Volume 52, pp. 281-294, 2020.
Pressure-robustness in quasi-optimal a priori estimates for the Stokes problem
Alexander Linke, Christian Merdon, and Michael Neilan
Abstract
Recent analysis of the divergence
constraint in the incompressible Stokes/Navier-Stokes problem
has stressed the importance
of equivalence classes of forces and how they play a fundamental role
for an accurate space discretization. Two forces in the momentum balance are
velocity-equivalent if they lead to the same velocity solution,
i.e., if and only if the forces differ by only a gradient field.
Pressure-robust space discretizations are designed to
respect these equivalence classes.
One way to achieve pressure-robust schemes
is to introduce a non-standard discretization of the right-hand side
forcing term for any inf-sup stable mixed finite element method.
This modification leads to pressure-robust and optimal-order
discretizations, but
a proof was only available for smooth situations and remained open in the case of minimal regularity, where it cannot be
assumed that the vector Laplacian of the velocity is at least
square-integrable. This contribution closes this gap by
delivering a general estimate for the consistency error that
depends only on the regularity of the data term.
Pressure-robustness of the estimate is achieved by the fact that
the new estimate only depends on the
Full Text (PDF) [310 KB], BibTeX , DOI: 10.1553/etna_vol52s281
Key words
incompressible Stokes equations, mixed finite elements methods, a-priori error estimates, stability estimates, pressure-robustness
AMS subject classifications
65N12, 65N30, 76D07