An efficient numerical scheme for precise time integration of a diffusion-dissolution/precipitation chemical system

A numerical scheme based on an operator splitting method and a dense output event location algorithm is proposed to integrate a diffusion-dissolution/precipitation chemical initial-boundary value problem with jumping nonlinearities. The numerical analysis of the scheme is carried out and it is proved to be of order 2 in time. This global order estimate is illustrated numerically on a test case.

     

Consistency,stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems

Summary. In an abstract framework we present a formalism which specifies the notions of consistency and stability of Petrov-Galerkin methods used to approximate nonlinear problems which are, in many practical situations, strongly nonlinear elliptic problems. This formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems. This theory is illustrated with the example: in a two dimensional domain with Dirichlet boundary conditions. Received June 10, 1992 / Revised version received February 28, 1994     

Adaptive finite element for semi-linear convection–diffusion problems

In this article a strategy of adaptive finite element for semi-linear problems, based on minimizing a residual-type estimator, is reported. We get an a posteriori error estimate which is asymptotically exact when the mesh size h tends to zero. By considering a model problem, the quality of this estimator is checked. It is numerically shown that without constraint on the mesh size h, the efficiency of the a posteriori error estimate can fail dramatically. This phenomenon is analysed and an algorithm which equidistributes the local estimators under the constraint h ⩽ h _max is proposed. This algorithm allows to improve the computed solution for semi-linear convection–diffusion problems, and can be used for stabilizing the Lagrange finite element method for linear convection–diffusion problems. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotic decomposition of a singular perturbation problem with unbounded energyDécomposition asymptotique d'une perturbation singulière d'énergie sans limite

We consider a singular perturbation with unbounded energy. We propose here an effective method of finite element computation, fit for accounting for the linear behavior of the solution. The Hilbert space of the variational formulation, H²₀(0,1), is replaced by a simpler subspace containing an asymptotic solution of the initial problem. Error estimates are derived by eliminating some degrees of freedom and a numerical experiment is developped. To cite this article: F. Fontvieille et al., C. R. Mecanique 330 (2002) 507–512.     

Resolution of the Transport equation subject to constraint

We present a method for solving the Transport equation when its solution has to belong to a constrained set which is not required to be convex. An autonomous formulation of the characteristics method allows us to use the tangency condition which has been introduced for ordinary differential equations. Thus we obtain a sufficient condition for existence of solutions, which shows the interplay between the geometry of the constraints set K   and the velocity field β$\beta$. A numerical method is proposed for solving the problem when the sufficient condition is not satisfied. A numerical experiment is presented showing the efficiency of the algorithm proposed.     

Solutions for Linear Conservation Laws with Velocity Fields in $$L^{\rm \infty}$$

A Space-Time Integrated Least Squares (STILS) method is derived for solving the linear conservation law with a velocity field in . An existence and uniqueness result is given for the solution of this equation. A maximum principle is established and finally a comparison with a renormalized solution is presented.