Numer Algor DOI 10.1007/s11075-016-0214-8 ORIGINAL PAPER Asymptotic behavior and finite element error estimates of Kelvin-Voigt viscoelastic fluid flow model Sudeep Kundu 1 · Saumya Bajpai 2 · Amiya K. Pani 1 Received: 14 June 2016 / Accepted: 20 September 2016 © Springer Science+Business Media New York 2016 Abstract In this article, the convergence of the solution of the Kelvin-Voigt vis- coelastic fluid flow model to its steady state solution with exponential rate is established under the uniqueness assumption. Then, a semidiscrete Galerkin method for spatial direction keeping time variable continuous is considered and asymptotic behavior of the semidiscrete solution is derived. Moreover, optimal error estimates are achieved for large time using steady state error estimates. Based on linearized backward Euler method, asymptotic behavior for the fully discrete solution is studied and optimal error estimates are derived for large time. All the results are even valid for κ → 0, that is, when the Kelvin-Voigt model converges to the Navier-Stokes sys- tem. Finally, some numerical experiments are conducted to confirm our theoretical findings. Keywords Viscoelastic fluid · Kelvin-Voigt model · Exponential decay · Long time behavior · Asymptotic analysis · Finite element method · Error analysis · Optimal order convergence · Numerical experiments Mathematics Subject Classifications (2010) 35B40 · 65M60 · 76D05 · 93D20  Amiya K. Pani akp@math.iitb.ac.in Sudeep Kundu sudeep.kundu85@gmail.com Saumya Bajpai mymsciitm@gmail.com 1 Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India 2 Institute of Infrastructure Technology Research and Management, Ahmedabad-380026, India