TY - JOUR T1 - An Adaptive Multigrid Method for Semilinear Elliptic Equations AU - Xu , Fei AU - Huang , Qiumei AU - Chen , Shuangshuang AU - Bing , Tao JO - East Asian Journal on Applied Mathematics VL - 4 SP - 683 EP - 702 PY - 2019 DA - 2019/10 SN - 9 DO - http://doi.org/10.4208/eajam.061118.070419 UR - https://global-sci.org/intro/article_detail/eajam/13327.html KW - Semilinear elliptic problem, adaptive multigrid method, convergence, optimal complexity. AB -
An adaptive multigrid method for semilinear elliptic equations based on adaptive multigrid methods and on multilevel correction methods is developed. The solution of a semilinear problem is reduced to a series of linearised elliptic equations on the sequence of adaptive finite element spaces and semilinear elliptic problems on a very low dimensional space. The corresponding linear elliptic equations are solved by an adaptive multigrid method. The convergence and optimal complexity of the algorithm is proved and illustrating numerical examples are provided. The method requires only the Lipschitz continuity of the nonlinear term. This approach can be extended to other nonlinear problems, including Navier-Stokes problems and phase field models.