Journal of the
Korean Mathematical Society JKMS

In this paper, a fully discrete numerical scheme for the viscoelastic Oldroyd flow is considered with an introduced auxiliary variable. Our scheme is based on the finite element approximation for the spatial discretization and the backward Euler scheme for the time discretization. The integral term is discretized by the right trapezoidal rule. Firstly, we present the corresponding equivalent form of the considered model, and show the relationship between the origin problem and its equivalent system in finite element discretization. Secondly, unconditional stability and optimal error estimates of fully discrete numerical solutions in various norms are established. Finally, some numerical results are provided to confirm the established theoretical analysis and show the performances of the considered numerical scheme.

Keywords: The viscoelastic Oldroyd flow, unconditional stability, optimal error estimates

MSC numbers: Primary 65M10, 65N30, 76Q10

Supported by: This work was financially supported by NSF of China (No.11971152) and NSF of Henan Province (202300410167).