Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes-problem
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- by A. Linke, C. Merdon, M. Neilan and F. Neumann PDF
- Math. Comp. 87 (2018), 1543-1566 Request permission
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.References
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Additional Information
- A. Linke
- Affiliation: Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Germany
- MR Author ID: 853921
- Email: Alexander.Linke@wias-berlin.de
- C. Merdon
- Affiliation: Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Germany
- MR Author ID: 901357
- Email: Christian.Merdon@wias-berlin.de
- M. Neilan
- Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania
- MR Author ID: 824091
- Email: neilan@pitt.edu
- F. Neumann
- Affiliation: Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Germany
- Email: felix.neumann@alumni.hu-berlin.de
- Received by editor(s): February 20, 2017
- Published electronically: February 6, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 87 (2018), 1543-1566
- MSC (2010): Primary 35J25, 65N30, 76D07
- DOI: https://doi.org/10.1090/mcom/3344
- MathSciNet review: 3787384