Least-Squares Finite Element Method for the Steady Upper-Convected Maxwell Fluid

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 L2 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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Zhou, S. and Hou, L. (2015) Least-Squares Finite Element Method for the Steady Upper-Convected Maxwell Fluid. Advances in Pure Mathematics, 5, 233-239. doi: 10.4236/apm.2015.55024.

  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  Braess, D. (2007) Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511618635 