Optimal error estimates for semidiscrete Galerkin approximations to equations of motion described by Kelvin–Voigt viscoelastic fluid flow model

Abstract

In this paper, the finite element Galerkin method is applied to the equations of motion arising in the Kelvin-Voigt viscoelastic fluid flow model, when the forcing function is in L ∞ (L 2 ). Some a priori estimates for the exact solution, which are valid uniformly in time as t → ∞ and even uniformly in the retardation time κ an κ → 0, are derived. It is shown that the semidiscrete method admits a global attractor. Further, with the help of a priori bounds and Sobolev-Stokes projection, optimal error estimates for the velocity in L ∞ (L 2 ) and L ∞ (H 1 )-norms and for the pressure in L ∞ (L 2 )-norm are established. Since the constants involved in error estimates have an exponential growth in time, therefore, in the last part of the article, under certain uniqueness condition, the error bounds are established which are valid uniformly in time. Finally, some numerical experiments are conducted which confirm our theoretical findings.

A priori error estimates of fully discrete finite element Galerkin method for Kelvin–Voigt viscoelastic fluid flow model

“…Now, Lemma 4.5 provides the estimate for the time derivative ζ t . Here again, the estimate differs from the estimate of ζ t in Lemma 5.3 of [26] in terms of involvement of weight function σ and additional power of κ. As stated earlier, the presence of σ(t) and additional power of κ in the estimate play a crucial role in making error estimates independent of κ.…”

“…Here, it can be noted that they have achieved an improvement in the error estimates in powers of κ as the constants in error bounds depend only on κ −1/2 . As an extension to the work in [26], Pany et al [27], [28] have employed a linearized first order backward Euler method and a second order backward difference scheme for the time discretization of the problem (1.1)- (1.3) with f ∈ L ∞ (L 2 ) and have derived a priori bounds for the discrete solution in the Dirichlet norm using a combination of discrete Gronwall's lemma and Stolz-Cesaro's classical result for sequences. Then, making use of these a priori estimates for the solution, they have established fully discrete optimal error estimates for the velocity and pressure, which hold true uniformly in time under uniqueness assumption.…”

“…We present below in Lemma 2.1, some a priori bounds for the weak solution pair (u, p) which will be used in our subsequent error analysis. Since the estimates in (2.6) and (2.7) are already derived in [26], we only provide proof of (2.8).…”

“…Then, making use of these a priori estimates for the solution, they have established fully discrete optimal error estimates for the velocity and pressure, which hold true uniformly in time under uniqueness assumption. In [28], the author has also mentioned that assuming the solution is smooth enough, that is, u 0 ∈ H 3 ∩ H 1 0 with ∆u 0 = 0 on ∂Ω, the optimal error estimates independent of κ can be achieved following the similar analysis as in [26]- [28]. For the articles related to the finite element analysis of the problem (1.1)-(1.3) with the right-hand side forcing function f = 0, one may refer to [2]- [5].…”

One‐level and multilevel space‐time finite element method for the viscoelastic Kelvin‐Voigt model

“…Theorem (see previous studies) Under the assumptions of (A1) and the domain is a convex polygon, then problem has a uniqueness solution. Furthermore, it holds …”