High Order Compact Scheme with Multigrid Local Mesh
Refinement Procedure for Convection Diffusion Problems

Jun Zhang and Haiwei Sun
Department of Computer Science
University of Kentucky
773 Anderson Hall
Lexington, KY 40506-0046, USA

Jennifer J. Zhao
Department of Mathematics and Statistics
University of Michigan at Dearborn
Dearborn, MI 48128-1491, USA


We derive a new fourth order compact finite difference scheme which allows different meshsize in different coordinate directions for the two dimensional convection diffusion equation. A multilevel local mesh refinement strategy is used to deal with the local singularity problem. A corresponding multilevel multigrid method is designed to solve the resulting sparse linear system. Numerical experiments are conducted to show that the local mesh refinement strategy works well with the high order compact discretization scheme to recover high order accuracy for the computed solution. Our solution method is also shown to be effective and robust with respect to the level of mesh refinement and the anisotropy of the problems.

Key words: convection diffusion equation, singularity and boundary layer, local mesh refinement, multigrid method

Mathematics Subject Classification: 65N06, 65N55, 65F10

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This paper has been published in Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 41-42, pp. 4661--4674 (2002).

Technical Report 330-01, Department of Computer Science, University of Kentucky, Lexington, KY, 2001. This research was supported in part by the U.S. National Science Foundation under the grant CCR-9902022, CCR-9988165, CCR-0092532, and CCR-0117602.