Towards Pressure-Robust Mixed Methods for the Incompressible Navier–Stokes Equations
In this contribution, we review classical mixed methods for the incompressible Navier–Stokes equations that relax the divergence constraint and are discretely inf-sup stable. Though the relaxation of the divergence constraint was claimed to be harmless since the beginning of the 1970s, Poisson locki...
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| Vydáno v: | Journal of computational methods in applied mathematics Ročník 18; číslo 3; s. 353 - 372 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Minsk
De Gruyter
01.07.2018
Walter de Gruyter GmbH |
| Témata: | |
| ISSN: | 1609-4840, 1609-9389 |
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| Abstract | In this contribution, we review classical mixed methods
for the incompressible Navier–Stokes equations that relax the divergence constraint
and are discretely inf-sup stable. Though the relaxation of the divergence constraint was claimed to be harmless since
the beginning of the 1970s,
Poisson locking is just replaced by another more subtle kind of locking phenomenon, which
is sometimes called
and led in computational practice to the exclusion
of mixed methods with low-order pressure approximations like the Bernardi–Raugel
or the Crouzeix–Raviart finite element methods. Indeed, divergence-free mixed methods
and classical mixed methods behave
qualitatively in a different way:
divergence-free mixed methods are
, which means that, e.g., their velocity error is independent
of the continuous pressure. The lack of pressure robustness in classical
mixed methods can be traced back to a consistency error
of an appropriately defined discrete Helmholtz projector.
Numerical analysis and numerical examples reveal that
mixed methods must be discretely inf-sup stable and
pressure-robust, simultaneously. Further, a recent discovery shows that
locking-free,
pressure-robust mixed methods do not have to be divergence free.
Indeed, relaxing the divergence constraint in the velocity trial functions
is harmless, if the relaxation of the divergence constraint in
some velocity test functions is repaired, accordingly.
Thus, inf-sup stable, pressure-robust mixed methods will potentially
allow in future to reduce the approximation order of the discretizations used in computational
practice, without compromising the accuracy. |
|---|---|
| AbstractList | In this contribution, we review classical mixed methods for the incompressible Navier–Stokes equations that relax the divergence constraint and are discretely inf-sup stable. Though the relaxation of the divergence constraint was claimed to be harmless since the beginning of the 1970s, Poisson locking is just replaced by another more subtle kind of locking phenomenon, which is sometimes called poor mass conservation and led in computational practice to the exclusion of mixed methods with low-order pressure approximations like the Bernardi–Raugel or the Crouzeix–Raviart finite element methods. Indeed, divergence-free mixed methods and classical mixed methods behave qualitatively in a different way: divergence-free mixed methods are pressure-robust, which means that, e.g., their velocity error is independent of the continuous pressure. The lack of pressure robustness in classical mixed methods can be traced back to a consistency error of an appropriately defined discrete Helmholtz projector. Numerical analysis and numerical examples reveal that really locking-free mixed methods must be discretely inf-sup stable and pressure-robust, simultaneously. Further, a recent discovery shows that locking-free, pressure-robust mixed methods do not have to be divergence free. Indeed, relaxing the divergence constraint in the velocity trial functions is harmless, if the relaxation of the divergence constraint in some velocity test functions is repaired, accordingly. Thus, inf-sup stable, pressure-robust mixed methods will potentially allow in future to reduce the approximation order of the discretizations used in computational practice, without compromising the accuracy. In this contribution, we review classical mixed methods for the incompressible Navier–Stokes equations that relax the divergence constraint and are discretely inf-sup stable. Though the relaxation of the divergence constraint was claimed to be harmless since the beginning of the 1970s, Poisson locking is just replaced by another more subtle kind of locking phenomenon, which is sometimes called and led in computational practice to the exclusion of mixed methods with low-order pressure approximations like the Bernardi–Raugel or the Crouzeix–Raviart finite element methods. Indeed, divergence-free mixed methods and classical mixed methods behave qualitatively in a different way: divergence-free mixed methods are , which means that, e.g., their velocity error is independent of the continuous pressure. The lack of pressure robustness in classical mixed methods can be traced back to a consistency error of an appropriately defined discrete Helmholtz projector. Numerical analysis and numerical examples reveal that mixed methods must be discretely inf-sup stable and pressure-robust, simultaneously. Further, a recent discovery shows that locking-free, pressure-robust mixed methods do not have to be divergence free. Indeed, relaxing the divergence constraint in the velocity trial functions is harmless, if the relaxation of the divergence constraint in some velocity test functions is repaired, accordingly. Thus, inf-sup stable, pressure-robust mixed methods will potentially allow in future to reduce the approximation order of the discretizations used in computational practice, without compromising the accuracy. In this contribution, we review classical mixed methods for the incompressible Navier–Stokes equations that relax the divergence constraint and are discretely inf-sup stable. Though the relaxation of the divergence constraint was claimed to be harmless since the beginning of the 1970s, Poisson locking is just replaced by another more subtle kind of locking phenomenon, which is sometimes called poor mass conservation and led in computational practice to the exclusion of mixed methods with low-order pressure approximations like the Bernardi–Raugel or the Crouzeix–Raviart finite element methods. Indeed, divergence-free mixed methods and classical mixed methods behave qualitatively in a different way: divergence-free mixed methods are pressure-robust , which means that, e.g., their velocity error is independent of the continuous pressure. The lack of pressure robustness in classical mixed methods can be traced back to a consistency error of an appropriately defined discrete Helmholtz projector. Numerical analysis and numerical examples reveal that really locking-free mixed methods must be discretely inf-sup stable and pressure-robust, simultaneously. Further, a recent discovery shows that locking-free, pressure-robust mixed methods do not have to be divergence free. Indeed, relaxing the divergence constraint in the velocity trial functions is harmless, if the relaxation of the divergence constraint in some velocity test functions is repaired, accordingly. Thus, inf-sup stable, pressure-robust mixed methods will potentially allow in future to reduce the approximation order of the discretizations used in computational practice, without compromising the accuracy. |
| Author | Ahmed, Naveed Linke, Alexander Merdon, Christian |
| Author_xml | – sequence: 1 givenname: Naveed surname: Ahmed fullname: Ahmed, Naveed email: naveed.ahmed@wias-berlin.de organization: Weierstrass Institute, Mohrenstr. 9, 10117Berlin, Germany – sequence: 2 givenname: Alexander surname: Linke fullname: Linke, Alexander email: linke@wias-berlin.de organization: Weierstrass Institute, Mohrenstr. 9, 10117Berlin, Germany – sequence: 3 givenname: Christian surname: Merdon fullname: Merdon, Christian email: christian.merdon@wias-berlin.de organization: Weierstrass Institute, Mohrenstr. 9, 10117Berlin, Germany |
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| Snippet | In this contribution, we review classical mixed methods
for the incompressible Navier–Stokes equations that relax the divergence constraint
and are discretely... In this contribution, we review classical mixed methods for the incompressible Navier–Stokes equations that relax the divergence constraint and are discretely... |
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| SubjectTerms | 65M60 65N30 76D05 A Priori Error Estimates Approximation Computational fluid dynamics Divergence Finite element method Fluid flow Helmholtz Projector Incompressible Navier–Stokes Equations Locking Mathematical analysis Mixed Finite Element Methods Mixed methods research Navier-Stokes equations Numerical analysis Pressure Robustness Robustness (mathematics) Velocity errors |
| Title | Towards Pressure-Robust Mixed Methods for the Incompressible Navier–Stokes Equations |
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