On a three step two‐grid finite element method for the Oldroyd model of order one

Abstract

In this work, an optimal error analysis of a three step two-grid method for the equations of motion arising in the 2𝐷 Oldroyd model of order one is discussed. The model, which can be thought of as an integral perturbation of Navier-Stokes equations (NSE), represents linear viscoelastic fluid flows. This non-linear model is analyzed here using a three step numerical scheme. In the first step the problem is solved on a coarse grid, and we use this course grid solution to linearize the problem and solve it in second step, on a finer grid. Third step is a correcting step done on the finer grid. Optimal error estimates for the velocity in 𝐿 ∞ (𝐋 2 ) and 𝐿 ∞ (𝐇 1 )-norms and for the pressure in 𝐿 ∞ (𝐿 2 )-norm in the semidiscrete case are established. Estimates are shown to be uniform under the uniqueness assumption. Then, based on backward Euler method, a completely discrete scheme is analyzed and optimal a priori error estimates are derived. All the analysis are carried out for the non-smooth initial data. Finally we present some numerical results to validate our theoretical results. These examples show that the three step two-grid method is efficient than solving a nonlinear problem directly, as is expected. K E Y W O R D Sbackward Euler method, Oldroyd model of order one, optimal and uniform error estimates, two-grid method, uniform estimates 1

Generalized Navier–Stokes Equations with Non-Homogeneous Boundary Conditions

“…From the well-known results on the unique solvability of the steady-state Stokes equations with non-homogeneous Dirichlet boundary conditions (see, e.g., [3], Chap. I) it follows that there exists a unique vector function v ∈ H 2 σ (Ω) satisfying system (13). Moreover, the following estimate holds…”

“…Let S : Ḣ3/2 (Γ) → H 2 σ (Ω) be the data-to-solution mapping for the Stokes problem (13), that is, S ζ def = v. Taking into account estimate (14), we obtain…”

“…The simplified Jeffreys model (Jeffreys-Oldroyd type viscoelastic fluids) when δ = 1, µ 0 > 0, µ 1 = 0, η(s, t) ≡ ae b(s−t) (see [10][11][12][13]); • The Oskolkov integro-differential system (generalized Kelvin-Voigt viscoelastic fluids) when δ = 1, µ 0 > 0, µ 1 > 0, η ≡ 0 (see [14,15]).…”

A discontinuous Galerkin finite element method for the Oldroyd model of order one

“…For time discrete schemes, we refer to other works [5][6][7] and the references therein. In addition to these, there are works based on stability analysis of time discrete schemes [8,9], projection methods [10,11], stabilized methods [12], characteristics finite element method [13], penalty method [14], and two-grid method [15] that have been carried out for this model.…”