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Error analysis of a fully discrete finite element variational multiscale method for time‐dependent incompressible Navier–Stokes equations
Author(s) -
Shang Yueqiang
Publication year - 2013
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.21787
Subject(s) - mathematics , discretization , finite element method , backward euler method , projection (relational algebra) , partial differential equation , navier–stokes equations , compressibility , gauss , euler's formula , projection method , mathematical analysis , mathematical optimization , algorithm , dykstra's projection algorithm , physics , quantum mechanics , thermodynamics , engineering , aerospace engineering
A finite element variational multiscale method based on two local Gauss integrations is applied to solve numerically the time‐dependent incompressible Navier–Stokes equations. A significant feature of the method is that the definition of the stabilization term is derived via two local Guass integrations at element level, making it more efficient than the usual projection‐based variational multiscale methods. It is computationally cheap and gives an accurate approximation to the quantities sought. Based on backward Euler and Crank–Nicolson schemes for temporal discretization, we derive error bounds of the fully discrete solution which are first and second order in time, respectively. Numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the proposed method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013