Accepted

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

Pengtao Sun, Long Chen, and Jinchao Xu

Journal of Scientific Computing

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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.