Abstract
The disproportionality in the problem parameters of the convection-diffusion-reaction equation may lead to the formation of layer structures in some parts of the problem domain which are difficult to resolve by the standard numerical algorithms. Therefore the use of a stabilized numerical method is inevitable. In this work, we employ and compare three classical stabilized finite element formulations, namely, the Streamline-Upwind Petrov-Galerkin (SUPG), Galerkin/Least-Squares (GLS), and Subgrid Scale (SGS) methods, and a recent Link-Cutting Bubble (LCB) strategy proposed by Brezzi and his coworkers for the numerical solution of the convection-diffusion-reaction equation, especially in the case of small diffusion. On the other hand, we also consider the pseudo residual-free bubble (PRFB) method as another alternative that is based on enlarging the finite element space by a set of appropriate enriching functions. We compare the performances of these stabilized methods on several benchmark problems. Numerical experiments show that the proposed methods are comparable and display good performance, especially in the convection-dominated regime. However, as the problem turns into reaction-dominated case, the PRFB method is slightly better than the other well-known and extensively used stabilized finite element formulations as they start to exhibit oscillations.
Highlights
The modeling of real world problems to predict physical quantities is an important subject in science and engineering
As the problem turns into reaction-dominated case, the pseudo residual-free bubble (PRFB) method is slightly better than the other well-known and extensively used stabilized finite element formulations as they start to exhibit oscillations
We remark that the Link-Cutting Bubble (LCB) method is employed only for one-dimensional problems as it is not designed for higher dimensions
Summary
The modeling of real world problems to predict physical quantities is an important subject in science and engineering. The lack of formalism in derivation of the stability parameters used in stabilized formulations has been seen as a major drawback of those strategies Another important and a more recent class of the stable discretization for the CDR problem is based on augmenting the finite element space by a set of suitable enriching functions. The locations of the subgrid nodes are of critical importance and 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] Their location is determined by minimizing the residual of a local differential problem defining the bubbles, with respect to the L1-norm.
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