Analysis of a new stabilized finite element method for the reaction–convection–diffusion equations with a large reaction coefficient

Duan, Huo Yuan; Hsieh, Po Wen; Tan, Roger C. E.; Yang, Shao-Ming

doi:10.1016/j.cma.2012.07.018

Cited by 17 publications

(23 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In our paper, the analysis is more general, e.g., we are dealing with the test function with convection in the stabilization term; in our paper, we also establish a more refined and better results, comparing the error bound results in Theorem 2 of our paper with the results in Theorem 4 and Remark 4 of Ref. [24]; in our paper, we have provided a better L2 error estimates. Moreover, in [24], no results are available for the case σ = 0.…”

Section: Theoremmentioning

confidence: 77%

“…[24]; in our paper, we have provided a better L2 error estimates. Moreover, in [24], no results are available for the case σ = 0. Most importantly, the method in this paper is novel, more applicable, and performs better than the method in Ref.…”

Section: Theoremmentioning

confidence: 86%

“…The analysis in this article is inspired by the one in [24]. In our paper, the analysis is more general, e.g., we are dealing with the test function with convection in the stabilization term; in our paper, we also establish a more refined and better results, comparing the error bound results in Theorem 2 of our paper with the results in Theorem 4 and Remark 4 of Ref.…”

Section: Theoremmentioning

confidence: 88%

“…The theoretical and numerical results show that the method in Ref. [24] can well solve the convection-dominated problem with the coexistence of small diffusivity and large reaction. For this type of problem,…”

Section: Introductionmentioning

confidence: 94%

See 3 more Smart Citations

A new stabilized finite element method for advection‐diffusion‐reaction equations

Duan

Qiu

2015

Numerical Methods Partial

View full text Add to dashboard Cite

In this article, a new stabilized finite element method is proposed and analyzed for advection-diffusionreaction equations. The key feature is that both the mesh-dependent Péclet number and the mesh-dependent Damköhler number are reasonably incorporated into the newly designed stabilization parameter. The error estimates are established, where, up to the regularity-norm of the exact solution, the explicit-dependence of the diffusivity, advection, reaction, and mesh size (or the dependence of the mesh-dependent Péclet number and the mesh-dependent Damköhler number) is revealed. Such dependence in the error bounds provides a mathematical justification on the effectiveness of the proposed method for any values of diffusivity, advection, dissipative reaction, and mesh size. Numerical results are presented to illustrate the performance of the method.

show abstract

Section: Theoremmentioning

confidence: 77%

Section: Theoremmentioning

confidence: 86%

Section: Theoremmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 94%

See 2 more Smart Citations

A new stabilized finite element method for advection‐diffusion‐reaction equations

Duan

Qiu

2015

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

“…A major difficulty encountered in numerically solving the ADDEs is instability due to the existence of the advection terms 11) . A large reaction term is also a source of numerical instability 12) . Another difficulty, an issue particular to the ADDEs on connected graphs, is consistent and efficient treatment of the IBCs.…”

Section: Introductionmentioning

confidence: 99%

Mathematical Analysis on a Conforming Finite Element Scheme for Advection-Dispersion-Decay Equations on Connected Graphs

2014

View full text Add to dashboard Cite

Theoretical stability and error analysis on a Conforming Petrov-Galerkin Finite Element (CPGFE) scheme with the fitting technique for solving the Advection-Dispersion-Decay Equations (ADDEs) on connected graphs is performed. This paper is the first research paper that applies the concept of the discrete Green's function (DGF) to error analysis on a numerical scheme for the ADDEs on connected graphs. Firstly, the stability analysis shows that the scheme is unconditionally stable in space for steady problems and is stable in both space and time for unsteady problems if the temporal term is appropriately discretized with a lumping technique. Secondly, basic properties of the DGF on connected graphs, which provide key mathematical tools in the error analysis, are presented. The error analysis with the DGF reveals a direct relationship between the regularity conditions on the known functions and accuracy of the scheme, explicitly indicating that the accuracy of the scheme is strongly influenced by the accuracy of the discretized known functions. The error analysis also shows that the scheme is uniformly-convergent in the L ¥ -error norm with respect to the diffusivity, which cannot be achieved in the conventional numerical schemes. This unique and remarkable property is a significant advantage of the present CPGFE scheme over the conventional ones.

show abstract

A New Edge Stabilization Method for the Convection-Dominated Diffusion-Convection Equations

Duan

2018

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Analysis of a new stabilized finite element method for the reaction–convection–diffusion equations with a large reaction coefficient

Cited by 17 publications

References 18 publications

A new stabilized finite element method for advection‐diffusion‐reaction equations

A new stabilized finite element method for advection‐diffusion‐reaction equations

Mathematical Analysis on a Conforming Finite Element Scheme for Advection-Dispersion-Decay Equations on Connected Graphs

A New Edge Stabilization Method for the Convection-Dominated Diffusion-Convection Equations

Contact Info

Product

Resources

About