TY - JOUR
T1 - Simplex-averaged finite element methods for \bfitH (GRAD), \bfitH (CURL), and \bfitH (DIV) convection-diffusion problems
AU - Wu, Shuonan
AU - Xu, Jinchao
N1 - Funding Information:
\ast Received by the editors November 16, 2018; accepted for publication (in revised form) November 20, 2019; published electronically February 27, 2020. https://doi.org/10.1137/18M1227196 \bfF \bfu \bfn \bfd \bfi \bfn \bfg : The work of the first author was partially supported by a startup grant from Peking University. The work of the second author was partially supported by the US Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program grant DE-SC0014400. \dagger School of Mathematical Sciences, Peking University, Beijing 100871, China ([email protected]. edu.cn, http://dsec.pku.edu.cn/\sim snwu). \ddagger Department of Mathematics, Pennsylvania State University, University Park, PA 16802 (xu@ math.psu.edu, http://www.math.psu.edu/xu/).
Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.
PY - 2020
Y1 - 2020
N2 - This paper is devoted to the construction and analysis of the finite element approximations for the H(D) convection-diffusion problems, where D can be chosen as grad, curl, or div in the three dimension (3D) case. An essential feature of these constructions is to properly average the PDE coefficients on the subsimplexes. The schemes are of the class of exponential fitting methods that result in special upwind schemes when the diffusion coefficient approaches to zero. Their well-posedness are established for sufficiently small mesh size assuming that the convection-diffusion problems are uniquely solvable. Convergence of first order is derived under minimal smoothness of the solution. Some numerical examples are given to demonstrate the robustness and effectiveness for general convection-diffusion problems.
AB - This paper is devoted to the construction and analysis of the finite element approximations for the H(D) convection-diffusion problems, where D can be chosen as grad, curl, or div in the three dimension (3D) case. An essential feature of these constructions is to properly average the PDE coefficients on the subsimplexes. The schemes are of the class of exponential fitting methods that result in special upwind schemes when the diffusion coefficient approaches to zero. Their well-posedness are established for sufficiently small mesh size assuming that the convection-diffusion problems are uniquely solvable. Convergence of first order is derived under minimal smoothness of the solution. Some numerical examples are given to demonstrate the robustness and effectiveness for general convection-diffusion problems.
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U2 - 10.1137/18M1227196
DO - 10.1137/18M1227196
M3 - Article
AN - SCOPUS:85093762138
SN - 0036-1429
VL - 58
SP - 884
EP - 906
JO - SIAM Journal on Numerical Analysis
JF - SIAM Journal on Numerical Analysis
IS - 1
ER -