Applications of the pseudo residual-free bubbles to the stabilization of the convection–diffusion–reaction problems in 2D

Abstract

A stabilized finite element method is studied herein for two-dimensional convection-diffusion-reaction problems. The method is based on the residual-free bubbles (RFB) method. However we replace the RFB functions by their cheap, yet efficient approximations computed on a specially chosen subgrid, which retain the same qualitative behavior. Since the correct spot of subgrid points plays a crucial role in the approximation, it is important to determine their optimal locations, which we do it through a minimization process with respect to the L 1 -norm. The resulting numerical method has similar stability features with the well-known stabilized methods in the literature for the whole range of problem parameters and this fact is also confirmed by numerical experiments.

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.…”

“…2 ) (see [15]). Thus the positions of , = 1, 2, 3, are determined by taking 1 = * 1 , 2 = max{1 − 1 , * * 2 } and 3 = max{1 − 1 , * * 3 }.…”

“…In this section, we shall display the computation of the pseudo residual-free bubbles and the construction of the associated subgrid nodes on which the bubble functions are approximated (see [15] for details). We note that the configuration of the subgrid nodes is of crucial importance and therefore should be determined in a special manner.…”

“…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].…”