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Volume 35, Issue 1
On the Monotonicity of $Q^2$ Spectral Element Method for Laplacian on Quasi-Uniform Rectangular Meshes

Logan J. Cross & Xiangxiong Zhang

DOI: 10.4208/cicp.OA-2023-0206

Commun. Comput. Phys., 35 (2024), pp. 160-180.

Published online: 2024-01

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  • Abstract

The monotonicity of discrete Laplacian implies discrete maximum principle, which in general does not hold for high order schemes. The $Q^2$ spectral element method has been proven monotone on a uniform rectangular mesh. In this paper we prove the monotonicity of the $Q^2$ spectral element method on quasi-uniform rectangular meshes under certain mesh constraints. In particular, we propose a relaxed Lorenz’s condition for proving monotonicity.

  • Keywords

Inverse positivity, discrete maximum principle, high order accuracy, monotonicity, discrete Laplacian, quasi uniform meshes, spectral element method.

  • AMS Subject Headings

65N30, 65N06, 65N12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-35-160, author = {Cross , Logan J. and Zhang , Xiangxiong}, title = {On the Monotonicity of $Q^2$ Spectral Element Method for Laplacian on Quasi-Uniform Rectangular Meshes}, journal = {Communications in Computational Physics}, year = {2024}, volume = {35}, number = {1}, pages = {160--180}, abstract = {

The monotonicity of discrete Laplacian implies discrete maximum principle, which in general does not hold for high order schemes. The $Q^2$ spectral element method has been proven monotone on a uniform rectangular mesh. In this paper we prove the monotonicity of the $Q^2$ spectral element method on quasi-uniform rectangular meshes under certain mesh constraints. In particular, we propose a relaxed Lorenz’s condition for proving monotonicity.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0206}, url = {http://global-sci.org/intro/article_detail/cicp/22899.html} }
TY - JOUR T1 - On the Monotonicity of $Q^2$ Spectral Element Method for Laplacian on Quasi-Uniform Rectangular Meshes AU - Cross , Logan J. AU - Zhang , Xiangxiong JO - Communications in Computational Physics VL - 1 SP - 160 EP - 180 PY - 2024 DA - 2024/01 SN - 35 DO - http://doi.org/10.4208/cicp.OA-2023-0206 UR - https://global-sci.org/intro/article_detail/cicp/22899.html KW - Inverse positivity, discrete maximum principle, high order accuracy, monotonicity, discrete Laplacian, quasi uniform meshes, spectral element method. AB -

The monotonicity of discrete Laplacian implies discrete maximum principle, which in general does not hold for high order schemes. The $Q^2$ spectral element method has been proven monotone on a uniform rectangular mesh. In this paper we prove the monotonicity of the $Q^2$ spectral element method on quasi-uniform rectangular meshes under certain mesh constraints. In particular, we propose a relaxed Lorenz’s condition for proving monotonicity.

Cross , Logan J. and Zhang , Xiangxiong. (2024). On the Monotonicity of $Q^2$ Spectral Element Method for Laplacian on Quasi-Uniform Rectangular Meshes. Communications in Computational Physics. 35 (1). 160-180. doi:10.4208/cicp.OA-2023-0206
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