Convection Dominated Problem

Convection-Dominated Convection-Diffusion Equation

"Accurate modeling of the interaction between convective and diffusive processes is the most ubiquitous and challenging task in the numerical approximation of partial differential equations."

From K.W. Morton Numerical Solution of Convection-Diffusion Problems

• L. Chen and J. Xu. Stability and accuracy of adapted finite element methods for singularly perturbed problems. Numerische Mathematik, 109(2): 167 - 191, 2008.
• L. Chen and J. Xu. An optimal streamline diffusion finite element method for a singularly perturbed problem. In AMS Contemporary Mathematics Series: Recent Advances in Adaptive Computation, volume 383, pages 236-246, Hangzhou, 2005.

In these two papers, it is shown that an unstable scheme on uniform grid can be made stable on a properly adapted grid and the accuracy of the adapted scheme may depend on (perhaps surprisingly) the uniformity on the grid where the solution is smooth. A new streamline diffusion method with a second order convergence rate that is uniform with respect to the size of convection is developed and analyzed

• L. Chen, Y. Wang and J. Wu. Stability of a Streamline Diffusion Finite Element Methods for Turning Point Problems. Journal of Computational and Applied Mathematics, 220, 712 - 724, 2008.

The method and analysis is extended to turning-point problems.

• L. Chen, P. Sun, and J. Xu. Multilevel homotopic adaptive finite element methods for convection dominated problems. In The Proceedings for 15th Conferences for Domain Decomposition Methods, Lecture Notes in Computational Science and Engineering 40, pages 459-468. Springer, 2004.
• P. Sun, L. Chen, and J. Xu. Numerical studies of adaptive finite element methods for two dimensional convection-dominated problems. 43(1), 24-43. Journal of Scientific Computing. 2010.

A new homotopic multilevel adaptive techniques for convection dominated problem is proposed and shown to be able to efficiently capture the solutions' singularities arising in boundary or interior layers.