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3-dimensional physically consistent diffusion in anisotropic media with memory

Michele Caputo (1998)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Some data on the flow of fluids exhibit properties which may not be interpreted with the classic theory of propagation of pressure and of fluids [21] based on the classic D’Arcy’s law which states that the flux is proportional to the pressure gradient. In order to obtain a better representation of the flow and of the pressure of fluids the law of D’Arcy is here modified introducing a memory formalisms operating on the flow as well as on the pressure gradient which implies a filtering of the pressure...

A continuous finite element method with face penalty to approximate Friedrichs' systems

Erik Burman, Alexandre Ern (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

A continuous finite element method to approximate Friedrichs' systems is proposed and analyzed. Stability is achieved by penalizing the jumps across mesh interfaces of the normal derivative of some components of the discrete solution. The convergence analysis leads to optimal convergence rates in the graph norm and suboptimal of order ½ convergence rates in the L2-norm. A variant of the method specialized to Friedrichs' systems associated with elliptic PDE's in mixed form and reducing the number...

A discrete kinetic approximation for the incompressible Navier-Stokes equations

Maria Francesca Carfora, Roberto Natalini (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we introduce a new class of numerical schemes for the incompressible Navier-Stokes equations, which are inspired by the theory of discrete kinetic schemes for compressible fluids. For these approximations it is possible to give a stability condition, based on a discrete velocities version of the Boltzmann H-theorem. Numerical tests are performed to investigate their convergence and accuracy.

A domain decomposition analysis for a two-scale linear transport problem

François Golse, Shi Jin, C. David Levermore (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present a domain decomposition theory on an interface problem for the linear transport equation between a diffusive and a non-diffusive region. To leading order, i.e. up to an error of the order of the mean free path in the diffusive region, the solution in the non-diffusive region is independent of the density in the diffusive region. However, the diffusive and the non-diffusive regions are coupled at the interface at the next order of approximation. In particular, our algorithm avoids iterating...

A Domain Decomposition Analysis for a Two-Scale Linear Transport Problem

François Golse, Shi Jin, C. David Levermore (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a domain decomposition theory on an interface problem for the linear transport equation between a diffusive and a non-diffusive region. To leading order, i.e. up to an error of the order of the mean free path in the diffusive region, the solution in the non-diffusive region is independent of the density in the diffusive region. However, the diffusive and the non-diffusive regions are coupled at the interface at the next order of approximation. In particular, our algorithm avoids iterating...

A Petrov-Galerkin approximation of convection-diffusion and reaction-diffusion problems

Josef Dalík (1991)

Applications of Mathematics

A general construction of test functions in the Petrov-Galerkin method is described. Using this construction; algorithms for an approximate solution of the Dirichlet problem for the differential equation - ϵ u n + p u ' + q u = f are presented and analyzed theoretically. The positive number ϵ is supposed to be much less than the discretization step and the values of p , q . An algorithm for the corresponding two-dimensional problem is also suggested and results of numerical tests are introduced.

A Survey on Mathematical Modelling of Deposition in Waxy Crude Oils

A. Fasano, L. Fusi, S. Correra, M. Margarone (2011)

Mathematical Modelling of Natural Phenomena

Waxy Crude Oils (WCO’s) are characterized by the presence of heavy paraffins in sufficiently large concentrations. They exhibit quite complex thermodynamical and rheological behaviour and present the peculiar property of giving rise to the formation of segregated wax deposits, when temperature falls down the so called WAT, or Wax Appearance Temperature. In extreme cases, segregated waxes may lead to pipeline occlusion due to deposition on cold walls....

An Adaptive Multi-level method for Convection Diffusion Problems

Martine Marion, Adeline Mollard (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this article we introduce an adaptive multi-level method in space and time for convection diffusion problems. The scheme is based on a multi-level spatial splitting and the use of different time-steps. The temporal discretization relies on the characteristics method. We derive an a posteriori error estimate and design a corresponding adaptive algorithm. The efficiency of the multi-level method is illustrated by numerical experiments, in particular for a convection-dominated problem.

An existence proof for the stationary compressible Stokes problem

A. Fettah, T. Gallouët, H. Lakehal (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper, we prove the existence of a solution for a quite general stationary compressible Stokes problem including, in particular, gravity effects. The Equation Of State gives the pressure as an increasing superlinear function of the density. This existence result is obtained by passing to the limit on the solution of a viscous approximation of the continuity equation.

Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems

Philippe Angot, Vít Dolejší, Miloslav Feistauer, Jiří Felcman (1998)

Applications of Mathematics

We present the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume barycentric mesh, whereas the diffusion term is discretized by piecewise linear nonconforming triangular finite elements. Under the assumption that the triangulations are of weakly...

Asymptotic dynamics in double-diffusive convection

Mikołaj Piniewski (2008)

Applicationes Mathematicae

We consider the double-diffusive convection phenomenon and analyze the governing equations. A system of partial differential equations describing the convective flow arising when a layer of fluid with a dissolved solute is heated from below is considered. The problem is placed in a functional analytic setting in order to prove a theorem on existence, uniqueness and continuous dependence on initial data of weak solutions in the class ( [ 0 , ∞ ) ; H ) ∩ L ² l o c ( ℝ + ; V ) . This theorem enables us to show that the infinite-dimensional...

Basic principles of mixed Virtual Element Methods

F. Brezzi, Richard S. Falk, L. Donatella Marini (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The aim of this paper is to give a simple, introductory presentation of the extension of the Virtual Element Method to the discretization of H(div)-conforming vector fields (or, more generally, of (n − 1) − Cochains). As we shall see, the methods presented here can be seen as extensions of the so-called BDM family to deal with more general element geometries (such as polygons with an almost arbitrary geometry). For the sake of simplicity, we limit ourselves to the 2-dimensional case, with the aim...

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