. Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2011, 200 (37-40) Abstract. We derive a posteriori error estimates for a class of second-order monotone quasi-linear diffusion-type problems approximated by piecewise affine, continuous finite elements. Our estimates yield a guaranteed and fully computable upper bound on the error measured by the dual norm of the residual, as well as a global error lower bound, up to a generic constant independent of the nonlinear operator. They are thus fully robust with respect to the nonlinearity, thanks to the choice of the error measure. They are also locally efficient, albeit in a different norm, and hence suitable for adaptive mesh refinement. Moreover, they allow to distinguish, estimate separately, and compare the discretization and linearization errors. Hence, the iterative (Newton-Raphson, quasi-Newton) linearization can be stopped whenever the linearization error drops to the level at which it does not affect significantly the overall error. This can lead to important computational savings, as performing an excessive number of unnecessary linearization iterations can be avoided. Numerical experiments for the p-Laplacian illustrate the theoretical developments.

Abstract.We analyze residual and hierarchical a posteriori error estimates for nonconforming finite element approximations of elliptic problems with variable coefficients. We consider a finite volume box scheme equivalent to a nonconforming mixed finite element method in a Petrov-Galerkin setting. We prove that all the estimators yield global upper and local lower bounds for the discretization error. Finally, we present results illustrating the efficiency of the estimators, for instance, in the simulation of Darcy flows through heterogeneous porous media.Mathematics Subject Classification. 65N15, 65N60, 75N12, 76905.

We analyse a non-conforming finite-element method to approximate advection-diffusion-reaction equations. The method is stabilized by penalizing the jumps of the solution and those of its advective derivative across mesh interfaces. The a priori error analysis leads to (quasi-)optimal estimates in the mesh size (sub-optimal by order 1 2 in the L 2 -norm and optimal in the broken graph norm for quasi-uniform meshes) keeping the Péclet number fixed. Then, we investigate a residual a posteriori error estimator for the method. The estimator is semi-robust in the sense that it yields lower and upper bounds of the error which differ by a factor equal at most to the square root of the Péclet number. Finally, to illustrate the theory we present numerical results including adaptively generated meshes.

A nonconforming (Crouzeix-Raviart) finite element method with subgrid viscosity is analyzed to approximate advection-diffusion-reaction equations. The error estimates are quasi-optimal in the sense that keeping the Péclet number fixed, the estimates are suboptimal of order 1 2 in the mesh size for the L 2 -norm and optimal for the advective derivative on quasi-uniform meshes. The method is also reformulated as a finite volume box scheme providing a reconstruction formula for the diffusive flux with local conservation properties. Numerical results are presented to illustrate the error analysis.