An a posterior error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow

In this paper we derive an a posteriori error estimate for the numerical approximation of the solution of a system modeling the flow of two incompressible and immiscible fluids in a porous medium. We take into account the capillary pressure, which leads to a coupled system of two equations: parabolic and elliptic. The parabolic equation may become degenerate, i.e., the nonlinear diffusion coefficient may vanish over regions that are not known a priori. We first show that, under appropriate assumptions, the energy-type-norm differences between the exact and the approximate nonwetting phase saturations, the global pressures, and the Kirchhoff transforms of the nonwetting phase saturations can be bounded by the dual norm of the residuals. We then bound the dual norm of the residuals by fully computable a posteriori estimators. Our analysis covers a large class of conforming, vertex-centered finite volume-type discretizations with fully implicit time stepping. As an example, we focus here on two approaches: a "mathematical" scheme derived from the weak formulation, and a phase-by-phase upstream weighting "engineering" scheme. Finally, we show how the different error components, namely the space discretization error, the time discretization error, the linearization error, the algebraic solver error, and the quadrature error can be distinguished and used for making the calculations efficient

Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids

Abstract Discontinuous Galerkin methods handle very well general polygonal and nonmatching meshes. We present in this Note a H(div)-conforming reconstruction of the flux on such meshes in the setting of an elliptic problem. We exploit the local conservation property of discontinuous Galerkin methods and solve local Neumann problems by means of the Raviart-Thomas-Nédélec mixed finite element method. Our reconstruction can be used in a guaranteed a posteriori error estimate and it is also of an independent interest when the approximate flux is to be used subsequently in a transport problem. To cite this article: A. Ern, M. Vohralík, C. R. Acad. Sci. Paris, Ser. I 340 (2005). Résumé Reconstruction du flux et estimations a posteriori pour la méthode de Galerkine discontinue sur des maillages non-coïncidants avec mailles polygonales Les méthodes de Galerkine discontinues sont bien adaptées pour traiter des maillages non-coïncidants avec mailles polygonales. Nous présentons dans cette Note une reconstruction H(div)-conforme du flux sur de tels maillages pour un problème elliptique. Nous exploitons la propriété de conservativité locale des méthodes de Galerkine discontinues afin de résoudre des problèmes locaux de Neumann approchés par deséléments finis mixtes de Raviart-Thomas-Nédélec. Notre reconstruction peutêtre utilisée pour une estimation garantie d'erreur a posteriori etégalement afin d'évaluer une vitesse approchée pour un problème de transport. Pour citer cet article : A. Ern, M. Vohralík, C. R. Acad. Sci. Paris, Ser. I 340 (2005)

Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids

Abstract Discontinuous Galerkin methods handle very well general polygonal and nonmatching meshes. We present in this Note a H(div)-conforming reconstruction of the flux on such meshes in the setting of an elliptic problem. We exploit the local conservation property of discontinuous Galerkin methods and solve local Neumann problems by means of the Raviart-Thomas-Nédélec mixed finite element method. Our reconstruction can be used in a guaranteed a posteriori error estimate and it is also of an independent interest when the approximate flux is to be used subsequently in a transport problem. To cite this article: A. Ern, M. Vohralík, C. R. Acad. Sci. Paris, Ser. I 340 (2005). Résumé Reconstruction du flux et estimations a posteriori pour la méthode de Galerkine discontinue sur des maillages non-coïncidants avec mailles polygonales Les méthodes de Galerkine discontinues sont bien adaptées pour traiter des maillages non-coïncidants avec mailles polygonales. Nous présentons dans cette Note une reconstruction H(div)-conforme du flux sur de tels maillages pour un problème elliptique. Nous exploitons la propriété de conservativité locale des méthodes de Galerkine discontinues afin de résoudre des problèmes locaux de Neumann approchés par deséléments finis mixtes de Raviart-Thomas-Nédélec. Notre reconstruction peutêtre utilisée pour une estimation garantie d'erreur a posteriori etégalement afin d'évaluer une vitesse approchée pour un problème de transport. Pour citer cet article : A. Ern, M. Vohralík, C. R. Acad. Sci. Paris, Ser. I 340 (2005)

On the Unilateral Contact Between Membranes. Part 1: Finite Element Discretization and Mixed Reformulation

The contact between two membranes can be described by a system of variational inequalities, where the unknowns are the displacements of the membranes and the action of a membrane on the other one. We first perform the analysis of this system. We then propose a discretization, where the displacements are approximated by standard finite elements and the action by a local postprocessing. Such a discretization admits an equivalent mixed reformulation. We prove the well-posedness of the discrete problem and establish optimal a priori error estimates

Conserving Soil and Water for Society: Sharing Solutions MODELLING OF GROUNDWATER FLOW IN FRACTURED ROCK: THEORETICAL APPROACH AND PRACTICAL APPLICATIONS

The poster gives an overview of the present knowledge achieved by our research group in the field of modelling of the fluid flow in a fractured rock environment. First, we introduce possible approaches. Then we describe a process of generation of a mesh. Our system is based on the discrete stochastic network approach. It means that particular fractures are represented as 2D entities (polygons in our case) placed in 3D space. The resulting mesh has the same statistical characteristic (density of fractures, their orientation, permeability etc.) as real fractured environment in the rock massif. The second part of the poster describes a numerical model of the groundwater flow. This model solves the problem using linear (Darcy) flow. A Mixed-hybrid FEM is used for approximation of the PDE’s. In the third part we show application of our method to a real-world hydrogeological problem-simulation of the injection test and communication between drillholes PTP3 and PTP4a in the Krušné Hory Mountains in the Czech Republic. Additional Keywords: fractured rock, numerical modelling, Darcy law, and finite element method