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

Abstract

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 ...

A Linear Stabilized Incompressible Magnetohydrodynamic Problem with Magnetic Pressure

“…The MHD model equation with spatial discretization ( 13)-( 16) will be employed in this section. The notations and norms from [12,23] are utilized to introduce the mixed finite element (FE) strategy. Several fundamental vector function spaces are given by…”

“…In order to solve the above-described problems, this work aims to develop a stabilization mechanism. We refer to [11,12] for the related stabilization technique, which has been implemented in the literature for different model issues.…”

“…The key to overcoming such a challenge is to include a pressure stabilization term in the variational formulation. The implementation of stabilization techniques in combination with coupling equations in the viscoelastic field has been the subject of recent finite element methods research (see [11,12]).…”