Daniela Capatina

Coupling of Darcy-Forchheimer and Compressible Navier-Stokes Equations with Heat Transfer

This paper is devoted to the coupling of a 2D reservoir model with a 1.5D vertical wellbore model, both written in axisymmetric form. The physical problems are, respectively, described by the Darcy-Forchheimer and the compressible Navier-Stokes equations, together with an exhaustive energy equation. Each model was previously studied and its finite element discretization was validated. The two weak problems are bound together by means of transmission conditions at the perforations, yielding a nonstandard mixed formulation. A technical analysis is then carried out and the well-posedness of the time-discretized coupled problem, in both the continuous and the discrete cases, is established. Numerical tests including physical cases are presented, validating the coupled code.

Local Flux Reconstructions for Standard Finite Element Methods on Triangular Meshes

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

Robust Local Flux Reconstruction for Various Finite Element Methods

H(div)

Numerical modelling of multi-component multi-phase flows in petroleum reservoirs with heat transfer

-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.

Stabilized Finite Element Formulation with Domain Decomposition for Incompressible Flows

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.

Stopping Criteria Based on Locally Reconstructed Fluxes

This article is devoted to the modelling and the finite volume approximation of multi-component multi-phase flows in a petroleum reservoir, from both a dynamic and a thermal point of view. We introduce an adequate energy equation and the corresponding thermodynamics, which are next integrated in the isothermal GPRS code. Numerical examples and comparisons with the isothermal GPRS code are presented.

Variational approach for the multiscale modeling of an estuarian river. Part 1 : Derivation and numerical approximation of a 2D horizontal model

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 Peclet 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.

A dG method for the Stokes equations related to nonconforming approximations

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.