Back
 APM  Vol.5 No.5 , April 2015
Least-Squares Finite Element Method for the Steady Upper-Convected Maxwell Fluid
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 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.
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.
References

[1]   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

[2]   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

[3]   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

[4]   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

[5]   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

[6]   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

[7]   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

[8]   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

[9]   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

[10]   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.

[11]   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

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

 
 
Top