A second order in time local projection stabilized Lagrange–Galerkin method for Navier–Stokes equations at high Reynolds numbers

We present a stabilized Backward Difference Formula of order 2-Lagrange Galerkin method to integrate the incompressible Navier–Stokes equations at high Reynolds numbers (Re). The stabilization of the conventional Lagrange–Galerkin method is done via a local projection technique for inf–sup stable finite elements. We prove that for a finite time TT the a priori error estimate for velocity in a mesh dependent norm is OO(hm+hm+Δt2Δt2), whereas the error for pressure in the l2(L2(D))l2(L2(D)) norm is O(hm+Δt2)O(hm+Δt2), with error constants that are independent of the Re−1Re−1; here, mm denotes the degree of the polynomials of the velocity finite element space. The size of the stabilization parameters is calculated from the velocity error estimate in a way that the error is optimal when the solution is sufficiently smooth. Numerical examples at high Reynolds numbers show the robustness of our method.