Published

Numerical studies of adaptive finite element methods for two dimensional convection-dominated problems

Pengtao Sun, Long Chen, and Jinchao Xu

Journal of Scientific Computing, 43(1):24-43, 2010.

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ABSTRACT: In this paper, we study the stability and accuracy of adaptive finite element methods for the convection-dominated convection-diffusion-reaction problem in two dimensions. Through various numerical examples, we show that the mesh adaptivity driven by improving 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 anisotropic mesh adaptation with the homotopy of the diffusion coefficient. Numerical experiments show that our MHAFEM can efficiently capture the solutions' singularities arising in boundary or interior layers and produce accurate solutions.