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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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A pressure-robust divergence free finite element basis for the Stokes equations
This paper considered divergence-free basis methods to solve the viscous Stokes equations. A discrete divergence-free subspace was constructed to reduce the saddle point problem of the Stokes problem to a smaller-sized symmetric and positive definite system solely depending on the velocity components. Then, the system could decouple the unknowns in velocity and pressure and solve them independently. However, such a scheme may not ensure an accurate numerical solution to the velocity. In order to obtain satisfactory accuracy, we used a velocity reconstruction technique to enhance the divergence-free scheme to achieve the desired pressure and viscosity robustness. Numerical results were presented to demonstrate the robustness and accuracy of this discrete divergence-free method.
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- PAR ID:
- 10626429
- Publisher / Repository:
- AIMs Press
- Date Published:
- Journal Name:
- Electronic Research Archive
- Volume:
- 32
- Issue:
- 10
- ISSN:
- 2688-1594
- Page Range / eLocation ID:
- 5633 to 5648
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3934/era.2024261
