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Volume 12, Issue 3
An Adaptive Immersed Finite Element Method with Arbitrary Lagrangian-Eulerian Scheme for Parabolic Equations in Time Variable Domains

Zhiming Chen, Zedong Wu & Yuanming Xiao

Int. J. Numer. Anal. Mod., 12 (2015), pp. 567-591.

Published online: 2015-12

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  • Abstract

We first propose an adaptive immersed finite element method based on the a posteriori error estimate for solving elliptic equations with non-homogeneous boundary conditions in general Lipschitz domains. The underlying finite element mesh need not fit the boundary of the domain. Optimal a priori error estimate of the proposed immersed finite element method is proved. The immersed finite element method is then used to solve parabolic problems in time variable domains together with an arbitrary Lagrangian-Eulerian (ALE) time discretization scheme. An a posteriori error estimate for the fully discrete immersed finite element method is derived which can be used to adaptively update the time step sizes and finite element meshes at each time step. Numerical experiments are reported to support the theoretical results.

  • Keywords

Immersed finite element, adaptive, a posteriori error estimate, time variable domain.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-12-567, author = {Zhiming Chen, Zedong Wu and Yuanming Xiao}, title = {An Adaptive Immersed Finite Element Method with Arbitrary Lagrangian-Eulerian Scheme for Parabolic Equations in Time Variable Domains}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2015}, volume = {12}, number = {3}, pages = {567--591}, abstract = {

We first propose an adaptive immersed finite element method based on the a posteriori error estimate for solving elliptic equations with non-homogeneous boundary conditions in general Lipschitz domains. The underlying finite element mesh need not fit the boundary of the domain. Optimal a priori error estimate of the proposed immersed finite element method is proved. The immersed finite element method is then used to solve parabolic problems in time variable domains together with an arbitrary Lagrangian-Eulerian (ALE) time discretization scheme. An a posteriori error estimate for the fully discrete immersed finite element method is derived which can be used to adaptively update the time step sizes and finite element meshes at each time step. Numerical experiments are reported to support the theoretical results.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/502.html} }
TY - JOUR T1 - An Adaptive Immersed Finite Element Method with Arbitrary Lagrangian-Eulerian Scheme for Parabolic Equations in Time Variable Domains AU - Zhiming Chen, Zedong Wu & Yuanming Xiao JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 567 EP - 591 PY - 2015 DA - 2015/12 SN - 12 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/502.html KW - Immersed finite element, adaptive, a posteriori error estimate, time variable domain. AB -

We first propose an adaptive immersed finite element method based on the a posteriori error estimate for solving elliptic equations with non-homogeneous boundary conditions in general Lipschitz domains. The underlying finite element mesh need not fit the boundary of the domain. Optimal a priori error estimate of the proposed immersed finite element method is proved. The immersed finite element method is then used to solve parabolic problems in time variable domains together with an arbitrary Lagrangian-Eulerian (ALE) time discretization scheme. An a posteriori error estimate for the fully discrete immersed finite element method is derived which can be used to adaptively update the time step sizes and finite element meshes at each time step. Numerical experiments are reported to support the theoretical results.

Zhiming Chen, Zedong Wu and Yuanming Xiao. (2015). An Adaptive Immersed Finite Element Method with Arbitrary Lagrangian-Eulerian Scheme for Parabolic Equations in Time Variable Domains. International Journal of Numerical Analysis and Modeling. 12 (3). 567-591. doi:
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