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A Novel Arbitrary Lagrangian–Eulerian Finite Element Method for a Mixed Parabolic Problem in a Moving Domain

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Abstract

In this paper, a novel arbitrary Lagrangian–Eulerian (ALE) mapping, thus a novel ALE-mixed finite element method (FEM), is developed and analyzed for a type of mixed parabolic equations in a moving domain. By means of a specific stabilization technique, the mixed finite element of a stable Stokes-pair is utilized to discretize this problem on the ALE description, and, stability and a nearly optimal convergence results are obtained for both semi- and fully discrete ALE finite element approximations. Numerical experiments are carried out to validate all theoretical results. The developed novel ALE–FEM can be also similarly extended to a transient porous (Darcy’s) fluid flow problem in a moving domain as well as to Stokes/Darcy- or Stokes/Biot moving interface problem in the future.

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Acknowledgements

The authors were partially supported by NSF Grant DMS-1418806.

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  1. Department of Mathematical Sciences, University of Nevada Las Vegas, 4505 Maryland Parkway, Las Vegas, NV, 89154, USA

    Rihui Lan & Pengtao Sun

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  1. Rihui Lan
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  2. Pengtao Sun
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Correspondence to Pengtao Sun.

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Lan, R., Sun, P. A Novel Arbitrary Lagrangian–Eulerian Finite Element Method for a Mixed Parabolic Problem in a Moving Domain. J Sci Comput 85, 9 (2020). https://doi.org/10.1007/s10915-020-01315-9

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  • DOI: https://doi.org/10.1007/s10915-020-01315-9

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