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

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 define...

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Bibliographic Details
Published in:Numerical algorithms 2024-05, Vol.96 (1), p.185-210
Main Authors: Xu, Fei, Wang, Bingyi, Xie, Manting
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Boundary conditions
Boundary value problems
Computer Science
Decomposition
Dirichlet problem
Estimates
Finite element analysis
Multigrid methods
Neumann problem
Numeric Computing
Numerical Analysis
Original Paper
Partial differential equations
Theory of Computation
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Summary: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.
ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-023-01643-5