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

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.