## A Finite Element Method for Domain Decomposition with Non-Matching Grids (2001)

Venue: | MATH. MODEL. ANAL. NUMER |

Citations: | 51 - 7 self |

### BibTeX

@TECHREPORT{Becker01afinite,

author = {Roland Becker and Peter Hansbo and Rolf Stenberg},

title = {A Finite Element Method for Domain Decomposition with Non-Matching Grids},

institution = {MATH. MODEL. ANAL. NUMER},

year = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this note, we propose and analyse a method for handling interfaces between non-matching grids based on an approach suggested by Nitsche [15] for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson's equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.

### Citations

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Galerkin Finite Element Methods for Parabolic Problems
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Citation Context ... ∥∥∥∥ −1/2,h,Γi ] ≥ 2∑ i=1 [( 1− CI 2ε ) ‖∇vi‖2L2(Ωi) + ( γ − ε 2 ) ‖[[v]]‖21/2,h,Γi ] ≥ C1‖v‖21,h ≥ C2|‖v‖|21,h, for γ > ε/2 and choosing ε > CI/2. The following interpolation estimates holds, cf. =-=[20]-=-. Lemma 2.7. Suppose that u ∈ Hs(Ω), with 3/2 < s ≤ p + 1. Then it holds inf v∈V h ‖u− v‖1,h ≤ Chs−1 ‖u‖Hs(Ω) (25) and inf v∈V h |‖u− v‖|1,h ≤ Chs−1 ‖u‖Hs(Ω) . (26) 214 R. BECKER ET AL. We are now abl... |

236 |
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Citation Context ...ned, e.g., by Lions [16]. • Direct procedures, using Lagrange multiplier techniques to achieve continuity. Different variants have been proposed, e.g., by Le Tallec and Sassi [15], and Bernadi et al. =-=[8]-=-. The multiplier method has the advantage of directly yielding a solvable global system. However, in the latter method, new unknowns (the multipliers) must be introduced and solved for. The method mus... |

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Citation Context ...originally introduced for the purpose of solving Dirichlet problems without enforcing the boundary conditions in the definition of the finite element spaces. This method has later been used by Arnold =-=[2]-=- for the discretization of second order elliptic equations by discontinuous finite elements. In earlier papers [18, 19] we have pointed out the close connection between Nitsche’s method and stabilized... |

105 | A feed-back approach to error control in finite element methods: Basic analysis and examples
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(Show Context)
Citation Context ...s to approximately match the exact error. In Figures 3 and 4 we show the first and last (adapted) meshes resulting from equilibrating the error distribution over the set of elements (for details, see =-=[7,14]-=-). In Figure 5 we show the exact and estimated L2-errors on the sequence of meshes, which show a reasonable agreement. For more exact error control, more computational effort must be invested. A FINIT... |

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Citation Context ...proximate solution or its normal derivative or combinations thereof should be continuous across interfaces. This forms the basis for the standard Schwarz alternating method as defined, e.g., by Lions =-=[16]-=-. • Direct procedures, using Lagrange multiplier techniques to achieve continuity. Different variants have been proposed, e.g., by Le Tallec and Sassi [15], and Bernadi et al. [8]. The multiplier meth... |

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Über ein Variationsprinzip zur Lösung von Dirichlet Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen
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Citation Context ...l choices of multiplier spaces (such as mortar elements, cf. [8]), or then stabilization techniques (cf. [3, 4, 9]) must be used. In this paper, we consider a third possibility, i.e. Nitsche’s method =-=[17]-=-, which was originally introduced for the purpose of solving Dirichlet problems without enforcing the boundary conditions in the definition of the finite element spaces. This method has later been use... |

47 |
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Citation Context ...s to approximately match the exact error. In Figures 3 and 4 we show the first and last (adapted) meshes resulting from equilibrating the error distribution over the set of elements (for details, see =-=[7,14]-=-). In Figure 5 we show the exact and estimated L2-errors on the sequence of meshes, which show a reasonable agreement. For more exact error control, more computational effort must be invested. A FINIT... |

30 |
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Citation Context ...th s > 3/2. A FINITE ELEMENT METHOD FOR DOMAIN DECOMPOSITION WITH NON-MATCHING GRIDS 211 With this assumption it holds ∂ui/∂ni ∈ L2(Γ) and the two problems (1) and (4) are equivalent (see for example =-=[1]-=-) with: u|Ωi = ui, i = 1, 2. (5) In the following we will therefore write u = (u1, u2) ∈ V1 × V2 with the continuous spaces Vi = { vi ∈ H1(Ωi) : ∂vi/∂ni ∈ L2(Γ), vi|∂Ω∩∂Ωi = 0 } , i = 1, 2. To formula... |

24 |
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Citation Context ...he definition of the finite element spaces. This method has later been used by Arnold [2] for the discretization of second order elliptic equations by discontinuous finite elements. In earlier papers =-=[18, 19]-=- we have pointed out the close connection between Nitsche’s method and stabilized methods and proposed it as a mortaring method. In this paper we will give a more detailed analysis of this domain deco... |

21 | A residual based error estimator for mortar finite element discretisations
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Citation Context ... is a constant C such that |‖u− uh‖|1,h ≤ C‖u− uh‖1,h. (34) We also remark that an analog assumption is used in the context of a posteriori error estimates for the mortar element method, see Wohlmuth =-=[21]-=-. The a posteriori estimate is now the following. Theorem 3.2. Suppose that the Assumption 3.1 is valid. Then there is a positive constant C such that |‖u− uh‖|1,h ≤ CE(uh). (35) Proof. We denote e = ... |

18 |
Domain decomposition with non matching grids and adaptive nite element techniques
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(Show Context)
Citation Context ...lternating method as defined, e.g., by Lions [16]. • Direct procedures, using Lagrange multiplier techniques to achieve continuity. Different variants have been proposed, e.g., by Le Tallec and Sassi =-=[15]-=-, and Bernadi et al. [8]. The multiplier method has the advantage of directly yielding a solvable global system. However, in the latter method, new unknowns (the multipliers) must be introduced and so... |

16 |
Stabilization of Galerkin Methods and Applications to Domain Decomposition
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Citation Context ...olved for. The method must then either satisfy the inf-sup condition, which necessitates special choices of multiplier spaces (such as mortar elements, cf. [8]), or then stabilization techniques (cf. =-=[3, 4, 9]-=-) must be used. In this paper, we consider a third possibility, i.e. Nitsche’s method [17], which was originally introduced for the purpose of solving Dirichlet problems without enforcing the boundary... |

16 | Nitsche type mortaring for some elliptic problem with corner singularities
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Citation Context ... smooth, i.e. that u ∈ Hs(Ω) with s > 3/2. The analysis can be extended for less regular solutions if the corner and transmission singularities are carefully taken into account, cf. the recent papers =-=[12, 13]-=-. 3. A POSTERIORI error estimates We will first consider control of the error e = u− uh in the mesh dependent energy norm |‖ · ‖|1,h. We define the local and global estimators as EK(uh)2 = h2K ‖f + ∆u... |

14 | Stabilization techniques for domain decomposition methods with non-matching grids. IAN-CNR Report 1037, Instituto di analisi Numerica Pavia
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- 1997
(Show Context)
Citation Context ...olved for. The method must then either satisfy the inf-sup condition, which necessitates special choices of multiplier spaces (such as mortar elements, cf. [8]), or then stabilization techniques (cf. =-=[3, 4, 9]-=-) must be used. In this paper, we consider a third possibility, i.e. Nitsche’s method [17], which was originally introduced for the purpose of solving Dirichlet problems without enforcing the boundary... |

14 | The Nitsche mortar finite element method for transmission problems with singularities
- Heinrich, Nicaise
(Show Context)
Citation Context ... smooth, i.e. that u ∈ Hs(Ω) with s > 3/2. The analysis can be extended for less regular solutions if the corner and transmission singularities are carefully taken into account, cf. the recent papers =-=[12, 13]-=-. 3. A POSTERIORI error estimates We will first consider control of the error e = u− uh in the mesh dependent energy norm |‖ · ‖|1,h. We define the local and global estimators as EK(uh)2 = h2K ‖f + ∆u... |

12 |
Boundary Lagrange multipliers in finite element methods: error analysis in natural norms
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- 1992
(Show Context)
Citation Context ...olved for. The method must then either satisfy the inf-sup condition, which necessitates special choices of multiplier spaces (such as mortar elements, cf. [8]), or then stabilization techniques (cf. =-=[3, 4, 9]-=-) must be used. In this paper, we consider a third possibility, i.e. Nitsche’s method [17], which was originally introduced for the purpose of solving Dirichlet problems without enforcing the boundary... |

9 |
On weakly imposed boundary conditions for second order problems
- Freund, Stenberg
- 1995
(Show Context)
Citation Context ...tive values of γ: ah(v, v) ≥ C|‖v‖|21,h ∀v ∈ V h, ∀γ > 0. If the Laplace operator is a part of a problem which is not symmetric, then it might be practical to use this nonsymmetric bilinear form. See =-=[10, 11]-=- for applications to convection diffusion problems. Remark 2.12. For linear elements ∇vi (vi ∈ V hi ) is constant on each element and hence for K ∈ T ih with the base E ∈ Gih we have ∥∥∥∥ ∂vi∂ni ∥∥∥∥ ... |

8 |
Finite element approximation of the Dirichlet problem using the boundary penalty method, Numer Math 49
- Barrett, Elliott
- 1986
(Show Context)
Citation Context ...r of O(h−2) for a quasiuniform mesh. Pure penalty methods, in contrast, are not consistent, and optimal error estimates require degrading the condition number for higher polynomial approximation (cf. =-=[5]-=-). Remark 2.11. The form ah(·, ·) in (14) is symmetric and positive definite, which is natural as the problem to be approximated has the same properties. With this there exists fast solvers, such as p... |

5 |
Discontinuous Galerkin methods for convection-diffusion problems with arbitrary Péclet number
- Becker, Hansbo
(Show Context)
Citation Context ...e method. Although we discuss its application to domain decomposition, the same technique is also suited for other applications, e.g., • to handle diffusion terms in the discontinuous Galerkin method =-=[2, 6]-=-; • to simplify mesh generation (different parts can be meshed independently from each other); • finite element methods with different polynomial degree on adjacent elements; • new finite element meth... |

3 | Mortaring by a method of J.A - Stenberg - 1998 |

3 |
Mortaring by a method of J.A. Nitsche, in Computational Mechanics
- Stenberg
- 1998
(Show Context)
Citation Context ...he definition of the finite element spaces. This method has later been used by Arnold [2] for the discretization of second order elliptic equations by discontinuous finite elements. In earlier papers =-=[18, 19]-=- we have pointed out the close connection between Nitsche’s method and stabilized methods and proposed it as a mortaring method. In this paper we will give a more detailed analysis of this domain deco... |

1 |
Space-time finite element methods for second order problems: an algorithmic approach
- Freund
- 1996
(Show Context)
Citation Context ...tive values of γ: ah(v, v) ≥ C|‖v‖|21,h ∀v ∈ V h, ∀γ > 0. If the Laplace operator is a part of a problem which is not symmetric, then it might be practical to use this nonsymmetric bilinear form. See =-=[10, 11]-=- for applications to convection diffusion problems. Remark 2.12. For linear elements ∇vi (vi ∈ V hi ) is constant on each element and hence for K ∈ T ih with the base E ∈ Gih we have ∥∥∥∥ ∂vi∂ni ∥∥∥∥ ... |