## Error reduction and convergence for an adaptive mixed finite element method (2005)

Venue: | Mathematics of Computation |

Citations: | 16 - 5 self |

### BibTeX

@ARTICLE{Carstensen05errorreduction,

author = {Carsten Carstensen and R. H. W. Hoppe},

title = {Error reduction and convergence for an adaptive mixed finite element method},

journal = {Mathematics of Computation},

year = {2005}

}

### OpenURL

### Abstract

Abstract. An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction property is established here for the Raviart–Thomas finite element method with a reduction factor ρ<1 uniformly for the L 2 norm of the flux errors. Our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels. The adaptive algorithm surprisingly does not require any particular mesh design, unlike the conforming finite element method. The new arguments are a discrete local efficiency and a quasi-orthogonality estimate. The proof does not rely on duality or on regularity. 1.

### Citations

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Citation Context ...oduct] (1.3) (pH,qH) L2 (Ω) +(uH, div qH) L2 (Ω) =0 forallqH∈RT0(TH), (div pH,vH) L2 (Ω) = −(f,vH) L2 (Ω) for all vH ∈ P0(TH). Details on the lowest-order Raviart–Thomas finite element space RT0 (TH) =-=[8]-=- can be found below in Section 2; P0(TH) denotes the piecewise constants. MATLAB Received by the editor April 11, 2004. 2000 Mathematics Subject Classification. Primary 65N30, 65N50. 1033 c○2006 Ameri... |

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Citation Context ...ethod consists of successive loops of the following sequence: (1.1) SOLVE → ESTIMATE → MARK → REFINE. The a posteriori error control in the step ESTIMATE has been developed over the last decades (cf. =-=[1, 3, 6, 12, 17]-=- and the references therein). The convergence analysis of the full algorithm (1.1), however, is restricted to the conforming finite element method [15, 16]. This paper investigates convergence propert... |

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Citation Context ...ethod consists of successive loops of the following sequence: (1.1) SOLVE → ESTIMATE → MARK → REFINE. The a posteriori error control in the step ESTIMATE has been developed over the last decades (cf. =-=[1, 3, 6, 12, 17]-=- and the references therein). The convergence analysis of the full algorithm (1.1), however, is restricted to the conforming finite element method [15, 16]. This paper investigates convergence propert... |

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Citation Context ...eveloped over the last decades (cf. [1, 3, 6, 12, 17] and the references therein). The convergence analysis of the full algorithm (1.1), however, is restricted to the conforming finite element method =-=[15, 16]-=-. This paper investigates convergence properties of such a loop for the mixed finite element method (MFEM) in a 2D model Poisson problem (1.2) f +∆u =0 inΩandu =0on∂Ω. Given a (coarse) mesh TH, a shap... |

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Citation Context ...integral mean of f over the patch ωE := int(T+ ∪ T−) ofarea|ωE| = |T+| + |T−| and let EH denote the set of all interior edges in TH. The bulk criterion in the step MARK was introduced and analyzed in =-=[7, 11, 15]-=- for displacement-based AFEMs. Here, it leads to a selection of a subset M of edges EH such that (1.5) θη 2 H ≤ � E∈M for some universal constant 0 <θ<1. It came much as a surprise to the authors that... |

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Citation Context ...ethod consists of successive loops of the following sequence: (1.1) SOLVE → ESTIMATE → MARK → REFINE. The a posteriori error control in the step ESTIMATE has been developed over the last decades (cf. =-=[1, 3, 6, 12, 17]-=- and the references therein). The convergence analysis of the full algorithm (1.1), however, is restricted to the conforming finite element method [15, 16]. This paper investigates convergence propert... |

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Citation Context ...≤ ηT and Tol is θ times the largest of such contributions. In the context of AMFEM, data oscillations have not been involved so far. We refer to [5] for algorithmic details and MATLAB routines and to =-=[2, 10, 18, 13]-=- for empirical examples. It is the authors’ overall impression that the AMFEM is very robust in changing algorithmic details in practice. The numerical experiments in [15, 16] with a realization of (2... |

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Citation Context ...ons and documentations of the step SOLVE are provided in [5]. In this paper, for the ease of discussion, the step ESTIMATE is the postprocessing to compute the residual-based explicit error estimator =-=[2, 9, 18]-=- (1.4) ηH := ( � η 2 E) 1/2 with η 2 E := hE�[pH]E� 2 L2 (E) . E∈EH Here and throughout, [pH] denotes the jump [pH]:=pH|T+ − pH|T− of the discrete flux over an interior edge E := T+ ∩ T− of length hE ... |

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Citation Context ...ons and documentations of the step SOLVE are provided in [5]. In this paper, for the ease of discussion, the step ESTIMATE is the postprocessing to compute the residual-based explicit error estimator =-=[2, 9, 18]-=- (1.4) ηH := ( � η 2 E) 1/2 with η 2 E := hE�[pH]E� 2 L2 (E) . E∈EH Here and throughout, [pH] denotes the jump [pH]:=pH|T+ − pH|T− of the discrete flux over an interior edge E := T+ ∩ T− of length hE ... |

21 | Local problems on stars: A posteriori error estimators, convergence, and performance
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Citation Context ...eveloped over the last decades (cf. [1, 3, 6, 12, 17] and the references therein). The convergence analysis of the full algorithm (1.1), however, is restricted to the conforming finite element method =-=[15, 16]-=-. This paper investigates convergence properties of such a loop for the mixed finite element method (MFEM) in a 2D model Poisson problem (1.2) f +∆u =0 inΩandu =0on∂Ω. Given a (coarse) mesh TH, a shap... |

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Citation Context ...ons and documentations of the step SOLVE are provided in [5]. In this paper, for the ease of discussion, the step ESTIMATE is the postprocessing to compute the residual-based explicit error estimator =-=[2, 9, 18]-=- (1.4) ηH := ( � η 2 E) 1/2 with η 2 E := hE�[pH]E� 2 L2 (E) . E∈EH Here and throughout, [pH] denotes the jump [pH]:=pH|T+ − pH|T− of the discrete flux over an interior edge E := T+ ∩ T− of length hE ... |

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Citation Context ...≤ ηT and Tol is θ times the largest of such contributions. In the context of AMFEM, data oscillations have not been involved so far. We refer to [5] for algorithmic details and MATLAB routines and to =-=[2, 10, 18, 13]-=- for empirical examples. It is the authors’ overall impression that the AMFEM is very robust in changing algorithmic details in practice. The numerical experiments in [15, 16] with a realization of (2... |

17 |
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Citation Context ...ouzeix–Raviart FEM solution of (DHu N H,DHv N H) L2 (Ω) =(fH,v N H) L2 (Ω) for all v N H ∈ V N H . The discrete fluxes pN H := DHuN H and pH from (1.3) are related. is different from its � Lemma 3.1 (=-=[14, 5]-=-). Let fT± := T± f(x) dx/|T±| and let xT± := mid(T±)denote the barycenter of T±. Then there holds pH|T± (x)=DHu N 1 H|T± − 2 fT± (x − xT± ) for x ∈ T±. � In this context, fH ∈ P0(TH) andfh∈ P0(Th) den... |

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Citation Context ...on the continuous level consists of two steps: a Poisson solve and an update formula. The substitute of the Poisson solve by some AFEM allows a perturbation of the convergence on the continuous level =-=[4]-=-. The advantage is that even unstable finite element schemes can be employed. The disadvantage is the possibly slow convergence of the Uzawa algorithm relative to multilevel solver [13]. 3. Notation a... |

6 |
Three MATLAB implementations of the lowest-order RaviartThomas MFEM with a posteriori error control
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(Show Context)
Citation Context ...❅ T bisec(T) bisec2ℓ(T) bisec2r(T) bisec3(T) red(T) bisec5(T) Figure 1. Possible refinements of one triangle T in the step REFINE. implementations and documentations of the step SOLVE are provided in =-=[5]-=-. In this paper, for the ease of discussion, the step ESTIMATE is the postprocessing to compute the residual-based explicit error estimator [2, 9, 18] (1.4) ηH := ( � η 2 E) 1/2 with η 2 E := hE�[pH]E... |