Two-Level Finite Element Approximation for Oseen Viscoelastic Fluid Flow

Abstract

In this paper, a two-level finite element method for Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law is presented. With the newly proposed algorithm, solving a large system of the constitutive equations will not be much more complex than the solution of one linearized equation. The viscoelastic fluid flow constitutive equation consists of nonlinear terms, which are linearized by taking a known velocity b(x), and transforms into the Oseen viscoelastic fluid flow model. Since Oseen viscoelastic fluid flow is already linear, we use a two-level method with a new technique. The two-level approach is consistent and efficient to study the coupled system which contains nonlinear terms. In the first step, the solution on the coarse grid is derived, and the result is used to determine the solution on the fine mesh in the second step. The decoupling algorithm takes two steps to solve a linear system on the fine mesh. The stability of the algorithm is derived for the temporal discretization and obtains the desired error bound. Two numerical experiments are executed to show the accuracy of the theoretical analysis. The approximations of the stress tensor, velocity vector, and pressure field are P1-discontinuous, P2-continuous and P1-continuous finite elements respectively.

Finite Element Approximation of Optimal Control Problem with Weighted Extended B-Splines

“…Equating (10) and (11) to zero, leads to Equations (6) and (7). Assuming Equation 12is non-negative gives Equation (8).…”

“…The equivalent problems of the variational form of Equations (6)- (8) in the sense of distribution are the set of partial differential equations together with inequality constraint:…”

“…It can easily be estimated from the increasing publishing trend of research articles over the last few decades that FEM has proved itself a well programmed and commonly used method in an extensive range of engineering applications in general, for example in structural analysis, heat transfer and fluid mechanics etc. Different mathematicians have presented FEM for PDEs as well as for OCPs [5][6][7][8][9][10][11][12].…”

SUPG Approximation for the Oseen Viscoelastic Fluid Flow with Stabilized Lowest-Equal Order Mixed Finite Element Method

“…In this example, the theoretical convergence rates are examined by applying fluid flow across a domain Ω = [0, 1] × [0, 1] with parameters a = 0, λ = 5.0 and α = 0.5, respectively. Different authors used this experimental pattern for Stokes and Navier-Stokes equations [23,36,40,41], where the function b(x) was chosen to be the exact solution of velocity u. In this context, a right hand-side function is added to the (8) and f in (9) is studied with the help of following true solution.…”

“…We consider a third example to test the proposed scheme, which is a well-known problem for viscoelastic flow "4-to-1 contraction channel flow problem" which has a huge application in polymeric liquid industries and studied by many authors [20,43]. Moreover, 4-to-1 has been widely used in the literature to show the convergence, stability, and behavior of the streamlines of the contraction channel and the behavior of pressure [36,41]. The geometry of the 4-to-1 contraction commonly occurs in the forming of 'die' for the viscoelastic fluid.…”