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Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes-Problem


Authors: A. Linke, C. Merdon, M. Neilan and F. Neumann
Journal: Math. Comp. 87 (2018), 1543-1566
MSC (2010): Primary 35J25, 65N30, 76D07
DOI: https://doi.org/10.1090/mcom/3344
Published electronically: February 6, 2018
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Abstract: 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.


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Additional Information

A. Linke
Affiliation: Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Germany
Email: Alexander.Linke@wias-berlin.de

C. Merdon
Affiliation: Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Germany
Email: Christian.Merdon@wias-berlin.de

M. Neilan
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania
Email: neilan@pitt.edu

F. Neumann
Affiliation: Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Germany
Email: felix.neumann@alumni.hu-berlin.de

DOI: https://doi.org/10.1090/mcom/3344
Keywords: Incompressible Stokes equations, mixed finite element methods, nonconforming discretizations, pressure-robustness, a priori error estimates, Helmholtz projector
Received by editor(s): February 20, 2017
Published electronically: February 6, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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