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Unconditional superconvergence analysis of an H 1 ‐galerkin mixed finite element method for nonlinear Sobolev equations
Author(s) -
Shi Dongyang,
Wang Junjun,
Yan Fengna
Publication year - 2018
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22189
Subject(s) - mathematics , superconvergence , sobolev space , uniqueness , norm (philosophy) , finite element method , nonlinear system , discontinuous galerkin method , galerkin method , mathematical analysis , physics , quantum mechanics , political science , law , thermodynamics
An efficient H 1 ‐Galerkin mixed finite element method (MFEM) is presented with E Q 1 rotand zero order Raviart‐Thomas elements for the nonlinear Sobolev equations. On one hand, the existence and uniqueness of the solutions of the semidiscrete approximation scheme are proved and the super close results of order O ( h 2 ) for the original variable u in a broken H 1 norm and the auxiliary variableq → = a ( u ) ∇ u t + b ( u ) ∇ u in H ( div ; Ω ) norm are deduced without the boundedness of the numerical solution inL ∞ ‐norm. Conversely, a linearized Crank‐Nicolson fully discrete scheme with the unconditional super close property O ( h 2 + τ 2 ) is also developed through a new approach, while previous literature always require certain time step conditions (see the references below). Finally, a numerical experiment is included to illustrate the feasibility of the proposed method. Here h is the subdivision parameter and τ is the time step.