Research ArticleOpen AccessGoogle Scholar indexed
Least-Squares Finite Element Method for the Steady Upper-Convected Maxwell Fluid
School of Science, Hebei University of Engineering, Handan, China
Department of Mathematics, Shanghai University, Shanghai, China
- 1 School of Science, Hebei University of Engineering, Handan, China
- 2 Department of Mathematics, Shanghai University, Shanghai, China
Advances in Pure Mathematics·Volume 05 (2015)·Pages 233–239·Published 3 April 2015·DOI10.4236/apm.2015.55024
Copy link · social · email
Abstract
In this paper, a least-squares finite element method for the upper-convected Maxell (UCM) fluid is proposed. We first linearize the constitutive and momentum equations and then apply a least-squares method to the linearized version of the viscoelastic UCM model. The L 2 least-squares functional involves the residuals of each equation multiplied by proper weights. The corresponding homogeneous functional is equivalent to a natural norm. The error estimates of the finite element solution are analyzed when the conforming piecewise polynomial elements are used for the unknowns.
KeywordsUpper-Convected Maxwell FluidLeast-Squares Finite Element MethodViscoelastic Fluid Model
- Chen, T.F., Lee, H. and Liu, C.C. (2013) Numerical Approximation of the Oldroyd-B Model by the Weighted Least- Squares/Discontinuous Galerkin Method. Numerical Methods for Partial Differential Equations, 29, 531-548. http://dx.doi.org/10.1002/num.21719
- Cai, Z. and Ku, J. (2006) The L2 Norm Error Estimates for the Div Least-Squares method. SIAM Journal on Numerical Analysis, 44, 1721-1734. http://dx.doi.org/10.1137/050636504
- Zhou, S.L. and Hou, L. (2015) Decoupled Algorithm for Solving Phan-Thien-Tanner Viscoelastic Fluid by Finite Element Method. Computer & Mathematics with Applications, 69, 423-437. http://dx.doi.org/10.1016/j.camwa.2015.01.006
- Cai, Z., Lazarov, R. and Manteuffel, T.A. and McCormick, S.F. (1994) First-Order System Least Squares for Second- Order Partial Differential Equations: Part I. SIAM Journal on Numerical Analysis, 31, 1785-1799. http://dx.doi.org/10.1137/0731091
- Cai, Z., Lee, B. and Wang, P. (2004) Least-Squares Methods for Incompressible Newtonian Fluid Flow: Linear Stationary Problems. SIAM Journal on Numerical Analysis, 42, 843-859. http://dx.doi.org/10.1137/S0036142903422673
- Lee, H.C. and Chen, T.F. (2015) Adaptive Least-Squares Finite Element Approximations to Stokes Equations. Journal of Computational and Applied Mathematics, 280, 396-412. http://dx.doi.org/10.1016/j.cam.2014.11.041
- Bochev, P.B. and Gunzburger, M.D. (1995) Least-Squares Methods for the Velocity-Pressure-Stress Formulation of the Stokes Equations. Computer Methods in Applied Mechanics and Engineering, 126, 267-287. http://dx.doi.org/10.1016/0045-7825(95)00826-M
- Lee, H.C. (2014) An Adaptively Refined Least-Squares Finite Element Method for Generalized Newtonian Fluid Flows Using the Carreau Model. SIAM Journal on Scientific Computing, 36, A193-A218. http://dx.doi.org/10.1137/130912682
- Fan, Y., Tanner, R.I. and Phan-Thien, N. (1999) Galerkin/Least-Square Finite-Element Methods for Steady Viscoelastic Flows. Journal of Non-Newtonian Fluid Mechanics, 84, 233-256. http://dx.doi.org/10.1016/S0377-0257(98)00154-2
- Cai, Z., Manteuffel, T.A. and McCormich, S.F. (1995) First-Order System Least Squares for Velocity-Vorticity-Pres- sure from of the Stokes Equations, with Application to Linear Elasticity. Electronic Transactions on Numerical Analysis, 3, 150-159.
- Cai, Z. and Westphal, C.R. (2009) An Adaptive Mixed Least-Squares Finite Element Method for Viscoelastic Fluids of Oldroyd Type. Journal of Non-Newtonian Fluid Mechanics, 159, 72-80. http://dx.doi.org/10.1016/j.jnnfm.2009.02.004