The stability and convergence of a linearly extrapolated second order backward difference (BDF2-LE) time-stepping scheme for solving viscoelastic fluid flow in Rd\documentclass[12pt]{minimal}
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\begin{document}$\mathbb{R}^{d}$\end{document}, d=2,3\documentclass[12pt]{minimal}
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\begin{document}$d=2,3$\end{document}, are presented in this paper. The time discretization is based on the implicit scheme for the linear term and the two-step linearly extrapolated scheme for the nonlinear term. Mixed finite element (MFE) method is applied for the spatial discretization. The approximations of stress tensor σ, velocity vector u and pressure p are Pm\documentclass[12pt]{minimal}
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\begin{document}$P_{m}$\end{document}-discontinuous, Pk\documentclass[12pt]{minimal}
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\begin{document}$P_{k}$\end{document}-continuous and Pq\documentclass[12pt]{minimal}
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\begin{document}$P_{q}$\end{document}-continuous elements, respectively. Upwinding needed for convection of σ is made by a discontinuous Galerkin (DG) FE method. For the time step △t small enough, the existence of an approximate solution is proven. If m,k⩾d2\documentclass[12pt]{minimal}
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\begin{document}$m, k \geqslant \frac{d}{2}$\end{document}, q+1⩾d2\documentclass[12pt]{minimal}
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\begin{document}$q+1\geqslant\frac{d}{2}$\end{document}, and △t⩽C0hd4\documentclass[12pt]{minimal}
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\begin{document}$\triangle t \leqslant C_{0} h^{\frac{d}{4}}$\end{document}, then the discrete H1\documentclass[12pt]{minimal}
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\begin{document}$H^{1}$\end{document} and L2\documentclass[12pt]{minimal}
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\begin{document}$L^{2}$\end{document} errors for the velocity and stress, and L2\documentclass[12pt]{minimal}
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\begin{document}$L^{2}$\end{document} error for the pressure, are bounded by C(△t2+hmin{m,k,q+1})\documentclass[12pt]{minimal}
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\begin{document}$C(\triangle t^{2}+h^{\min\{m,k,q+1\}})$\end{document}, where h denotes the mesh size. The derived theoretical results are supported by numerical tests.
