Narrow Results

The aim of this paper is to show that, for a linear second-order hyperbolic equation discretized by the backward Euler scheme in time and continuous piecewise affine finite elements in space, the adaptation of the time steps can be combined with spatial mesh adaptivity in an optimal way. We derive a priori and a posteriori error estimates which admit, as much as it is possible, the decoupling of the errors committed in the temporal and spatial discretizations.

We consider the finite element discretization of the Navier–Stokes equations locally coupled with the equation for the turbulent kinetic energy through an eddy viscosity. We prove a posteriori error estimates which allow to automatically determine the zone where the turbulent kinetic energy must be inserted in the Navier–Stokes equations and also to perform mesh adaptivity in order to optimize the discretization of these equations. Numerical results confirm the interest of such an approach.