Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes-problem. (English) Zbl 1392.35116
Summary: Nearly all classical inf-sup stable mixed finite element methods for the incompressible Stokes equations are not pressure-robust, i.e., the velocity error is dependent on the pressure. However, recent results show that pressure-robustness can be recovered by a nonstandard discretization of the right-hand side alone. This variational crime introduces a consistency error in the method which can be estimated in a straightforward manner provided that the exact velocity solution is sufficiently smooth. The purpose of this paper is to analyze the pressure-robust scheme with low regularity. The numerical analysis applies divergence-free \(H^1\)-conforming Stokes finite element methods as a theoretical tool. As an example, pressure-robust velocity and pressure a priori error estimates will be presented for the (first-order) nonconforming Crouzeix-Raviart element. A key feature in the analysis is the dependence of the errors on the Helmholtz projector of the right-hand side data, and not on the entire data term. Numerical examples illustrate the theoretical results.
MSC:
| 35J25 | Boundary value problems for second-order elliptic equations |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 76D07 | Stokes and related (Oseen, etc.) flows |
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