Two fast numerical methods for a generalized Oldroyd-B fluid model

This work proposes two fast and efficient numerical methods for a generalized Oldroyd-B fluid model with fractional derivatives. The fast evaluations of the Riemann–Liouville fractional derivatives are developed based on convolution quadrature generated by the backward Euler and the second-order backward difference methods. By using these fast algorithms in time and employing the finite element method in space, two fully discrete schemes are established for the considered problem. Further, we discuss the error estimates of these numerical schemes based on the property of the initial value and the right hand function instead of the regularity assumption of the exact solution. Finally, some numerical tests are presented to verify the effectiveness of the numerical schemes and confirm the theoretical results.