| Summary: | 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>-<inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>-<inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, respectively. However, it is well known that these elements do not fulfill the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>i</mi> <mi>n</mi> <mi>f</mi> </mrow> </semantics> </math> </inline-formula>-<inline-formula> <math display="inline"> <semantics> <mrow> <mi>s</mi> <mi>u</mi> <mi>p</mi> </mrow> </semantics> </math> </inline-formula> 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.
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