A modified nonlinear spectral Galerkin method for the equations of motion arising in the Kelvin–Voigt fluids

Abstract

In this paper, a variant of nonlinear Galerkin method is proposed and analysed for equations of motions arising in a Kelvin–Voigt model of viscoelastic fluids in a bounded spatial domain in Some new a priori bounds are obtained for the exact solution when the forcing function is independent of time or belongs to in time. As a consequence, existence of a global attractor is shown. For the spectral Galerkin scheme, existence of a unique discrete solution to the semidiscrete scheme is proved and again existence of a discrete global attractor is established. Further, optimal error estimate in and -norms are proved. Finally, a modified nonlinear Galerkin method is developed and optimal error bounds are derived. It is, further, shown that error estimates for both schemes are valid uniformly in time under uniqueness condition.

Keywords:

• viscoelastic fluids
• Kelvin–Voigt model
• nonlinear Galerkin method
• spectral approximations
• a priori error bound
• uniform convergence in time

AMS Subject Classifications:

• 65M15
• 37L65
• 74S25
• 76A10
• 76M22

Acknowledgements

The authors acknowledge the financial support provided by the DST-CNPq Indo-Brazil Project No. DST/INT/Brazil/RPO-05/2007 (Grant No. 490795/2007-2). Further, they thank the referees for their valuable comments.