Computable Error Estimators and Adaptive Techniques for Fluid Flow Problems

Abstract

The present paper reviews several methods for a posteriori error es- timation of finite element approximations. The first part (Section 1) is concerned with global error estimators which provide for estimates and bounds in global norms (so-called energy norms). In particular, explicit and implicit residual methods are described in detail. The second part of the paper (Section 2) deals with a more recent technique, in which the numerical error in finite element approximations of general problems are estimated in terms of quantities of interest. The concept of goal-oriented adaptivity which embodies mesh adaptation procedures designed to control error in specific quantities is also described. Both global and goal-oriented error estimation methods are illustrated on model problems based on the Poisson, convection-diffusion, Stokes and steady-state Navier-Stokes equations. The third part of the paper (Section 3) presents preliminary work on a posteriori error esti- mation for time-dependent problems, namely the parabolic heat equation and the incompressible Navier-Stokes equations. The methods deal with estimating the ap- proximation errors due to the discretization in space using either explicit or implicit error estimators.