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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
Abstract 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.