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Link-cutting bubbles for the stabilization of convection-diffusion-reaction problems (Articolo in rivista)

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Link-cutting bubbles for the stabilization of convection-diffusion-reaction problems (Articolo in rivista) (literal)

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Brezzi F.(1), Hauke G.(2), Marini L.D.(3), Sangalli G.(4) (2003)
Link-cutting bubbles for the stabilization of convection-diffusion-reaction problems
in Mathematical models and methods in applied sciences
(literal)

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The paper presents a numerical scheme, similar to the Residual-free Bubble FEM, which is able to compute the numerical solution of the steady-state advection-diffusion-reaction one-dimensional problem, in all regimes. In a subsequent work, the scheme has been used for solving a time-dependent problem, coupling it with a finite difference scheme for the time discretization. Indeed, its good properties make him suitable for this purpose. The usability of this methodology for multi-dimensional problems is still under study. (literal)

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(1) UNI-Pavia; (2) Politecnico, Saragoza; (3) UNI-Pavia; (4) IMATI-CNR (literal)

Titolo

Link-cutting bubbles for the stabilization of convection-diffusion-reaction problems (literal)

Abstract

It is known that the addition and elimination of suitable bubble functions can result in a stabilized scheme of the SUPG-type. Residual-Free Bubbles (RFB), in particular, can assure a quasi-optimal stabilized scheme, but they are difficult to compute in one dimension and nearly impossible to compute in 2 and 3 dimensions, unless in special limit cases. Strongly convection-dominated problems (without reaction terms) are one of these cases, where it is possible to find reasonably simple computable bubbles that provide a stabilizing effect as good as that of true RFB. Here, although in a one-dimensional framework, we analyze the case in which a non-negligible reaction term is present, and we provide a simple recipe for spotting a suitable bubble space (adding two bubbles per element) that provides a very good stabilizing effect. The method adapts very well to all regimes with continuous transitions from one regime to another. It is clear that the one-dimensional case, in itself, has no real interest. We believe, however, that the discussion can cast some light on the interaction between convection and reaction that could be useful in future works dealing with multidimensional, more realistic problems. (literal)

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