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Superconvergence analysis of a linearized three‐step backward differential formula finite element method for nonlinear Sobolev equation
Author(s) -
Wang Junjun,
Li Qingfu
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.5589
Subject(s) - superconvergence , mathematics , sobolev space , finite element method , mathematical analysis , bilinear interpolation , norm (philosophy) , galerkin method , discontinuous galerkin method , nonlinear system , partial differential equation , bounded function , statistics , physics , quantum mechanics , political science , law , thermodynamics
A linearized three‐step backward differential formula (BDF) Galerkin finite element method (FEM) is developed for nonlinear Sobolev equation with bilinear element. Temporal error and spatial error are discussed through introducing a time‐discrete system. Solutions of the time‐discrete system are bounded in H 2 ‐norm by the temporal error. Superconvergence results of order O ( h 2 + τ 3 ) in H 1 ‐norm for the original variable are deduced based on the spatial error. Some new tricks are utilized to get higher order of the temporal error and the spatial error. At last, two numerical examples are provided to support the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.