Convergence Analysis for Incompressible Generalized Newtonian Fluid Flows with Nonstandard Anisotropic Growth Conditions

Abstract

We study equations to describe incompressible generalized Newtonian fluids, where the extra stress tensor satisfies a nonstandard anisotropic asymptotic growth condition. An implicit finite element discretization, and a simple, fully practical fixed-point scheme with proper thresholding criterion are proposed, and convergence towards weak solutions of the limiting problem is shown. Computational experiments are included, which motivate nontrivial fluid flow behavior.

Finite-element-based discretizations of the incompressible Navier-Stokes equations with multiplicative random forcing

“…Since in our special case we obviously have by (SII 3 ) the convergence 8) we infer that the process u from (7.6) satisfies the following equation, for all ϕ ϕ ϕ ∈ Hence, in order to prove that the limiting process u is a strong solution of Problem 1.1, we have to prove that…”

Convergence analysis for a finite element approximation of a steady model for electrorheological fluids

“…We observe that in [14] a time-dependent system with the same stress-tensor (1.3) and (smoothed) convective terms is studied and the convergence of the finite element approximation is shown without convergence rate. To our knowledge no quantitative estimate on the convergence rate for problems with variable exponents is known, while recent results for the p(·)-Laplacian (i.e.…”

Fully discrete finite element approximation of unsteady flows of implicitly constituted incompressible fluids

“…In [30] an a posteriori analysis is performed for implicitly constituted fluid flow models using discretely divergence-free finite element functions. In the unsteady case no convergence result is available for implicitly constituted fluid flow models, and even those contributions that are focussed on explicit constitutive laws, such as [12], for example, are restricted to the case when q > 2(d+1) d+2 .…”

“…The approximation of the stress is then explicit and continuous in D D Du. l ∈ N: A time stepping based on the implicit Euler method is introduced similarly as, e.g., in [12,39], see Subsection 3.3. n ∈ N: The velocity u is approximated by a Galerkin approximation in finite element spaces in the spatial variable, see Section 3. m ∈ N: The regularizing term 1 m |u| 2q ′ −2 u is added to the equation to gain admissibility of the approximate solutions if q ≤ 3d+2 d+2 and to enable us to use the bound on b(·, ·, ·) in (3.16), without imposing the restriction q > 3d+2 d+2 . This results in a fully discrete approximation.…”