A posteriori bounds for linear functional outputs of Crouzeix–Raviart finite element discretizations of the incompressible Stokes problem

Abstract

A finite element technique is presented for the efficient generation of lower and upper bounds to outputs which are linear functionals of the solutions to the incompressible Stokes equations in two space dimensions. The finite element discretization is effected by Crouzeix–Raviart elements, the discontinuous pressure approximation of which is central to this approach. The bounds are based upon the construction of an augmented Lagrangian: the objective is a quadratic ‘energy’ reformulation of the desired output, the constraints are the finite element equilibrium equations (including the incompressibility constraint), and the inter-sub-domain continuity conditions on velocity. Appealing to the dual max–min problem for appropriately chosen candidate Lagrange multipliers then yields inexpensive bounds for the output associated with a fine-mesh discretization. The Lagrange multipliers are generated by exploiting an associated coarse-mesh approximation. In addition to the requisite coarse-mesh calculations, the bound technique requires the solution of only local sub-domain Stokes problems on the fine mesh. The method is illustrated for the Stokes equations, in which the outputs of interest are the flow rate past and the lift force on a body immersed in a channel. Copyright © 2000 John Wiley & Sons, Ltd.