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Superconvergence analysis for nonlinear reaction‐diffusion equation with BDF‐FEM
Author(s) -
Wang Junjun,
Shi Dongyang
Publication year - 2020
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.6229
Subject(s) - superconvergence , mathematics , norm (philosophy) , finite element method , uniqueness , reaction–diffusion system , mathematical analysis , monotonic function , nonlinear system , galerkin method , partial differential equation , discontinuous galerkin method , law , physics , quantum mechanics , political science , thermodynamics
An implicit backward differential formula (BDF) scheme is constructed for nonlinear reaction‐diffusion equation, and superconvergence results are studied with the Galerkin finite element method (FEM). The existence and uniqueness of the numerical solution are given by using of the function's monotonicity. Splitting technique is utilized to get rid of the ratio between the time step τ and the subdivision parameter h . Temporal error estimates inH 2 ‐norm are derived, which implies the boundedness of the solutions about the time‐discrete equations inH 2 ‐norm. Unconditional spatial error estimates inL 2 ‐norm are deduced, which help bound the numerical solutions inL ∞ ‐norm. The unconditional superconvergent property ofu ninH 1 ‐norm with order O ( h 2 + τ 2 ) is obtained by the relationship between the classic Ritz projection operator and the the corresponding interpolation operator. The global superconvergent property is deduced through the above results. Two numerical examples show the validity of the theoretical analysis.