Numerical Studies of Adaptive Finite Element Methods for Two Dimensional Convection-Dominated Problems |
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Authors: | Pengtao Sun Long Chen Jinchao Xu |
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Affiliation: | 1.Department of Mathematical Sciences,University of Nevada, Las Vegas,Las Vegas,USA;2.Department of Mathematics,University of California, Irvine,Irvine,USA;3.Department of Mathematics,Pennsylvania State University,University Park,USA |
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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 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. |
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