Algorithms for adaptive mesh refinement using a residual error estimator are proposed for fluid flow problems in a finite volume framework. The residual error estimator, referred to as the ℜ-parameter is used to derive refinement and coarsening criteria for the adaptive algorithms. An adaptive strategy based on the ℜ-parameter is proposed for continuous flows, while a hybrid adaptive algorithm employing a combination of error indicators and the ℜ-parameter is developed for discontinuous flows. Numerical experiments for inviscid and viscous flows on different grid topologies demonstrate the effectiveness of the proposed algorithms on arbitrary polygonal grids.

The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equationin $\mathbb{R}^d$ , d=2 or 3,using backward Euler's scheme. For this discretization, we derive a residual indicator, which usea spatial residual indicator based on thejumps of normal and tangential derivatives of the nonconformingapproximation and a time residual indicator based on the jump of broken gradients at each time step.Lower and upper bounds form the main results with minimal assumptions on the mesh.Numerical experiments and a space-time adaptive algorithm confirm the theoretical predictions.

We consider a posteriori error estimators that can be applied to anisotropic tetrahedral finite element meshes, i.e. meshes where the aspect ratio of the elements can be arbitrarily large.Two kinds of Zienkiewicz–Zhu (ZZ) type error estimators are derivedwhich originate from different backgrounds. In the course of the analysis, the first estimator turns out to be a special case of the second one, and both estimators can be expressed using some recovered gradient.The advantage of keeping two different analyses of the estimators is that they allow different and partially novel investigations and results. Both rigorous analytical approaches yield the equivalence of each ZZ error estimator to a known residual error estimator. Thus reliability and efficiency of the ZZ error estimation is obtained.The anisotropic discretizations require analytical tools beyond the standard isotropic methods. Particular attention is paid to the requirements on the anisotropic mesh.The analysis is complemented and confirmed by extensive numerical examples. They show that good results can be obtained for a large class of problems, demonstrated exemplary for the Poisson problem and a singularly perturbed reaction diffusion problem.

Singularly perturbed problems often yield solutions with strong directional features,e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation.To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.