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An L 2 finite element approximation for the incompressible Navier–Stokes equations
Author(s) -
Lee Eunjung,
Choi Wonjoon,
Ha Heonkyu
Publication year - 2020
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.22478
Subject(s) - mathematics , finite element method , navier–stokes equations , linearization , mathematical analysis , compressibility , pressure correction method , nonlinear system , convergence (economics) , mixed finite element method , physics , thermodynamics , quantum mechanics , economics , economic growth
Abstract This paper utilizes the Picard method and Newton's method to linearize the stationary incompressible Navier–Stokes equations and then uses an LL* approach, which is a least‐squares finite element method applied to the dual problem of the corresponding linear system. The LL* approach provides an L 2 ‐approximation to a given problem, which is not typically available with conventional finite element methods for nonlinear second‐order partial differential equations. We first show that the proposed combination of linearization scheme and LL* approach provides an L 2 ‐approximation to the stationary incompressible Navier–Stokes equations. The validity of L 2 ‐approximation is proven through the analysis of the weak problem corresponding to the linearized Navier–Stokes equations. Then, the convergence is analyzed, and numerical results are presented.