Two types of spurious oscillations at layers diminishing methods for convection–diffusion–reaction equations on surface

Abstract

In this article, we consider the steady convection–diffusion–reaction equation posed on general surface in convection-dominated regimes, which is an important problem in many models for simulating substances transport on ultrathin material and solid surfaces. Two types of spurious oscillations at layers diminishing (SOLD) methods are considered when the streamline diffusion method cannot completely eliminates the spurious oscillations at layers for convection-dominated problem with nonsmooth solution. One is the edge stabilization method that uses least square stabilization of the gradient jumps a across element boundaries, the other, a Petrov–Galerkin method, is called the Mizukami–Hughes method. An error analysis of the edge stabilization method on surface is provided. Finally, numerical experiments including the comparisons among these methods are presented to illustrate the accuracy and efficiency of the SOLD methods.

Acknowledgements

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped us to improve the results of this article.

Additional information

Funding

This work is in parts supported by the Excellent Doctor Innovation Program of Xinjiang University (No. XJUBSCX-2016006), the Graduate Student Research Innovation Program of Xinjiang (No. XJGRI2017013), the Research Fund from Key Laboratory of Xinjiang Province (No. 2017D04030), the NSF of China (Nos. 11671345, 11571385), the Special Project on High-performance Computing under the National Key R&D Program (No. 2016YFB0200604), and Guangdong Natural Science Foundation (No. 2017A030313017).