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Local and parallel multigrid method for semilinear Neumann problem with nonlinear boundary condition

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Abstract

A novel local and parallel multigrid method is proposed in this study for solving the semilinear Neumann problem with nonlinear boundary condition. Instead of solving the semilinear Neumann problem directly in the fine finite element space, we transform it into a linear boundary value problem defined in each level of a multigrid sequence and a small-scale semilinear Neumann problem defined in a low-dimensional correction subspace. Furthermore, the linear boundary value problem can be efficiently solved using local and parallel methods. The proposed process derives an optimal error estimate with linear computational complexity. Additionally, compared with existing multigrid methods for semilinear Neumann problems that require bounded second order derivatives of nonlinear terms, ours only needs bounded first order derivatives. A rigorous theoretical analysis is proposed in this paper, which differs from the maturely developed theories for equations with Dirichlet boundary conditions.

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Authors and Affiliations

  1. Beijing Institute for Scientific and Engineering Computing, Faculty of Science, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Fei Xu & Bingyi Wang

  2. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    Manting Xie

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  1. Fei Xu
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  2. Bingyi Wang
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Correspondence to Manting Xie.

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This work was supported by the Beijing Municipal Natural Science Foundation (Grant No. 1232001) General projects of science and technology plan of Beijing Municipal Education Commission (Grant No. KM202110005011) and the National Natural Science Foundation of China (Grant Nos. 11801021).

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Xu, F., Wang, B. & Xie, M. Local and parallel multigrid method for semilinear Neumann problem with nonlinear boundary condition. Numer Algor 96, 185–210 (2024). https://doi.org/10.1007/s11075-023-01643-5

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