A unified framework for a posteriori error estimation for the Stokes problem
In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on -conforming velocity reconstruction and -conforming, locally conservative flux (stress) reconstruction. It gives guaranteed, fully computable global upper bounds as well as local...
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| Published in | Numerische Mathematik Vol. 122; no. 4; pp. 725 - 769 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer-Verlag
01.12.2012
Springer Verlag |
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| Online Access | Get full text |
| ISSN | 0029-599X 0945-3245 |
| DOI | 10.1007/s00211-012-0472-x |
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| Abstract | In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on
-conforming velocity reconstruction and
-conforming, locally conservative flux (stress) reconstruction. It gives guaranteed, fully computable global upper bounds as well as local lower bounds on the energy error. In order to apply this framework to a given numerical method, two simple conditions need to be checked. We show how to do this for various conforming and conforming stabilized finite element methods, the discontinuous Galerkin method, the Crouzeix–Raviart nonconforming finite element method, the mixed finite element method, and a general class of finite volume methods. The tools developed and used include a new simple equilibration on dual meshes and the solution of local Poisson-type Neumann problems by the mixed finite element method. Numerical experiments illustrate the theoretical developments. |
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| AbstractList | In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on
-conforming velocity reconstruction and
-conforming, locally conservative flux (stress) reconstruction. It gives guaranteed, fully computable global upper bounds as well as local lower bounds on the energy error. In order to apply this framework to a given numerical method, two simple conditions need to be checked. We show how to do this for various conforming and conforming stabilized finite element methods, the discontinuous Galerkin method, the Crouzeix–Raviart nonconforming finite element method, the mixed finite element method, and a general class of finite volume methods. The tools developed and used include a new simple equilibration on dual meshes and the solution of local Poisson-type Neumann problems by the mixed finite element method. Numerical experiments illustrate the theoretical developments. In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on $[H^1_0(\Om)]^d$-conforming velocity reconstruction and $\HH(\dv,\Om)$-conforming, locally conservative flux (stress) reconstruction. It gives guaranteed, fully computable global upper bounds as well as local lower bounds on the energy error. In order to apply this framework to a given numerical method, two simple conditions need to be checked. We show how to do this for various conforming and conforming stabilized finite element methods, the discontinuous Galerkin method, the Crouzeix--Raviart nonconforming finite element method, the mixed finite element method, and a general class of finite volume methods. Numerical experiments illustrate the theoretical developments. |
| Author | Hannukainen, Antti Vohralík, Martin Stenberg, Rolf |
| Author_xml | – sequence: 1 givenname: Antti surname: Hannukainen fullname: Hannukainen, Antti organization: Department of Mathematics and Systems Analysis, Aalto University – sequence: 2 givenname: Rolf surname: Stenberg fullname: Stenberg, Rolf organization: Department of Mathematics and Systems Analysis, Aalto University – sequence: 3 givenname: Martin surname: Vohralík fullname: Vohralík, Martin email: vohralik@ann.jussieu.fr organization: Laboratoire Jacques-Louis Lions, UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, CNRS, UMR 7598 |
| BackLink | https://hal.science/hal-00470131$$DView record in HAL |
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| Keywords | 65N15 76S05 76M12 Discontinuous Galerkin method Mixed finite element method Stokes problem A posteriori error estimate Conforming finite element method Unified framework Guaranteed upper bound Finite volume method Nonconforming finite element method |
| Language | English |
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| Snippet | In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on
-conforming velocity reconstruction... In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on $[H^1_0(\Om)]^d$-conforming velocity... |
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| SubjectTerms | Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical and Computational Physics Simulation Theoretical |
| Title | A unified framework for a posteriori error estimation for the Stokes problem |
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