Adaptive Flux-Only Least-Squares Finite Element Methods for Linear Transport Equations
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Adaptive Flux-Only Least-Squares Finite Element Methods for Linear Transport Equations Qinjie Liu1 · Shun Zhang1 Received: 22 July 2019 / Revised: 27 April 2020 / Accepted: 18 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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. Keywords Least-squares finite element method · Linear transport equation · Error estimate · Discontinuous solution · Overshooting · Adaptive LSFEM
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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Shun Zhang [email protected] Qinjie Liu [email protected]
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Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR, China 0123456789().: V,-vol
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Page 2 of 22
Journal of Scientific Computing
(2020) 84:26
1 Introduction Consider the following linear transport equation in conservative form: ∇ · (βu) + γ u = f in , u = g on − ,
(1.1)
with − the inflow boundary and β an advection field. In Section 2, we present detailed descriptions of the equation. Let μ = γ + ∇ · β, we can have an equivalent non-conservative reformulation: β · ∇u + μu = f
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