TY - JOUR
T1 - Numerical studies of adaptive finite element methods for two dimensional convection-dominated problems
AU - Sun, Pengtao
AU - Chen, Long
AU - Xu, Jinchao
N1 - Funding Information:
Acknowledgements Pengtao Sun was supported by Research Development Award of University of Nevada Las Vegas 2220-320-980C. Long Chen was supported in part by NSF Grant DMS-0811272, and in part by NIH Grant P50GM76516 and R01GM75309. Jinchao Xu was supported in part by NSF Grant DMS-0609727 and Alexander H. Humboldt Foundation.
PY - 2010/4
Y1 - 2010/4
N2 - 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.
AB - 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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U2 - 10.1007/s10915-009-9337-6
DO - 10.1007/s10915-009-9337-6
M3 - Article
AN - SCOPUS:77951104721
SN - 0885-7474
VL - 43
SP - 24
EP - 43
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
IS - 1
ER -