This paper deals with an algorithm for the solution of diffusion and/or convection equations where we mixed the method of characteristics and the finite element method. Globally it looks like one does one step of transport plus one step of diffusion (or projection) but the mathematics show that it is also an implicit time discretization of thePDE in Lagrangian form. We give an error bound (h+Δt+h×h/Δt in the interesting case) that holds also for the Navier-Stokes equations even when the Reynolds number is infinite (Euler equation).

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On the transport-diffusion algorithm and its applications to the Navier-Stokes equations

Finite element methods for incompressible viscous flow

SUMMARY This paper is devoted to a comparison of various iterative solvers for the Stokes problem, based on the preconditioned Uzawa approach. In the first section the basic equations and general results of gradient-like methods are recalled. Then a new class of preconditioners, whose optimality will be shown, is introduced. In the last section numerical experiments and comparisons with multigrid methods prove the quality of these schemes, whose discretization is detailed.

Some fast 3D finite element solvers for the generalized Stokes problem

Numerical methods for the navier-stokes equations. Applications to the simulation of compressible and incompressible viscous flows

An overview is given of new developments of the least squares finite element method (LSFEM) in fluid dynamics. Special emphasis is placed on the universality of LSFEM; the symmetry and positiveness of the algebraic systems obtained from LSFEM; the accommodation of LSFEM to equal order interpolations for incompressible viscous flows; and the natural numerical dissipation of LSFEM for convective transport problems and high speed compressible flows. The performance of LSFEM is illustrated by numerical examples.

A. Keramsi

Papers

A finite element method for Navier-Stokes equations