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A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements. (English) Zbl 0942.65113
For the numerical approximation of non-stationary fluid flow problems in a moving domain, the so-called arbitrary Lagrangian Eulerian (ALE) formulation has been studied since the beginning of the 80’s. The idea behind is very similar to the well-known Lagrangian coordinates in fluid mechanics.
In the paper under review, the authors present a general theory of the application of the ALE formulation to conservation laws using a finite element method for the numerical approximation. As it is known from the literature, usual finite volume or finite difference methods may cause instabilities. Therefore, the authors also focus on the stability properties of the method they use.
Finally, the authors analyze a scalar linear convection-diffusion problem in a moving domain and discuss conditions that are necessary for the numerical scheme to satisfy a so-called geometric conservation law.
Reviewer: E.Emmrich (Berlin)

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35L65 Hyperbolic conservation laws
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