Robust equilibration a posteriori error estimation for convection-diffusion-reaction problems

We study a posteriori error estimates for convection-diffusion-reaction problems with possibly dominating convection or reaction and inhomogeneous boundary conditions. For the conforming FEM discretisation with streamline diffusion stabilisation (SDM), we derive robust and efficient error estimators based on the reconstruction of equilibrated fluxes in an admissible discrete subspace of H (div, Ω). Error estimators of this type have become popular recently since they provide guaranteed error bounds without further unknown constants. The estimators can be improved significantly by some postprocessing and divergence correction technique. For an extension of the energy norm by a dual norm of some part of the differential operator, complete independence from the coefficients of the problem is achieved. Numerical benchmarks illustrate the very good performance of the error estimators in the convection dominated and the singularly perturbed cases

Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?

The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important problems in these fields are discussed

Equilibration a Posteriori Error Estimation for Convection–Diffusion–Reaction Problems

We study a posteriori error estimates for convection–diffusion–reaction problems with possibly dominating convection or reaction and inhomogeneous boundary conditions. For the conforming FEM discretisation with streamline diffusion stabilisation, we derive reliable and efficient error estimators based on the reconstruction of equilibrated fluxes in an admissible discrete subspace of $H(\mathrm{div},\Omega ). Error estimators of this type have become popular recently since they provide guaranteed error bounds without further unknown constants. The estimators can be improved significantly by some postprocessing and divergence correction technique. For an extension of the energy norm by a dual norm of the convection part of the differential operator, robustness of the error estimator with respect to the coefficients of the problem is achieved. Numerical benchmarks illustrate the good performance of the error estimators for singularly perturbed problems, in particular with dominating convection