A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems

Abstract

Computable a posteriori error bounds and related adaptive mesh-refining algorithms are provided for the numerical treatment of monotone stationary flow problems with a quite general class of conforming and nonconforming finite element methods. A refined residual-based error estimate generalises the works of Verfürth; Dari, Duran and Padra; Bao and Barrett. As a consequence, reliable and efficient averaging estimates can be established on unstructured grids. The symmetric formulation of the incompressible flow problem models certain nonNewtonian flow problems and the Stokes problem with mixed boundary conditions. A Helmholtz decomposition avoids any regularity or saturation assumption in the mathematical error analysis. Numerical experiments for the partly nonconforming method analysed by Kouhia and Stenberg indicate efficiency of related adaptive mesh-refining algorithms.