Defect correction method for time-dependent viscoelastic fluid flow

This paper studies a fully discrete Crank-Nicolson linear extrapolation stabilized finite element method for the natural convection problem, which is unconditionally stable and has second order temporal accuracy ofO(Δt2+hΔt+hm). A simple artificial viscosity stabilized of the linear system for the approximation of the new time level connected to antidiffusion of its effects at the old time level is used. An unconditionally stability and an a priori error estimate are derived for the fully discrete scheme. A series of numerical results are presented that validate our theoretical findings.

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“…Step 1.2. Derive the error inequalities of the momentum equation (40). From the definition of the modified Stokes projection (11)-(13), we have…”

Section: Error Estimatementioning

confidence: 99%

“…. As in [38][39][40][41], to obtain an analytical solution for the considered problem, a right-hand side function is added to the momentum equation of (1). The analytical solution is…”

Section: Example 1 (Analytical Solution)mentioning

confidence: 99%

The Crank-Nicolson Extrapolation Stabilized Finite Element Method for Natural Convection Problem

Zhang

Hou

2014

Mathematical Problems in Engineering

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“…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.…”

Section: Analytical Solution Testmentioning

confidence: 99%

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

et al. 2019

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In this article, a stabilized mixed finite element (FE) method for the Oseen viscoelastic fluid flow (OVFF) obeying an Oldroyd-B type constitutive law is proposed and investigated by using the Streamline Upwind Petrov-Galerkin (SUPG) method. To find the approximate solution of velocity, pressure and stress tensor, we choose lowest-equal order FE triples P1-P1-P1, respectively. However, it is well known that these elements do not fulfill the in f -sup condition. Due to the violation of the main stability condition for mixed FE method, the system becomes unstable. To overcome this difficulty, a standard stabilization term is added in finite element variational formulation. The technique is applied herein possesses attractive features, such as parameter-free, flexible in computation and does not require any higher-order derivatives. The stability analysis and optimal error estimates are obtained. Three benchmark numerical tests are carried out to assess the stability and accuracy of the stabilized lowest-equal order feature of the OVFF.case [8], and the time-dependent case of the same continuous interpolation was analysed by Baranger in [9].In Newtonian fluid flow, the Oseen equations are abridged to linearize the system. This is because the Oseen fluid flow model is the reduced linearized form of the Newtonian fluid which is described by the Navier-Stokes equation [10,11]. Moreover, the viscoelastic fluid flow model equations are non-linear equations in many terms. Hence, by taking the assumption of creeping fluid flow, the inertial part (u · ∇ u) of the momentum equation can be ignored. In this sense, the non-linearity arise only in the constitutive equation [12,13] which may be introduce in a linear form by fixing u(x) with a known velocity field b(x). The resulting system of equations can explicitly described the parameter space for α, λ and ∇b ∞ , which ensure the well-posedness of the continuous problem and its numerical approximation [14].In the FE framework, the main difficulty arises in the viscoelastic fluid flow due to its hyperbolic constitutive equation. It needs a stabilization (upwinding ) technique to approximate the FE solution. There are two main approaches that are mainly used to solve the Oldroyd-B model by using the OVFF technique; the discontinuous Galerkin (DG) approximation and the (SUPG) estimate to deal with the spurious oscillations of the hyperbolic constitutive equation; see [15][16][17][18][19][20]. Herein, we consider SUPG method to solve the OVFF as an upwinding technique [21]. To the best of our knowledge, V.J. Ervin et al. introduced the SUPG method in [22] to approximate the model equations with the standard FE or Hood-Taylor FE pair (P2 for velocity, P1 for pressure and P1 for continuous stress), where the existence and uniqueness of a solution to the problem was shown. Later, the same authors studied a defect correction method in [23], where they used the same standard FE (P2-P1) to approximate the velocity and pressure but for discontinuous stress. They have used the conforming ...

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“…consists of Newtonian part and viscoelastic part [27]. For notational convenience, we use Einstein's convention of summation and denote differentiation with comma as ∂u ∂x i is written as u ,i and ∂u ∂t is written u t .…”

Section: Model Problemmentioning

confidence: 99%

Two-Level Finite Element Approximation for Oseen Viscoelastic Fluid Flow

et al. 2018

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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.

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Defect correction method for time-dependent viscoelastic fluid flow

Cited by 10 publications

References 18 publications

The Crank-Nicolson Extrapolation Stabilized Finite Element Method for Natural Convection Problem

The Crank-Nicolson Extrapolation Stabilized Finite Element Method for Natural Convection Problem

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

Two-Level Finite Element Approximation for Oseen Viscoelastic Fluid Flow

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