A flux-free a posteriori error estimator for the incompressible Stokes problem using a mixed FE formulation

Author (s): Larsson, F; Díez, P and Huerta, A.
Journal: Computer Methods in Applied Mechanics and Engineering

Volume: 199
Pages: 2383 – 2402
Date: 2010

In this contribution, we present an a posteriori error estimator for the incom- pressible Stokes problem valid for a conventional mixed FE formulation. Due to the saddle-point property of the problem, conventional error estimators de- veloped for pure minimization problems cannot be utilized straight-forwardly. The new estimator is built up by two key ingredients. At first, a computed error approximation, exactly fulfilling the continuity equation for the error, is obtained via local Dirichlet problems. Secondly, we adopt the approach of solving local equilibrated flux-free problems in order to bound the remaining, incompressible, error. In this manner, guaranteed upper and lower bounds, of the velocity ”energy norm” of the error as well as goal-oriented (linear) output functionals, with respect to a reference (overkill) mesh are obtained. In particular, it should be noted that this approach requires no computa- tion of hybrid fluxes. Furthermore, the estimator is applicable to mixed FE formulations using continuous pressure approximations, such as the Mini and Taylor-Hood class of elements. In conclusion, a few simple numerical examples are presented, illustrating the accuracy of the error bounds.



  Author   = {Larsson, Fredrik and D{\'{\i}}ez, Pedro and Huerta, Antonio},
  Title    = {A flux-free a posteriori error estimator for the incompressible {S}tokes problem using a mixed {FE} formulation},
  Fjournal = {Computer Methods in Applied Mechanics and Engineering},
  Journal  = {Comput. Methods Appl. Mech. Eng.},
  Volume   = {199},
  Number   = {37-40},
  Pages    = {2383--2402},
  Year     = {2010},
  Doi      = {10.1016/j.cma.2010.03.011},
  Url      = {http://dx.doi.org/10.1016/j.cma.2010.03.011}}