Verlag der Österreichischen Akademie der Wissenschaften

Abstract:
Recent analysis of the divergenceconstraint in the incompressible Stokes/Navier-Stokes problemhas stressed the importanceof equivalence classes of forces and how they play a fundamental rolefor an accurate space discretization. Two forces in the momentum balance arevelocity-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 torespect these equivalence classes.One way to achieve pressure-robust schemesis to introduce a non-standard discretization of the right-hand sideforcing term for any inf-sup stable mixed finite element method.This modification leads to pressure-robust and optimal-orderdiscretizations, buta proof was only available for smooth situations and remained open in the case of minimal regularity, where it cannot beassumed that the vector Laplacian of the velocity is at leastsquare-integrable. This contribution closes this gap bydelivering a general estimate for the consistency error thatdepends only on the regularity of the data term.Pressure-robustness of the estimate is achieved by the fact thatthe new estimate only depends on the $L^2$-norm of theHelmholtz-Hodge projector of the data termand not on the $L^2$-norm of the entire data term. Numerical examples illustrate the theory.