We prove convergence and optimal complexity of an adaptive finite element algorithm on quadrilate... more We prove convergence and optimal complexity of an adaptive finite element algorithm on quadrilateral meshes. The local mesh refinement algorithm is based on regular subdivision of marked cells, leading to meshes with hanging nodes. In order to avoid multiple layers of these, a simple rule is defined, which leads to additional refinement. We prove an estimate for the complexity of this refinement technique. As in former work, we use an adaptive marking strategy which only leads to refinement according to an oscillation term, if it is dominant. In comparison to the case of triangular meshes, the a posteriori error estimator contains an additional term which implicitly measure the deviation of a given quadrilateral from a parallelogram. The well-known lower bound of the estimator for the case of conforming P 1 elements does not hold here. We instead prove decrease of the estimator, in order to establish convergence and complexity estimates
Computer Methods in Applied Mechanics and Engineering, 2015
We study the finite element formulation of general boundary conditions for incompressible flow pr... more 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 propose a three-field formulation for efficiently solving a twodimensional Stokes problem in t... more We propose a three-field formulation for efficiently solving a twodimensional Stokes problem in the case of nonstandard boundary conditions. More specifically, we consider the case where the pressure and either normal or tangential components of the velocity are prescribed at some given parts of the boundary. The proposed computational methodology consists in reformulating the considered boundary value problem via a mixed-type formulation where the pressure and the vorticity are the principal unknowns while the velocity is the Lagrange multiplier. The obtained formulation is then discretized and a convergence analysis is performed. A priori error estimates are established, and some numerical results are presented to highlight the perfomance of the proposed computational methodology.
In this work, the main phenomena of genesis and hydrocarbons migration in a sedimentary basin are... more In this work, the main phenomena of genesis and hydrocarbons migration in a sedimentary basin are presented. Then, a mathematical modelling of these characters is formulated. Pressure, hydrocarbons masses and water mass are the main unknowns which have held our attention for mathematical modelling. Phases miscibilities effects are also taken into account. Afterwards a numerical method to compute the mathematical model is described. The numerical method must be robust, stable and low cost in computing time. At last, numerical results are exposed in a black-oil flow case.
ABSTRACT We present here the main ideas and results concerning the derivation of a new quasi-3D h... more ABSTRACT We present here the main ideas and results concerning the derivation of a new quasi-3D hydrodynamical model, also called 2.5D model, within the framework of nonlinear weak formulations. The idea is to work in the sum of spaces concerning 2D models, one in the horizontal plane, the other in the vertical one. The new model takes into account the river’s geometry and provides a three-dimensional velocity and pressure. We present the finite element approximation of the model and some numerical results.
Vladimir Sabel'nikov [ ONERA ] Visiting scientist Shipeng Mao [ funding from UPPA, CAS, China ] F... more Vladimir Sabel'nikov [ ONERA ] Visiting scientist Shipeng Mao [ funding from UPPA, CAS, China ] Francesco-Javier Sayas [ funding from UPPA, University of Zaragoza, Spain ] 2.2. Highlights CONCHA has been created as an 'equipe INRIA' in april 2007.
... págs. 121-128. A tax-investment dynamic reaction model. M. Victoria Fernández, C. Sánchez. Re... more ... págs. 121-128. A tax-investment dynamic reaction model. M. Victoria Fernández, C. Sánchez. Resumen; Texto completo. págs. 129-134. On an analytical solution in the planar elliptic restricted three-body problem. Luis Floria Gimeno. Resumen; Texto completo. págs. 135-144. ...
The paper is devoted to the 2D hydrodynamical modeling and numerical approximation of an estuaria... more The paper is devoted to the 2D hydrodynamical modeling and numerical approximation of an estuarian river flow. A new 2D horizontal model is derived and analyzed as a conforming approximation of the 3D time-discretized problem. It provides the three-dimensional velocity field as well as the pressure, which remains an unknown of the problem. Thanks to the framework of weak formulations, we avoid any closure problem usually encountered in the shallow water system and we can next employ finite elements for the approximation of the 2D model. The discrete problem is shown to be well-posed and finally, numerical tests are presented. The developed code is validated by means of comparisons with the classical shallow water model as well as with measured data.
We develop a robust finite element method with domain decomposition for incompressible flows, all... more 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.
We propose here a new approach for deriving 2D and 1D hydrodynamical models, within the framework... more We propose here a new approach for deriving 2D and 1D hydrodynamical models, within the framework of mixed variational formulations. We thus obtain a 2Dhorizontal model, as well as a 2D-vertical and a 1D model taking into account the geometry of the river. We analyze here only the 3D model and the 2D-horizontal one, for which we propose (after time discretization) low-order conforming finite element approximations. All the variational problems are well-posed. We intend to justify next a posteriori estimators between these models.
Numerical Mathematics and Advanced Applications 2009, 2010
We are interested by the modeling and the numerical simulation of a quasi-3D river flow, within t... more We are interested by the modeling and the numerical simulation of a quasi-3D river flow, within the context of hydrodynamical multidimensional modeling and simula-tion of estuarian river flows. The ideal model to be employed is a 3D one, but due to the huge computational ...
We discuss some recent progress in the convergence analysis of adaptive finite element methods fo... more We discuss some recent progress in the convergence analysis of adaptive finite element methods for the Stokes equations. First we present a result concerning the quasi-optimality of low-order non-conforming methods. Both the case of the Crouzeix-Raviart element on triangular meshes, and the Rannacher-Turek element on parallelogram elements are covered. Numerical experiments are conducted in order to appreciate the different variants of the algorithm.
We propose a new goal-oriented adaptive finite element method, which is based on the weighting of... more We propose a new goal-oriented adaptive finite element method, which is based on the weighting of the residuals from the primal and dual problem at each step of the iteration. Our main result is the quasi-optimality of the resulting algorithm. Numerical experiments are reported, showing the convergence behavior of the algorithm.
We prove convergence and optimal complexity of an adaptive finite element algorithm on quadrilate... more We prove convergence and optimal complexity of an adaptive finite element algorithm on quadrilateral meshes. The local mesh refinement algorithm is based on regular subdivision of marked cells, leading to meshes with hanging nodes. In order to avoid multiple layers of these, a simple rule is defined, which leads to additional refinement. We prove an estimate for the complexity of this refinement technique. As in former work, we use an adaptive marking strategy which only leads to refinement according to an oscillation term, if it is dominant. In comparison to the case of triangular meshes, the a posteriori error estimator contains an additional term which implicitly measure the deviation of a given quadrilateral from a parallelogram. The well-known lower bound of the estimator for the case of conforming P 1 elements does not hold here. We instead prove decrease of the estimator, in order to establish convergence and complexity estimates
Computer Methods in Applied Mechanics and Engineering, 2015
We study the finite element formulation of general boundary conditions for incompressible flow pr... more 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 propose a three-field formulation for efficiently solving a twodimensional Stokes problem in t... more We propose a three-field formulation for efficiently solving a twodimensional Stokes problem in the case of nonstandard boundary conditions. More specifically, we consider the case where the pressure and either normal or tangential components of the velocity are prescribed at some given parts of the boundary. The proposed computational methodology consists in reformulating the considered boundary value problem via a mixed-type formulation where the pressure and the vorticity are the principal unknowns while the velocity is the Lagrange multiplier. The obtained formulation is then discretized and a convergence analysis is performed. A priori error estimates are established, and some numerical results are presented to highlight the perfomance of the proposed computational methodology.
In this work, the main phenomena of genesis and hydrocarbons migration in a sedimentary basin are... more In this work, the main phenomena of genesis and hydrocarbons migration in a sedimentary basin are presented. Then, a mathematical modelling of these characters is formulated. Pressure, hydrocarbons masses and water mass are the main unknowns which have held our attention for mathematical modelling. Phases miscibilities effects are also taken into account. Afterwards a numerical method to compute the mathematical model is described. The numerical method must be robust, stable and low cost in computing time. At last, numerical results are exposed in a black-oil flow case.
ABSTRACT We present here the main ideas and results concerning the derivation of a new quasi-3D h... more ABSTRACT We present here the main ideas and results concerning the derivation of a new quasi-3D hydrodynamical model, also called 2.5D model, within the framework of nonlinear weak formulations. The idea is to work in the sum of spaces concerning 2D models, one in the horizontal plane, the other in the vertical one. The new model takes into account the river’s geometry and provides a three-dimensional velocity and pressure. We present the finite element approximation of the model and some numerical results.
Vladimir Sabel'nikov [ ONERA ] Visiting scientist Shipeng Mao [ funding from UPPA, CAS, China ] F... more Vladimir Sabel'nikov [ ONERA ] Visiting scientist Shipeng Mao [ funding from UPPA, CAS, China ] Francesco-Javier Sayas [ funding from UPPA, University of Zaragoza, Spain ] 2.2. Highlights CONCHA has been created as an 'equipe INRIA' in april 2007.
... págs. 121-128. A tax-investment dynamic reaction model. M. Victoria Fernández, C. Sánchez. Re... more ... págs. 121-128. A tax-investment dynamic reaction model. M. Victoria Fernández, C. Sánchez. Resumen; Texto completo. págs. 129-134. On an analytical solution in the planar elliptic restricted three-body problem. Luis Floria Gimeno. Resumen; Texto completo. págs. 135-144. ...
The paper is devoted to the 2D hydrodynamical modeling and numerical approximation of an estuaria... more The paper is devoted to the 2D hydrodynamical modeling and numerical approximation of an estuarian river flow. A new 2D horizontal model is derived and analyzed as a conforming approximation of the 3D time-discretized problem. It provides the three-dimensional velocity field as well as the pressure, which remains an unknown of the problem. Thanks to the framework of weak formulations, we avoid any closure problem usually encountered in the shallow water system and we can next employ finite elements for the approximation of the 2D model. The discrete problem is shown to be well-posed and finally, numerical tests are presented. The developed code is validated by means of comparisons with the classical shallow water model as well as with measured data.
We develop a robust finite element method with domain decomposition for incompressible flows, all... more 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.
We propose here a new approach for deriving 2D and 1D hydrodynamical models, within the framework... more We propose here a new approach for deriving 2D and 1D hydrodynamical models, within the framework of mixed variational formulations. We thus obtain a 2Dhorizontal model, as well as a 2D-vertical and a 1D model taking into account the geometry of the river. We analyze here only the 3D model and the 2D-horizontal one, for which we propose (after time discretization) low-order conforming finite element approximations. All the variational problems are well-posed. We intend to justify next a posteriori estimators between these models.
Numerical Mathematics and Advanced Applications 2009, 2010
We are interested by the modeling and the numerical simulation of a quasi-3D river flow, within t... more We are interested by the modeling and the numerical simulation of a quasi-3D river flow, within the context of hydrodynamical multidimensional modeling and simula-tion of estuarian river flows. The ideal model to be employed is a 3D one, but due to the huge computational ...
We discuss some recent progress in the convergence analysis of adaptive finite element methods fo... more We discuss some recent progress in the convergence analysis of adaptive finite element methods for the Stokes equations. First we present a result concerning the quasi-optimality of low-order non-conforming methods. Both the case of the Crouzeix-Raviart element on triangular meshes, and the Rannacher-Turek element on parallelogram elements are covered. Numerical experiments are conducted in order to appreciate the different variants of the algorithm.
We propose a new goal-oriented adaptive finite element method, which is based on the weighting of... more We propose a new goal-oriented adaptive finite element method, which is based on the weighting of the residuals from the primal and dual problem at each step of the iteration. Our main result is the quasi-optimality of the resulting algorithm. Numerical experiments are reported, showing the convergence behavior of the algorithm.
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Papers by David Trujillo