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Multigrid method for coupled semilinear elliptic equation
Author(s) -
Xu Fei,
Ma Hongkun,
Zhai Jian
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5543
Subject(s) - multigrid method , mathematics , lipschitz continuity , finite element method , boundary value problem , partial differential equation , elliptic partial differential equation , sequence (biology) , elliptic curve , bounded function , mathematical analysis , elliptic operator , nonlinear system , physics , genetics , quantum mechanics , biology , thermodynamics
This paper introduces a kind of multigrid finite element method for the coupled semilinear elliptic equations. Instead of the common way of directly solving the coupled semilinear elliptic problems on some fine spaces, the presented method transforms the solution of the coupled semilinear elliptic problem into a series of solutions of the corresponding decoupled linear boundary value problems on the sequence of multilevel finite element spaces and some coupled semilinear elliptic problems on a very low dimensional space. The decoupled linearized boundary value problems can be solved by some multigrid iterations efficiently. The optimal error estimate and optimal computational work are proved theoretically and demonstrated numerically. Moreover, the requirement of bounded second‐order derivatives of the nonlinear term in the existing multigrid method is reduced to a Lipschitz continuous condition in the proposed method.