Unconditional stability and convergence of fully discrete FEM for the viscoelastic Oldroyd flow with an introduced auxiliary variable
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this paper, a fully discrete numerical scheme for the viscoelastic Oldroyd flow is considered with an introduced auxiliary variable. Our scheme is based on the finite element approximation for the spatial discretization and the backward Euler scheme for the time discretization. The integral term is discretized by the right trapezoidal rule. Firstly, we present the corresponding equivalent form of the considered model, and show the relationship between the origin problem and its equivalent system in finite element discretization. Secondly, unconditional stability and optimal error estimates of fully discrete numerical solutions in various norms are established. Finally, some numerical results are provided to confirm the established theoretical analysis and show the performances of the considered numerical scheme.
Abstract
In this paper, a fully discrete numerical scheme for the viscoelastic Oldroyd flow is considered with an introduced auxiliary variable. Our scheme is based on the finite element approximation for the spatial discretization and the backward Euler scheme for the time discretization. The integral term is discretized by the right trapezoidal rule. Firstly, we present the corresponding equivalent form of the considered model, and show the relationship between the origin problem and its equivalent system in finite element discretization. Secondly, unconditional stability and optimal error estimates of fully discrete numerical solutions in various norms are established. Finally, some numerical results are provided to confirm the established theoretical analysis and show the performances of the considered numerical scheme.
참고문헌 (46)
- M. M. Akhmatov, "Convergence difference schemes for equations of motion of an Oldroyd fluid", J. Soviet Math, 47(6), pp.2926~2933 (1989)
- J. R. Cannon, "A modified nonlinear Galerkin method for the viscoelastic fluid motion equations", Internat. J. Engrg. Sci, 37(13), pp.1643~1662 (1999)[DOI]
- E. Erturk, "Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers", Int. J. Numer Meth Fl, 48, pp.747~774 (2005)
- L. Ge, "Convergence analysis and error estimate for distributed optimal control problems governed by Stokes equations with velocity-constraint", Adv. Appl. Math. Mech, 14(1), pp.33~55 (2022)[DOI]
- U. Ghia, "High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method", J. Comput. Phy, 48, pp.387~411 (1982)
- F. X. Giraldo, "Semi-implicit formulations of the Navier-Stokes equations : application to nonhydrostatic atmospheric modeling", SIAM J. Sci. Comput, 32(6), pp.3394~3425 (2010)[DOI]
- V. Girault, "Finite element methods for Navier-Stokes equations" (1986), Springer-Verlag
- J. Guan, "A rotational velocity-correction projection method for unsteady incompressible magnetohydrodynamics equations", Comput. Math. Appl, 80(5), pp.809~821 (2020)[DOI]
- J. L. Guermond, "An overview of projection methods for incompressible flows", Comput. Methods Appl. Mech. Engrg, 195(44-47), pp.6011~6045 (2006)[DOI]
- Y. Guo, "Fully discrete finite element method based on second-order Crank-Nicolson", Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one, 2015, pp.20(8), 2583~2609
- E. Hansen, "Convergence of the implicit-explicit Euler scheme applied to perturbed dissipative evolution equations", Math. Comp, 82(284), pp.1975~1985 (2013)[DOI]
- Y. He, "The Euler implicit", explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data, 2008, pp.77(264), 2097~2124
- Y. He, "H2-stability of the first order fully discrete schemes for the time-dependent Navier-Stokes equations", J. Sci. Comput, 62(1), pp.230~264 (2015)[DOI]
- Y. He, "A penalty finite element method based on the Euler implicit", explicit scheme for the time-dependent Navier-Stokes equations, 2010, pp.235(3), 708~725
- Y. He, "Finite element approximation for the viscoelastic fluid motion problem", J. Comput. Appl. Math, 155(2), pp.201~222 (2003)
- J. G. Heywood, "Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization", SIAM J. Numer. Anal, 19(2), pp.275~311 (1982)[DOI]
- J. Jin, "H2-stability of the first order Galerkin method for the Boussinesq equations with smooth and non-smooth initial data", Comput. Math. Appl, 75(1), pp.248~288 (2018)[DOI]
- X. Li, "Error analysis of the SAV-MAC scheme for the Navier-Stokes equations", SIAM J. Numer. Anal, 58(5), pp.2465~2491 (2020)[DOI]
- X. Li, "On a SAV-MAC scheme for the Cahn-Hilliard-Navier-Stokes phasefield model and its error analysis for the corresponding Cahn-Hilliard-Stokes case", Math. Models Methods Appl. Sci, 30(12), pp.2263~2297 (2020)[DOI]
- X. Li, "New SAV-pressure correction methods for the Navier-Stokes equations : stability and error analysis", Math. Comp, 91(333), pp.141~167 (2021)[DOI]
- X. Li, "Energy stability and convergence of SAV block-centered finite difference method for gradient flows", Math. Comp, 88(319), pp.2047~2068 (2019)[DOI]
- L. Lin, "Numerical approximation of incompressible Navier-Stokes equations based on an auxiliary energy variable", J. Comput. Phys, 388, pp.1~22 (2019)[DOI]
- Z. Liu, "The exponential scalar auxiliary variable(E-SAV)approach for phase field models and its explicit computing", SIAM J. Sci. Comput, 42(3), pp.B630~B655 (2020)[DOI]
- J. Marti, "An explicit-implicit finite element model for the numerical solution of incompressible Navier-Stokes equations on moving grids", Comput. Methods Appl. Mech. Engrg, 350, pp.750~765 (2019)[DOI]
- H. Niu, "Numerical analysis of an optimal control problem governed by the stationary Navier-Stokes equations with global velocity-constrained", Commun. Comput. Phys, 24(5), pp.1477~1502 (2018)[DOI]
- A. K. Pani, "Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model", IMA J. Numer. Anal, 25(4), pp.750~782 (2005)[DOI]
- A. K. Pani, "On a linearized backward Euler method for the equations of motion of Oldroyd fluids of order one", SIAM J. Numer. Anal, 44(2), pp.804~825 (2006)[DOI]
- J. Shen, "On error estimates of projection methods for Navier-Stokes equations : firstorder schemes", SIAM J. Numer. Anal, 29(1), pp.57~77 (1992)[DOI]
- J. Shen, "Convergence and error analysis for the scalar auxiliary variable(SAV)schemes to gradient flows", SIAM J. Numer. Anal, 56(5), pp.2895~2912 (2018)[DOI]
- J. Shen, "The scalar auxiliary variable(SAV)approach for gradient flows", J. Comput. Phys, 353, pp.407~416 (2018)[DOI]
- X. Shen, "A rotational pressure-correction projection methods for unsteady incompressible magnetohydrodynamics equations", Appl. Math. Comput, 387, pp.21~ (2020)[DOI]
- P. E. Sobolevskiı, "Stabilization of viscoelastic fluid motion(Oldroyd’s mathematical model)", Diff. Integral Equ, 7(5-6), pp.1597~1612 (1994)
- J. Su, "The almost unconditional convergence of the Euler implicit", explicit scheme for the three dimensional nonstationary Navier-Stokes equations, 2017, pp.22(9), 3421~3438
- R. Temam, "Theory and numerical analysis, Studies in Mathematics and its Applications Vol. 2" (1977), North-Holland Publishing Co
- Y. Wang, "A second order in time incremental pressure correction finite element method for the Navier-Stokes", Darcy problem, 2018, pp.52(4), 1477~1500
- K. Wang, "Stabilized finite element method for the viscoelastic Oldroyd fluid flows", Numer. Algorithms, 60(1), pp.75~100 (2012)[DOI]
- Z. Yang, "An unconditionally energy-stable scheme based on an implicit auxiliary energy variable for incompressible two-phase flows with different densities involving only precomputable coefficient matrices", J. Comput. Phys, 393, pp.229~257 (2019)[DOI]
- X. Yang, "second order and unconditionally energy stable schemes for the viscous Cahn-Hilliard equation with hyperbolic relaxation using the invariant energy quadratization method", J. Comput. Appl. Math, 343, pp.80~97 (2018)[DOI]
- X. Yang, "Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method", J. Comput. Phys, 333, pp.104~127 (2017)[DOI]
- T. Zhang, "A large time stepping viscosity-splitting finite element method for the viscoelastic flow problem", Adv. Comput. Math, 41(1), pp.149~190 (2015)[DOI]
- T. Zhang, "Stability analysis of several first order schemes for the Oldroyd model with smooth and nonsmooth initial data", Numer. Methods Partial Differential Equations, 34(6), pp.2180~2216 (2018)[DOI]
- T. Zhang, "A stabilized characteristic finite element method for the viscoelastic Oldroyd fluid motion problem", Int. J. Numer. Anal. Model, 12(4), pp.617~635 (2015)
- T. Zhang, "Unconditional stability and optimal error estimates of Euler implicit", explicit-SAV scheme for the Navier-Stokes equations, 2022, pp.90(1), 20~
- J. Zhao, "Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach", Internat. J. Numer. Methods Engrg, 110(3), pp.279~300 (2017)[DOI]
- J. Zhou, "The error estimates of spectral methods for 1-dimension singularly perturbed problem", Appl. Math. Lett, 100, pp.106001~ (2020)[DOI]
- J. Zhou, "A posteriori error estimates of spectral approximations for second order partial differential equations in spherical geometries", J. Sci. Comput, 90(1), pp.30~ (2022)[DOI]
