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The numerical methods for the coupled fluid flow under the leak interface condition of the friction-type

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The friction-type (or called barrier-type) leak interface condition (FLIC) is proposed to model the viscous fluid through a perforated membrane with a threshold permeability, where the flow passes through the perforations only when the stress difference on the membrane is above a threshold. This work establishes a comprehensive study on several numerical approaches for the Stokes/Stokes coupled flow under the FLIC, including the projection, regularization (or called penalty), and domain decomposition methods. In the continuous sense, first, we revisit the well-posedness, introduce the Lagrange multiplier formulation, and show the convergence of the projection method, i.e., the Uzawa algorithm. Second, we approximate the variational inequality by the regularization technique. We derive the approximation error and discuss the applicability of the Picard iteration to the nonlinear regularization problem. And third, a domain decomposition algorithm is proposed to decouple the system into two Stokes problems with the Dirichlet and friction-type leak boundary conditions on \(\Gamma \), respectively. In the discrete sense, we apply the finite element method with P1b/P1-element and enforce the interface condition by mass-lumping technique. We obtain the error estimates of the finite element approximation. The convergence of the discrete version of the projection methods (including the Uzawa and Active/Inactive set algorithms), the regularization problem, and the domain decomposition algorithm are investigated as with the continuous cases. We carry out several numerical experiments to confirm the theoretical results.

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Authors and Affiliations

  1. Institute of Fundamental and Frontier Sciences, School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 610051, People’s Republic of China

    Guanyu Zhou

  2. School of Mathematics and Statistics and Xi’an Key Laboratory of Scientific Computation and Applied Statistics, Northwestern Polytechnical University, Xi’an, 710129, People’s Republic of China

    Feifei Jing

  3. The Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo, 153-8914, Japan

    Takahito Kashiwabara

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  1. Guanyu Zhou
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Correspondence to Feifei Jing.

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The research of Guanyu Zhou was supported by the National Natural Science Foundation of China (No. 12171071 and No. 12071061) and the Central Guidance on Local Science and Technology Development Fund of Sichuan Province (2021ZYD0002). The research of Feifei Jing is supported by the National Natural Science Foundation of China (No. 12001413 and No. 12171415) and the Postdoctoral Science Foundation of China (No. 2021M692647). The research of Takahito Kashiwabara was supported by JSPS Grant-in-Aid for Early-Career Scientists (No. 20K14357).

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Zhou, G., Jing, F. & Kashiwabara, T. The numerical methods for the coupled fluid flow under the leak interface condition of the friction-type. Numer. Math. 153, 729–773 (2023). https://doi.org/10.1007/s00211-023-01348-w

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