New approaches to error estimation and adaptivity for the Stokes and Oseen equations

SUMMARY We present a new approach to deliver reliable approximations of the norm of the residuals resulting from finite element solutions to the Stokes and Oseen equations. The method is based upon a global solve in a bubble space using iterative techniques. This provides an alternative to the classical equilibrated element residual methods for which it is necessary to construct proper boundary conditions for each local problem. The method is first used to develop a global a posteriori error estimator. It is then applied in a strategy to control the numerical error in specific outputs or quantities of interest which are functions of the solutions to the Stokes and Oseen equations. Copyright © 1999 John Wiley & Sons, Ltd. In recent years, the goal in a posteriori error estimation for computational processes has drifted from evaluating the numerical error in the classical energy norm to estimating it in terms of quantities of practical interest. Works in this field have been undertaken in [1‐3] with the objective to estimate and:or control the error by adapting the mesh parameters with respect to these quantities. The methodology, developed in [4], is extended here to non-self-adjoint problems, specifically the Oseen equations. It is based on the computation of functions which relate the influence of the residuals, viewed as the sources of error in the finite element approximations, onto the error quantity of interest. However, this technique relies on accurate evaluation of the norm of the residuals and on global error estimators in the ‘energy’ norm. We then propose a method which belongs to the family of Implicit Error Residual methods, for which norms of the residuals Rh in the momentum equation and Rh in the continuity equation are post-processed to provide meaningful error estimates. The computation of the norm of Rh is shown to be exact and cheap. The calculation of the norm of Rh is however more demanding. A new technique is developed which provides accurate approximations of Rh in a space of bubble functions which is gradually enriched through a global but inexpensive iterative process.