Volume 13, Issue 2
Modified Two-Grid Algorithm for Nonlinear Power-Law Conductivity in Maxwell's Problems with High Accuracy

Adv. Appl. Math. Mech., 13 (2021), pp. 481-502.

Published online: 2020-12

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• Abstract

In this paper, we develop the superconvergence analysis of two-grid algorithm by Crank-Nicolson finite element discrete scheme with the lowest Nédélec element for nonlinear power-law conductivity in Maxwell's problems. Our main contribution will have two parts. On the one hand, in order to overcome the difficulty of misconvergence of classical two-grid method by the lowest Nédélec element, we employ the Newton-type Taylor expansion at the superconvergent solutions for the nonlinear terms on coarse mesh, which is different from the numerical solution on the coarse mesh classically. On the other hand, we push the two-grid solution to high accuracy by the postprocessing interpolation technique. Such a design can improve the computational accuracy in space and decrease time consumption simultaneously. Based on this design, we can obtain the convergent rate $\mathcal{O}(\Delta t^2+h^2+H^{\frac{5}{2}})$ in three-dimension space, which means that the space mesh size satisfies $h=\mathcal{O}(H^\frac{5}{4})$. We also present two examples to verify our theorem.

65N30, 65N15

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@Article{AAMM-13-481, author = {Yao , Changhui and Li , Yanfei}, title = {Modified Two-Grid Algorithm for Nonlinear Power-Law Conductivity in Maxwell's Problems with High Accuracy}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {13}, number = {2}, pages = {481--502}, abstract = {

In this paper, we develop the superconvergence analysis of two-grid algorithm by Crank-Nicolson finite element discrete scheme with the lowest Nédélec element for nonlinear power-law conductivity in Maxwell's problems. Our main contribution will have two parts. On the one hand, in order to overcome the difficulty of misconvergence of classical two-grid method by the lowest Nédélec element, we employ the Newton-type Taylor expansion at the superconvergent solutions for the nonlinear terms on coarse mesh, which is different from the numerical solution on the coarse mesh classically. On the other hand, we push the two-grid solution to high accuracy by the postprocessing interpolation technique. Such a design can improve the computational accuracy in space and decrease time consumption simultaneously. Based on this design, we can obtain the convergent rate $\mathcal{O}(\Delta t^2+h^2+H^{\frac{5}{2}})$ in three-dimension space, which means that the space mesh size satisfies $h=\mathcal{O}(H^\frac{5}{4})$. We also present two examples to verify our theorem.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0371}, url = {http://global-sci.org/intro/article_detail/aamm/18494.html} }
TY - JOUR T1 - Modified Two-Grid Algorithm for Nonlinear Power-Law Conductivity in Maxwell's Problems with High Accuracy AU - Yao , Changhui AU - Li , Yanfei JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 481 EP - 502 PY - 2020 DA - 2020/12 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2019-0371 UR - https://global-sci.org/intro/article_detail/aamm/18494.html KW - Maxwell's equation, two-grid algorithm, Nédélec element, postprocessing, superconvergence. AB -

In this paper, we develop the superconvergence analysis of two-grid algorithm by Crank-Nicolson finite element discrete scheme with the lowest Nédélec element for nonlinear power-law conductivity in Maxwell's problems. Our main contribution will have two parts. On the one hand, in order to overcome the difficulty of misconvergence of classical two-grid method by the lowest Nédélec element, we employ the Newton-type Taylor expansion at the superconvergent solutions for the nonlinear terms on coarse mesh, which is different from the numerical solution on the coarse mesh classically. On the other hand, we push the two-grid solution to high accuracy by the postprocessing interpolation technique. Such a design can improve the computational accuracy in space and decrease time consumption simultaneously. Based on this design, we can obtain the convergent rate $\mathcal{O}(\Delta t^2+h^2+H^{\frac{5}{2}})$ in three-dimension space, which means that the space mesh size satisfies $h=\mathcal{O}(H^\frac{5}{4})$. We also present two examples to verify our theorem.

Changhui Yao & Yanfei Li. (1970). Modified Two-Grid Algorithm for Nonlinear Power-Law Conductivity in Maxwell's Problems with High Accuracy. Advances in Applied Mathematics and Mechanics. 13 (2). 481-502. doi:10.4208/aamm.OA-2019-0371
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