Volume 13, Issue 2
Error Analysis of SAV Finite Element Method to Phase Field Crystal Model

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 372-399.

Published online: 2020-03

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• Abstract

In this paper, we construct and analyze an energy stable scheme by combining the latest developed scalar auxiliary variable (SAV) approach and linear finite element method (FEM) for phase field crystal (PFC) model, and show rigorously that the scheme is first-order in time and second-order in space for the $L^2$ and $H^{-1}$ gradient flow equations. To reduce efficiently computational cost and capture accurately the phase interface, we give a simple adaptive strategy, equipped with a posteriori gradient estimator, i.e.,  $L^2$ norm of the recovered gradient. Extensive numerical experiments are presented to verify our theoretical results and to demonstrate the effectiveness and accuracy of our proposed method.

• Keywords

Linear finite element method, scalar auxiliary variable approach, phase field crystal model, error analysis, energy stability, adaptive method.

65M12, 65M50, 65M60, 35Q56

kaijiang@xtu.edu.cn (Kai Jiang)

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@Article{NMTMA-13-372, author = {Wang , Liupeng and Huang , Yunqing and Jiang , Kai }, title = {Error Analysis of SAV Finite Element Method to Phase Field Crystal Model}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {13}, number = {2}, pages = {372--399}, abstract = {

In this paper, we construct and analyze an energy stable scheme by combining the latest developed scalar auxiliary variable (SAV) approach and linear finite element method (FEM) for phase field crystal (PFC) model, and show rigorously that the scheme is first-order in time and second-order in space for the $L^2$ and $H^{-1}$ gradient flow equations. To reduce efficiently computational cost and capture accurately the phase interface, we give a simple adaptive strategy, equipped with a posteriori gradient estimator, i.e.,  $L^2$ norm of the recovered gradient. Extensive numerical experiments are presented to verify our theoretical results and to demonstrate the effectiveness and accuracy of our proposed method.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0110}, url = {http://global-sci.org/intro/article_detail/nmtma/15465.html} }
TY - JOUR T1 - Error Analysis of SAV Finite Element Method to Phase Field Crystal Model AU - Wang , Liupeng AU - Huang , Yunqing AU - Jiang , Kai JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 372 EP - 399 PY - 2020 DA - 2020/03 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0110 UR - https://global-sci.org/intro/article_detail/nmtma/15465.html KW - Linear finite element method, scalar auxiliary variable approach, phase field crystal model, error analysis, energy stability, adaptive method. AB -

In this paper, we construct and analyze an energy stable scheme by combining the latest developed scalar auxiliary variable (SAV) approach and linear finite element method (FEM) for phase field crystal (PFC) model, and show rigorously that the scheme is first-order in time and second-order in space for the $L^2$ and $H^{-1}$ gradient flow equations. To reduce efficiently computational cost and capture accurately the phase interface, we give a simple adaptive strategy, equipped with a posteriori gradient estimator, i.e.,  $L^2$ norm of the recovered gradient. Extensive numerical experiments are presented to verify our theoretical results and to demonstrate the effectiveness and accuracy of our proposed method.

Liupeng Wang, Yunqing Huang & Kai Jiang. (2020). Error Analysis of SAV Finite Element Method to Phase Field Crystal Model. Numerical Mathematics: Theory, Methods and Applications. 13 (2). 372-399. doi:10.4208/nmtma.OA-2019-0110
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