Convergence of a linearly extrapolated BDF2 finite element scheme for viscoelastic fluid flow

The stability and convergence of a linearly extrapolated second order backward difference (BDF2-LE) time-stepping scheme for solving viscoelastic fluid flow in R-d, d = 2, 3, 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 sigma, velocity vector u and pressure rho are P-m-discontinuous, P-k-continuous and P-q-continuous elements, respectively. Upwinding needed for convection of sigma is made by a discontinuous Galerkin (DG) FE method. For the time step Delta t small enough, the existence of an approximate solution is proven. If m, k >= d/2, q + 1 >= d/2, and Delta t <= C(0)h d/4, then the discrete H-1 and L-2 errors for the velocity and stress, and L-2 error for the pressure, are bounded by C(Delta t(2) + h(min{m, k, q+1})), where h denotes the mesh size. The derived theoretical results are supported by numerical tests.