We study the drag error for the Navier–Stokes equations approximated by conforming low-order finite elements. The numerical scheme uses a SUPG stabilization and a new Nitche’s type stabilization on the whole boundary. We introduce a definition of the discrete drag which contains additional terms resulting from the discrete formulat. We prove O(h2) convergence for the drag error and illustrate the theoretical results by numerical tests. The extension to other finite element methods is also discussed.

We are interested in the a posteriori error analysis based on locally reconstructed fluxes for the 2D Signorini problem. We start from a P 1-conforming approximation where the contact condition is treated by means of a Nitsche method. We propose an extension of a general approach previously developed for the Laplace operator, allowing to obtain H(div)-conforming conservative fluxes by a local post-process. The reconstructed flux yields an a posteriori error indicator, which is completed by two additional terms taking into account the non-linear contact condition. We then prove the reliability of the indicator, without any additional assumption.

We develop in this article a uniform framework for the computation of conservative local fluxes for some classical finite element methods of arbitrary order on triangular meshes: conforming (CG), nonconforming (NC), and discontinuous methods (DG). The computation of these H (div)-conforming fluxes is done by local postprocessing of the finite element solution, avoiding the solution of any mixed (local or global) problem. We prove optimal error estimates and study the relations between the fluxes for the different methods when the stabilization parameter of DG tends to infinity. In particular, we prove that the DG flux tends to either the CG or the NC flux, depending on the nature of the stabilization term, as it is well-known for the discrete solutions. The considered reconstructions coincide in particular cases with other approaches from the literature.

We study the finite element formulation of general boundary conditions for incompressible flow problems. Distinguishing between the contributions from the inviscid and viscid parts of the equations, we use Nitsche's method to develop a discrete weighted weak formulation valid for all values of the viscosity parameter, including the limit case of the Euler equations. In order to control the discrete kinetic energy, additional consistent terms are introduced. We treat the limit case as a (degenerate) system of hyperbolic equations, using a balanced spectral decomposition of the flux Jacobian matrix, in analogy with compressible flows. Then, following the theory of Friedrich's systems, the natural characteristic boundary condition is generalized to the considered physical boundary conditions. Several numerical experiments, including standard benchmarks for viscous flows as well as inviscid flows are presented.

We develop a robust finite element method with domain decomposition for incompressible flows, allowing for control of the kinetic energy. First, we introduce a streamline upwind Petrov--Galerkin stabilization, which preserves the scaling of the Navier--Stokes equations and yields robustness with respect to the Péclet number. In view of parallelization, we then generalize the method in order to take into account several subdomains with independent finite element spaces, discontinuous at the interfaces. The interface conditions are treated by a generalized Nitsche-type method, also respecting the correct scaling. Detailed numerical experiments are presented in order to confirm robustness of the method and study its dependence on the different numerical parameters.

We present a uniform approach to local reconstructions of the gradient of primal approximations by conforming, nonconforming and totally discontinuous finite elements of arbitrary order. We start from a hybrid formulation which covers all considered methods and whose Lagrange multipliers approximate the normal fluxes. It turns out that the multipliers can be computed locally and are next used to define local corrections of the flux. We also show that the DG solution and reconstructed flux with stabilisation parameter γ converge uniformly in h with the convergence rate 1∕γ towards the CG or NC ones, depending on the stabilisation.

We propose stopping criteria for the iterative solution of equations resulting from discretization by conforming, nonconforming, and total discontinuous finite element methods. A simple modification of error estimators based on locally reconstructed fluxes allows to split the estimator into a discretisation-based and an iteration-based part. Comparison of both then leads to stopping criteria which can be used in the framework of an adaptive algorithm.

In this note, we propose a modification of the NXFEM proposed in Hansbo and Hansbo (2002) [4] for the elliptic interface problem. It leads to a robust method not only with respect to the mesh-interface geometry, but also with respect to the diffusion parameters. (C) 2012 Published by Elsevier Masson SAS on behalf of Academie des sciences.

We deal with the hydrostatic Stokes approximation with non homogeneous Dirichlet boundary conditions. After having investigated the homogeneous case, we build a lifting operator of boundary values related to the divergence operator, and solve the non homogeneous problem in a cylindrical type domain.