5 research outputs found
Robust arbitrary order mixed finite element methods for the incompressible Stokes equations
Standard mixed finite element methods for the incompressible Navier-Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H1-conforming, divergence-free mixed finite element methods is rather difficult. Instead, we present a novel approach for the construction of arbitrary order mixed finite element methods which deliver pressure-independent velocity errors. The approach does not change the trial functions but replaces discretely divergence-free test functions in some operators of the weak formulation by divergence-free ones. This modification is applied to inf-sup stable conforming and nonconforming mixed finite element methods of arbitrary order in two and three dimensions. Optimal estimates for the incompressible Stokes equations are proved for the H1 and L2 errors of the velocity and the L2 error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results
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Robust arbitrary order mixed finite element methods for the incompressible Stokes equations
Standard mixed finite element methods for the incompressible
Navier-Stokes equations that relax the divergence constraint are not robust
against large irrotational forces in the momentum balance and the velocity
error depends on the continuous pressure. This robustness issue can be
completely cured by using divergence-free mixed finite elements which deliver
pressure-independent velocity error estimates. However, the construction of
H1-conforming, divergence-free mixed finite element methods is rather
difficult. Instead, we present a novel approach for the construction of
arbitrary order mixed finite element methods which deliver
pressure-independent velocity errors. The approach does not change the trial
functions but replaces discretely divergence-free test functions in some
operators of the weak formulation by divergence-free ones. This modification
is applied to inf-sup stable conforming and nonconforming mixed finite
element methods of arbitrary order in two and three dimensions. Optimal
estimates for the incompressible Stokes equations are proved for the H1 and
L2 errors of the velocity and the L2 error of the pressure. Moreover, both
velocity errors are pressure-independent, demonstrating the improved
robustness. Several numerical examples illustrate the results