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A fully discrete nonlinear Galerkin method for the 3D Navier–Stokes equations
Author(s) -
Guermond J.L.,
Prudhomme Serge
Publication year - 2008
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.20287
Subject(s) - mathematics , mathematical analysis , galerkin method , nonlinear system , superconvergence , norm (philosophy) , boundary value problem , partial differential equation , discontinuous galerkin method , homogenization (climate) , navier–stokes equations , finite element method , political science , compressibility , law , thermodynamics , aerospace engineering , biodiversity , ecology , physics , engineering , quantum mechanics , biology
The purpose of this paper is twofold: (i) We show that the Fourier‐based Nonlinear Galerkin Method (NLGM) constructs suitable weak solutions to the periodic Navier–Stokes equations in three space dimensions provided the large scale/small scale cutoff is appropriately chosen. (ii) If smoothness is assumed, NLGM always outperforms the Galerkin method by a factor equal to 1 in the convergence order of the H 1 ‐norm for the velocity and the L 2 ‐norm for the pressure. This is a purely linear superconvergence effect resulting from standard elliptic regularity and holds independently of the nature of the boundary conditions (whether periodicity or no‐slip BC is enforced). © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008