Choosing bubbles for advection-diffusion problems.

*(English)*Zbl 0819.65128This clearly written paper reviews one approach to numerical treatment of advection-diffusion equations with dominant advection terms. The authors first present the simple case of one-dimensional linear advection- diffusion. They present its classical centered difference or linear finite element approximation and illustrate by example the “wiggles” which can appear in its solution. They point out that these wiggles can be eliminated by “upwind” methods or by introducing an “artificial” diffusion term into the equation. Both of these approaches can result in reduced solution accuracy.

The authors then present an alternative approach to artificial diffusion. This alternative involves enriching the finite dimensional space of trial functions by including “bubble” functions, which are functions taking the value zero at all nodes of the finite elements. Bubble functions can be used to augment the usual finite element space of piecewise linear functions. The result is equivalent to an alternative prescription for artificial diffusion.

For linear advection-diffusion in multiple dimensions, many choices of bubble functions are available. The authors perform a calculation showing that one choice leads to the streamline upwind/Petrov-Galerkin (SUPG) method of A. N. Brooks and T. J. R. Hughes [Comput. Meth. Appl. Mech. Eng. 32, 199-259 (1982; Zbl 0497.76041 )], and remark that this is also true in more general situations. The authors then present a choice of bubble functions which appear natural from their viewpoint. Two numerical examples illustrate that the authors’ choice is competitive with SUPG. The authors remark that “more investigation is needed in order to reach an ‘optimal’ scheme”.

The authors then present an alternative approach to artificial diffusion. This alternative involves enriching the finite dimensional space of trial functions by including “bubble” functions, which are functions taking the value zero at all nodes of the finite elements. Bubble functions can be used to augment the usual finite element space of piecewise linear functions. The result is equivalent to an alternative prescription for artificial diffusion.

For linear advection-diffusion in multiple dimensions, many choices of bubble functions are available. The authors perform a calculation showing that one choice leads to the streamline upwind/Petrov-Galerkin (SUPG) method of A. N. Brooks and T. J. R. Hughes [Comput. Meth. Appl. Mech. Eng. 32, 199-259 (1982; Zbl 0497.76041 )], and remark that this is also true in more general situations. The authors then present a choice of bubble functions which appear natural from their viewpoint. Two numerical examples illustrate that the authors’ choice is competitive with SUPG. The authors remark that “more investigation is needed in order to reach an ‘optimal’ scheme”.

Reviewer: Myron Sussman (Bethel Park)

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N06 | Finite difference methods for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

76M30 | Variational methods applied to problems in fluid mechanics |