A large time stepping viscosity-splitting finite element method for the viscoelastic flow problem
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- 作者：Tong Zhang ; Damazio Pedro ; JinYun Yuan
- 关键词：Viscoelastic flow problem ; Viscosity ; splitting method ; Large time stepping ; Stability ; Convergence ; Optimal error estimates ; 65N15 ; 65N30 ; 76D07
- 刊名：Advances in Computational Mathematics
- 出版年：2015
- 出版时间：February 2015
- 年：2015
- 卷：41
- 期：1
- 页码：149-190
- 全文大小：829 KB
- 参考文献：1. Blasco, J., Codina, R.: Error estimates for an operator-splitting method for incompressible flows. Appl. Numer. Math. 51, 1-7 (2004) CrossRef
2. Blasco, J., Codina, R., Huerta, A.: A fractional-step method for the incompressible Navier-Stokes equations related to a predictor-multicorrector algorithm. Int. J. Numer. Methods Fluids 28, 1391-419 (1998) CrossRef
3. Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math. Comput. 22, 745-62 (1968) CrossRef
4. Chorin, A.J.: On the convergence of discrete approximations to the Navier-Stokes equations. Math. Comput. 23, 341-53 (1969) CrossRef
5. Codina, R.: Pressure stability in fractional step finite element methods for incompressible flows. J. Comput. Phys. 170, 112-40 (2001) CrossRef
6. Codina, R., Badia, S.: On some pressure segregation methods of fractional-step type for the finite element approximation of incompressible flow problems. Comput. Methods Appl. Mech. Eng. 195, 2900-918 (2006) CrossRef
7. Dukowicz, J.K., Dvinsky, A.S.: Approximate factorization as a high order splitting for the implicit incompressible flow equations. J. Comput. Phys. 102, 336-47 (1992) CrossRef
8. Feng, X.L., He, Y.N., Liu, D.M.: Convergence analysis of an implicit fractional-step method for the incompressible Navier-Stokes equations. Appl. Math. Model. 35, 5856-871 (2011) CrossRef
9. Fernandez-Cara, E., Marin Beltran, M.: The convergence of two numerical schemes for the Navier-Stokes equations. Numer. Math. 55, 33-0 (1989) CrossRef
10. Gervasio, P., Saleri, F.: Algebraic fractional-step schemes for time-dependent incompressible Navier-Stokes equations. J. Sci. Comput. 27, 257-69 (2006) CrossRef
11. Gervasio, P., Saleri, F., Veneziani, A.: Algebraic fractional-step schemes with spectral methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 214, 347-65 (2006) CrossRef
12. Girault, V., Raviart, P.A.: Finite Element Method for Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, Berlin (1987)
13. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems, 2nd edn.Springer-Verlag, New York (1984) CrossRef
14. Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engng. 195, 6011-045 (2006) CrossRef
15. Guermond, J.L., Quartapelle, L.: On the approximation of the unsteady Navier-Stokes equations by finite element projection methods. Numer. Math. 80, 207-38 (1998) CrossRef
16. Guillen-Gonzalez, F., Redondo-Neble, M.V.: New error estimates for a viscosity splitting scheme in time for the three-dimensional Navier-Stokes equations. IMA J. Numer. Anal. - 刊物类别：Computer Science
- 刊物主题：Numeric Computing
Calculus of Variations and Optimal Control
Mathematics
Algebra
Theory of Computation - 出版者：Springer U.S.
- ISSN：1572-9044
文摘
In this article, a large time stepping viscosity-splitting scheme is considered for the viscoelastic flow problem. The temporal term is decomposed into a sequence of two steps (using decomposition of the viscosity). For the first step, a linear elliptic problem is solved with explicit scheme for the convection term (a linear system with a constant coefficient matrix is obtained and the computation becomes easy). At the second step, the problem has the structure of the Stokes problem. Both two problems satisfy the homogeneous Derichlet boundary conditions for the velocities. The main novelties of this work are the stability of numerical solutions under the condition k 1 Δt ?1 with a positive constant k 1, and optimal error estimates for both velocity in L ?/sup>(H 1) and L 2(H 1) norms and pressure in L ?/sup>(L 2) and L 2(L 2) norms. In order to enlarge the time step, we introduce a diffusion term θ Δu in all steps of the schemes. Finally, some numerical results are provided to display the performance of our algorithm.