Schéma des différences finies pour un système d'équations paraboliques non linéaires avec dérivées mixtes
Second-order MUSCL schemes based on Dual Mesh Gradient Reconstruction (DMGR)
We discuss new MUSCL reconstructions to approximate the solutions of hyperbolic systems of conservations laws on 2D unstructured meshes. To address such an issue, we write two MUSCL schemes on two overlapping meshes. A gradient reconstruction procedure is next defined by involving both approximations coming from each MUSCL scheme. This process increases the number of numerical unknowns, but it allows to reconstruct very accurate gradients. Moreover a particular attention is paid on the limitation...
Semidiscrete and single step fully discrete finite element approximations for second order hyperbolic equations with nonsmooth solutions
Semi-discrétisation en temps pour les équations d'évolution paraboliques lorsque l'opérateur dépend du temps
Solution of contaminant transport with adsorption in porous media by the method of characteristics
A new approximation scheme is presented for the mathematical model of convection-diffusion and adsorption. The method is based on the relaxation method and the method of characteristics. We prove the convergence of the method and present some numerical experiments in 1D. The results can be applied to the model of contaminant transport in porous media with multi-site, equilibrium and non-equilibrium type of adsorption.
Solution of contaminant transport with adsorption in porous media by the method of characteristics
A new approximation scheme is presented for the mathematical model of convection-diffusion and adsorption. The method is based on the relaxation method and the method of characteristics. We prove the convergence of the method and present some numerical experiments in 1D. The results can be applied to the model of contaminant transport in porous media with multi-site, equilibrium and non-equilibrium type of adsorption.
Solution of degenerate parabolic variational inequalities with convection
Degenerate parabolic variational inequalities with convection are solved by means of a combined relaxation method and method of characteristics. The mathematical problem is motivated by Richard’s equation, modelling the unsaturated – saturated flow in porous media. By means of the relaxation method we control the degeneracy. The dominance of the convection is controlled by the method of characteristics.
Solution of degenerate parabolic variational inequalities with convection
Degenerate parabolic variational inequalities with convection are solved by means of a combined relaxation method and method of characteristics. The mathematical problem is motivated by Richard's equation, modelling the unsaturated – saturated flow in porous media. By means of the relaxation method we control the degeneracy. The dominance of the convection is controlled by the method of characteristics.
Solution of porous medium type systems by linear approximation schemes.
Solution of Stefan problems by fully discrete linear schemes.
Solution of the first problem of plane elasticity for multiply connected regions by the method of least squares on the boundary. I
Solution of the first problem of plane elasticity for multiply connected regions by the method of least squares on the boundary. II
Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles*
Caused by the problem of unilateral contact during vibrations of satellite solar arrays, the aim of this paper is to better understand such a phenomenon. Therefore, it is studied here a simplified model composed by a beam moving between rigid obstacles. Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of non-penetration Signorini's conditions. For this, starting from the works of Dumont and Paoli, we adapt to our beam...
Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles*
Caused by the problem of unilateral contact during vibrations of satellite solar arrays, the aim of this paper is to better understand such a phenomenon. Therefore, it is studied here a simplified model composed by a beam moving between rigid obstacles. Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of non-penetration Signorini's conditions. For this, starting from the works of Dumont and Paoli, we adapt to our beam...
Some new error estimates for finite element methods for second order hyperbolic equations using the Newmark method
We consider a family of conforming finite element schemes with piecewise polynomial space of degree in space for solving the wave equation, as a model for second order hyperbolic equations. The discretization in time is performed using the Newmark method. A new a priori estimate is proved. Thanks to this new a priori estimate, it is proved that the convergence order of the error is in the discrete norms of and , where and are the mesh size of the spatial and temporal discretization, respectively....
Some remarks concerning stabilization techniques for convection--diffusion problems
There are many methods and approaches to solving convection--diffusion problems. For those who want to solve such problems the situation is very confusing and it is very difficult to choose the right method. The aim of this short overview is to provide basic guidelines and to mention the common features of different methods. We place particular emphasis on the concept of linear and non-linear stabilization and its implementation within different approaches.
Source terms identification for time fractional diffusion equation.
Space-time discontinuos Galerkin method for solving nonstationary convection-diffusion-reaction problems
The paper presents the theory of the discontinuous Galerkin finite element method for the space-time discretization of a linear nonstationary convection-diffusion-reaction initial-boundary value problem. The discontinuous Galerkin method is applied separately in space and time using, in general, different nonconforming space grids on different time levels and different polynomial degrees and in space and time discretization, respectively. In the space discretization the nonsymmetric interior...
Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift
This paper is concerned with the analysis and implementation of spectral Galerkin methods for a class of Fokker-Planck equations that arises from the kinetic theory of dilute polymers. A relevant feature of the class of equations under consideration from the viewpoint of mathematical analysis and numerical approximation is the presence of an unbounded drift coefficient, involving a smooth convex potential that is equal to along the boundary of the computational domain . Using a symmetrization...
Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift
This paper is concerned with the analysis and implementation of spectral Galerkin methods for a class of Fokker-Planck equations that arises from the kinetic theory of dilute polymers. A relevant feature of the class of equations under consideration from the viewpoint of mathematical analysis and numerical approximation is the presence of an unbounded drift coefficient, involving a smooth convex potential U that is equal to +∞ along the boundary ∂D of the computational domain D. Using a symmetrization...