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An optimal nonlinear Galerkin method with mixed finite elements for the steady Navier‐Stokes equations
Author(s) -
He Yinnian,
Wang Aiwen,
Chen Zhangxin,
Li Kaitai
Publication year - 2003
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.10074
Subject(s) - galerkin method , mathematics , finite element method , nonlinear system , discontinuous galerkin method , partial differential equation , mathematical analysis , grid , navier–stokes equations , rate of convergence , pressure correction method , compressibility , space (punctuation) , convergence (economics) , geometry , mechanics , physics , channel (broadcasting) , quantum mechanics , economic growth , economics , thermodynamics , engineering , linguistics , philosophy , electrical engineering
An optimal nonlinear Galerkin method with mixed finite elements is developed for solving the two‐dimensional steady incompressible Navier‐Stokes equations. This method is based on two finite element spaces X H and X h for the approximation of velocity, defined on a coarse grid with grid size H and a fine grid with grid size h ≪ H , respectively, and a finite element space M h for the approximation of pressure. We prove that the difference in appropriate norms between the solutions of the nonlinear Galerkin method and a classical Galerkin method is of the order of H 5 . If we choose H = O ( h 2/5 ), these two methods have a convergence rate of the same order. We numerically demonstrate that the optimal nonlinear Galerkin method is efficient and can save a large amount of computational time. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 762–775, 2003.