique. The two-level approach is consistent and efficient to study the coupled system which contains nonlinear terms. In the first step, the solution on the coarse grid is derived, and the result is used to determine the solution on the fine mesh in the second step. The decoupling algorithm takes two steps to solve a linear system on the fine mesh. The stability of the algorithm is derived for the temporal discretization and obtains the desired error bound. Two numerical experiments are executed to show the accuracy of the theoretical analysis. The approximations of the stress tensor, velocity vector, and pressure field are P 1 -discontinuous, P 2 -continuous and P 1 -continuous finite elements respectively.

Suggested Citation

  • Nasrin Jahan Nasu & Md. Abdullah Al Mahbub & Shahid Hussain & Haibiao Zheng, 2018. "Two-Level Finite Element Approximation for Oseen Viscoelastic Fluid Flow," Mathematics, MDPI, vol. 6(5), pages 1-20, May.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:5:p:71-:d:144385
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    References listed on IDEAS

    as
    1. Aiwen Wang & Xin Zhao & Peihua Qin & Dongxiu Xie, 2012. "An Oseen Two-Level Stabilized Mixed Finite-Element Method for the 2D/3D Stationary Navier-Stokes Equations," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-12, April.
    2. Yunzhang Zhang & Yanren Hou & Ganshan Yang, 2011. "A Defect-Correction Method for Time-Dependent Viscoelastic Fluid Flow Based on SUPG Formulation," Discrete Dynamics in Nature and Society, Hindawi, vol. 2011, pages 1-25, July.
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