Considered are numerical integration schemes for nondissipative dynamical systems in which multiple time scales are present. It is assumed that one can do an explicit separation of the RHS "forces" into fast forces and slow forces such that (i) the fast forces contain the high frequency part of the solution, (ii) the fast forces are conservative, and (iii) the reduced problem consisting only of the fast forces can be integrated much more cheaply than the full problem. The fast forces are allowed to have low frequency components. Particular applications for which the schemes are intended include N-body problems (for which most of the forces are slow) and nonlinear wave phenomena (for which the fast forces can be propagated by spectral methods). The assumption of cheap integration of fast forces implies that the overall cost of integration is primarily determined by the step size used to sample the slow forces. A long-time-step method is one in which this step size exceeds half the period of the fastest normal mode present in the full system. An existing method that comes close to qualifying is the "impulse" method, also known as Verlet-I and r-RESPA. It is shown that it might fail, though, for a couple of reasons. First, it suffers a serious loss of accuracy if the step size is near a multiple of the period of a normal mode, and, second, it is unstable if the step size is near a multiple of half the period of a normal mode. Proposed in this paper is a "mollified" impulse method having an error bound that is independent of the frequency of the fast forces. It is also shown to possess superior stability properties. Theoretical results are supplemented by numerical experiments. The method is efficient and reasonably easy to implement.

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This paper studies inf-sup stable finite element discretizations of the evolutionary Navier-Stokes equations with a grad-div type stabilization. The analysis covers both the case in which the solution is assumed to be smooth and consequently has to satisfy nonlocal compatibility conditions as well as the practically relevant situation in which the nonlocal compatibility conditions are not satisfied. The constants in the error bounds obtained do not depend on negative powers of the viscosity. Taking into account the loss of regularity suffered by the solution of the Navier-Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data, error bounds of order O(h 2 ) in space are proved. The analysis is optimal for quadratic/linear inf-sup stable pairs of finite elements. Both the continuous-intime case and the fully discrete scheme with the backward Euler method as time integrator are analyzed.Keywords Incompressible Navier-Stokes equations; inf-sup stable finite element methods; grad-div stabilization; error bounds independent of the viscosity; nonlocal compatibility condition; backward Euler method * Instituto de Investigación en Matemáticas (IMUVA), Universidad de Valladolid, Spain.

The approximation of the time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin method with grad-div stabilization is studied. The main goal is to prove that adding a grad-div stabilization term to the Galerkin approximation has a stabilizing effect for small viscosity. Both the continuous-in-time and the fully discrete case (backward Euler method, the two-step BDF, and Crank-Nicolson schemes) are analyzed. In fact, error bounds are obtained that do not depend on the inverse of the viscosity in the case where the solution is sufficiently smooth. The bounds for the divergence of the velocity as well as for the pressure are optimal. The analysis is based on the use of a specific Stokes projection. Numerical studies support the analytical results.Keywords Time-dependent Oseen equations · Inf-sup stable pairs of finite element spaces · Grad-div stabilization · Backward Euler scheme · Two-step backward differentiation scheme (BDF2) · Crank-Nicolson scheme · Uniform error estimates