Journal of Scientific Computing

, Volume 43, Issue 1, pp 24-43

First online:

Numerical Studies of Adaptive Finite Element Methods for Two Dimensional Convection-Dominated Problems

  • Pengtao SunAffiliated withDepartment of Mathematical Sciences, University of Nevada, Las Vegas Email author 
  • , Long ChenAffiliated withDepartment of Mathematics, University of California, Irvine
  • , Jinchao XuAffiliated withDepartment of Mathematics, Pennsylvania State University

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In this paper, we study the stability and accuracy of adaptive finite element methods for the convection-dominated convection-diffusion-reaction problem in the two-dimension space. Through various numerical examples on a type of layer-adapted grids (Shishkin grids), we show that the mesh adaptivity driven by accuracy alone cannot stabilize the scheme in all cases. Furthermore the numerical approximation is sensitive to the symmetry of the grid in the region where the solution is smooth. On the basis of these two observations, we develop a multilevel-homotopic-adaptive finite element method (MHAFEM) by combining streamline diffusion finite element method, anisotropic mesh adaptation, and the homotopy of the diffusion coefficient. We use numerical experiments to demonstrate that MHAFEM can efficiently capture boundary or interior layers and produce accurate solutions.

Convection-dominated convection-diffusion-reaction problem Layer-adapted Shishkin grids Streamline diffusion finite element Anisotropic adaptive mesh Multilevel homotopic adaptive finite element method