Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data

Abstract

Abstract In this paper, a penalty formulation is proposed and analyzed in both continuous and finite element setups, for the two-dimensional Oldroyd model of order one, when the initial velocity is in 𝐇 0 1 {\mathbf{H}_{0}^{1}} . New regularity results which are valid uniformly in time as t → ∞ {t\to\infty} and in the penalty parameter ε as ε → 0 {\varepsilon\to 0} are derived for the solution of the penalized problem. Then, based on conforming finite elements to discretize the spatial variables and keeping temporal variable continuous, a semidiscrete problem is discussed and a uniform-in-time a priori bound of the discrete velocity in Dirichlet norm is derived with the help of a penalized discrete Stokes operator and a modified uniform Gronwall’s lemma. Further, optimal error estimates for the penalized velocity in 𝐋 2 {\mathbf{L}^{2}} as well in 𝐇 1 {\mathbf{H}^{1}} -norms and for the penalized pressure in L 2 {L^{2}} -norm have been established for the semidiscrete problem with non-smooth data. These error estimates hold uniformly in time under uniqueness assumption and also in the penalty parameter as it goes to zero. Our analysis relies on the suitable use of the inverse of the penalized Stokes operator, penalized Stokes–Volterra projection and judicious application of weighted time estimates with positivity property of the memory term. Finally, several numerical experiments are conducted on benchmark problems which confirm our theoretical findings.