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Adaptive Flux-Only Least-Squares Finite Element Methods for Linear Transport Equations

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In this paper, two flux-only least-squares finite element methods (LSFEM) for the linear hyperbolic transport problem are developed. The transport equation often has discontinuous solutions and discontinuous inflow boundary conditions, but with continuous normal component of the flux across the mesh interfaces. Continuous finite element spaces are used to approximate the solution in traditional LSFEMs. This will introduce unnecessary error and serious overshooting. In Liu and Zhang (Comput Methods Appl Mech Eng 366:113041, 2020), we reformulate the equation by introducing a new flux variable to separate the continuity requirements of the flux and the solution. Realizing that the Raviart-Thomas mixed element space has enough degrees of freedom to approximate both the flux and its divergence, we eliminate the solution from the system and get two flux-only formulations, and develop corresponding LSFEMs. The solution then is recovered by simple post-processing methods using its relation with the flux. These two versions of flux-only LSFEMs use less DOFs than the method we developed in Liu and Zhang (2020). Similar to the LSFEM developed in Liu and Zhang (2020), both flux-only LSFEMs can handle discontinuous solutions better than the traditional continuous polynomial approximations. We show the existence, uniqueness, a priori and a posteriori error estimates of the proposed methods. With adaptive mesh refinements driven by the least-squares a posteriori error estimators, the solution can be accurately approximated even when the mesh is not aligned with discontinuity. The overshooting phenomenon is very mild if a piecewise constant reconstruction of the solution is used. Extensive numerical tests are done to show the effectiveness of the methods developed in the paper.

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Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR, China

    Qinjie Liu & Shun Zhang

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  1. Qinjie Liu
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  2. Shun Zhang
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Correspondence to Shun Zhang.

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This work was supported in part by Hong Kong Research Grants Council under the GRF Grant Project No. CityU 11305319 and a China Sichuan Provincial Science and Technology Research Grant 2018JY0187 via Chengdu Research Institute of City University of Hong Kong.

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Liu, Q., Zhang, S. Adaptive Flux-Only Least-Squares Finite Element Methods for Linear Transport Equations. J Sci Comput 84, 26 (2020). https://doi.org/10.1007/s10915-020-01269-y

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  • DOI: https://doi.org/10.1007/s10915-020-01269-y

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