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Volume 31, Issue 5
Monotonicity Correction for the Finite Element Method of Anisotropic Diffusion Problems

Boyang Yu, Hongtao Yang, Yonghai Li & Guangwei Yuan

Commun. Comput. Phys., 31 (2022), pp. 1489-1524.

Published online: 2022-05

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  • Abstract

We apply the monotonicity correction to the finite element method for the anisotropic diffusion problems, including linear and quadratic finite elements on triangular meshes. When formulating the finite element schemes, we need to calculate the integrals on every triangular element, whose results are the linear combination of the two-point pairs. Then we decompose the integral results into the main and remaining parts according to coefficient signs of two-point pairs. We apply the nonlinear correction to the positive remaining parts and move the negative remaining parts to the right side of the finite element equations. Finally, the original stiffness matrix can be transformed into a nonlinear M-matrix, and the corrected schemes have the positivity-preserving property. We also give the monotonicity correction to the time derivative term for the time-dependent problems. Numerical experiments show that the corrected finite element method has monotonicity and maintains the convergence order of the original schemes in $H^1$-norm and $L^2$-norm, respectively.

  • AMS Subject Headings

65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-31-1489, author = {Yu , BoyangYang , HongtaoLi , Yonghai and Yuan , Guangwei}, title = {Monotonicity Correction for the Finite Element Method of Anisotropic Diffusion Problems}, journal = {Communications in Computational Physics}, year = {2022}, volume = {31}, number = {5}, pages = {1489--1524}, abstract = {

We apply the monotonicity correction to the finite element method for the anisotropic diffusion problems, including linear and quadratic finite elements on triangular meshes. When formulating the finite element schemes, we need to calculate the integrals on every triangular element, whose results are the linear combination of the two-point pairs. Then we decompose the integral results into the main and remaining parts according to coefficient signs of two-point pairs. We apply the nonlinear correction to the positive remaining parts and move the negative remaining parts to the right side of the finite element equations. Finally, the original stiffness matrix can be transformed into a nonlinear M-matrix, and the corrected schemes have the positivity-preserving property. We also give the monotonicity correction to the time derivative term for the time-dependent problems. Numerical experiments show that the corrected finite element method has monotonicity and maintains the convergence order of the original schemes in $H^1$-norm and $L^2$-norm, respectively.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0197}, url = {http://global-sci.org/intro/article_detail/cicp/20512.html} }
TY - JOUR T1 - Monotonicity Correction for the Finite Element Method of Anisotropic Diffusion Problems AU - Yu , Boyang AU - Yang , Hongtao AU - Li , Yonghai AU - Yuan , Guangwei JO - Communications in Computational Physics VL - 5 SP - 1489 EP - 1524 PY - 2022 DA - 2022/05 SN - 31 DO - http://doi.org/10.4208/cicp.OA-2021-0197 UR - https://global-sci.org/intro/article_detail/cicp/20512.html KW - The finite element method, nonlinear M-matrix, monotonicity correction, positivity-preserving property, two-point pair. AB -

We apply the monotonicity correction to the finite element method for the anisotropic diffusion problems, including linear and quadratic finite elements on triangular meshes. When formulating the finite element schemes, we need to calculate the integrals on every triangular element, whose results are the linear combination of the two-point pairs. Then we decompose the integral results into the main and remaining parts according to coefficient signs of two-point pairs. We apply the nonlinear correction to the positive remaining parts and move the negative remaining parts to the right side of the finite element equations. Finally, the original stiffness matrix can be transformed into a nonlinear M-matrix, and the corrected schemes have the positivity-preserving property. We also give the monotonicity correction to the time derivative term for the time-dependent problems. Numerical experiments show that the corrected finite element method has monotonicity and maintains the convergence order of the original schemes in $H^1$-norm and $L^2$-norm, respectively.

Boyang Yu, Hongtao Yang, Yonghai Li & Guangwei Yuan. (2022). Monotonicity Correction for the Finite Element Method of Anisotropic Diffusion Problems. Communications in Computational Physics. 31 (5). 1489-1524. doi:10.4208/cicp.OA-2021-0197
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