Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems

Abstract

It is known that the enrichment of the polynomial finite element space of degree 1 by bubble functions results in a stabilized scheme of the SUPG-type for the convection-diffusion-reaction problems. In particular, the residual-free bubbles (RFB) can assure stabilized methods, but they are usually difficult to compute, unless the configuration is simple. Therefore it is important to devise numerical algorithms that provide cheap approximations to the RFB functions, contributing a good stabilizing effect to the numerical method overall. Here we propose a stabilization technique based on the RFB method and particularly designed to treat the most interesting case of small diffusion. We replace the RFB functions by their cheap, yet efficient approximations which retain the same qualitative behavior. The approximate bubbles are computed on a suitable sub-grid, the choice of whose nodes are critical and determined by minimizing the residual of a local problem with respect to L 1 norm. The resulting numerical method has similar stability features with the RFB method for the whole range of problem parameters. This fact is also confirmed by numerical experiments. We also note that the location of the sub-grid nodes suggested by the strategy herein coincides with the one in Brezzi et al. (Math. Models Methods Appl. Sci. 13:445-461, 2003).

A Comparative Study on Stabilized Finite Element Methods for the Convection-Diffusion-Reaction Problems

“…Indeed, in a recent approach based on the RFB method, the bubble functions are replaced by their suitable approximate counterparts, the so-called pseudo bubbles [14,15]. The development and the details of that approach are given in the following section.…”

“…In that approach, the RFB functions are basically approximated by piecewise linear functions on an appropriately chosen simple subgrid established inside each element and then they are replaced by their approximate counterparts, the so-called pseudo residual-free bubbles, in the numerical computations. Here, the locations of the subgrid nodes are of critical importance and therefore they should be chosen specially so that the fine scale-effect of the exact solution can accurately be represented in the coarse scale numerical approximation [13][14][15]. In particular, their location is determined by minimizing the residual of a local differential problem defining the bubbles, with respect to the 1 -norm.…”

Finite difference approximations of multidimensional convection–diffusion–reaction problems with small diffusion on a special grid

“…However, it requires to solve a local differential equation which may not be easier than to solve the original one [15]. That observation has motivated the introduction of a further option so-called the Pseudo Residual-free Bubble (PRFB) method which approximates the bubble functions on a specially chosen subgrid [6,7,31,34,35]. Roughly speaking, such grid points can be constructed by minimizing the residual of a local differential equation with respect to L 1 norm so that small scale-effect of the exact solution could be accurately represented in the numerical approximation through the use of those approximate bubble functions [34].…”

“…That observation has motivated the introduction of a further option so-called the Pseudo Residual-free Bubble (PRFB) method which approximates the bubble functions on a specially chosen subgrid [6,7,31,34,35]. Roughly speaking, such grid points can be constructed by minimizing the residual of a local differential equation with respect to L 1 norm so that small scale-effect of the exact solution could be accurately represented in the numerical approximation through the use of those approximate bubble functions [34]. Alternatively, the Link-Cutting Bubbles (LCB) method that is based on the plain Galerkin variational formulation on a special grid was proposed by Brezzi et al in [5] and it could be viewed as a similar, yet interesting option for another stable discretization in 1D.…”

Finite difference approximations of multidimensional unsteady convection–diffusion–reaction equations

“…The main problem with this method is that it requires the solution of a local PDE. Inexpensive approximate solutions to this local problem were designed by several researchers [3][4][5]7,8]. Stabilized finite element methods for unsteady convectiondiffusion-reaction equations have been studied by several researchers.…”