## A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations

### Cached

### Download Links

Venue: | 475 (1994) MR 94j:65136 |

Citations: | 60 - 2 self |

### BibTeX

@INPROCEEDINGS{Verfürth_aposteriori,

author = {R. Verfürth},

title = {A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations},

booktitle = {475 (1994) MR 94j:65136},

year = {},

pages = {445}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Using the abstract framework of [9] we analyze a residual a posteriori error estimator for space-time finite element discretizations of quasilinear parabolic pdes. The estimator gives global upper and local lower bounds on the error of the numerical solution. The finite element discretizations in particular cover the so-called θ-scheme, which includes the implicit and explicit Euler methods and the Crank-Nicholson scheme. 1.

### Citations

1892 |
Sobolev Spaces
- Adams
- 1975
(Show Context)
Citation Context ...+ 1)-st diagonal Padé approximation. For k = 0 we in particular obtain the Crank-Nicholson scheme. By slightly modifying the basis functions of Yh we may also recover the popular θ-scheme for all θ ∈ =-=[0, 1]-=-. This in particular covers the explicit (θ = 0) and implicit (θ = 1) Euler schemes. We obtain global upper and local lower bounds for the error measured in an L r (0,T;L ρ (Ω))-norm. The upper and lo... |

999 | The Mathematical Theory of Finite Element Methods - Brenner, Scott - 1994 |

434 |
A review of a posteriori error estimation and adaptive mesh-refinement techniques
- Verfürth
- 1996
(Show Context)
Citation Context ... ≤ �∇ � U n �0,π. �∇�u n h�0,π ≤ 1 �r βπ n τ �−1,π. 1 3 min{απ, βπ}{�∇�u n h�0,π + �∇( � U n − �u n h)�0,π} ≤ �r n τ �−1,π ≤ �∇�u n h�0,π + �∇( � U n − �u n h)�0,π. Using standard arguments (cf. e.g. =-=[9]-=-) we infer from Lemmas 5.1 and 5.2 that �∇( � U n − �u n h)�0,π ≤ c{�η n h + � Θ n h}, �η n h ≤ C{�∇( � U n − �u n h)�0,π + � Θ n h} (9.4) (9.5) (9.6) with constants c and C which only depend on the p... |

254 |
Mathematical Analysis and Numerical Methods for Science and Technology vol 2
- Dautray, Lions
- 1988
(Show Context)
Citation Context ...ace equipped with the norm �u� L p (a,b;V ) = � � b a �u(·, t)� p V dt �u�L∞ (a,b;V ) = ess.sup �u(·, t)�V t∈(a,b) � 1/p , p < ∞, (cf. [6, Vol. 5, Chap. XVIII, §1]). Slightly changing the notation of =-=[6]-=-, we further introduce the Banach space W p (a, b; V, W ) = equipped with the norm �u� W p (a,b;V,W ) = � � b �u� W ∞ (a,b;V,W ) = ess.sup t∈(a,b) Here the partial derivative ∂u ∂t a � u ∈ L p (a, b; ... |

242 |
Approximation by finite element functions using local regularization
- Clément
- 1975
(Show Context)
Citation Context ...y ωx and consists of all elements in Th,n that share the vertex x. Denote by πx the L2 (ωx)-projection onto R1 defined by � � πxvw = vw ∀ w ∈ R1. ωx ωx Then the interpolation operator Ih,n of Clément =-=[5]-=- corresponding to Th,n is defined by Ih,nv = � x∈Nh,n λx(πxv)(x). (5.1) Due to condition (6) of Section 3 Ih,n maps L 1 (Ω) into a subspace of Xh,n. 5.1 Lemma. [9, Lemma 3.1] The following error estim... |

175 | Some a posteriori error estimates for elliptic partial differential equations - Bank, Weiser - 1985 |

154 | Adaptive finite element methods for parabolic problems iv: Nonlinear problems
- Eriksson, Johnson
- 1995
(Show Context)
Citation Context ... including the time-dependent incompressible Navier-Stokes equations, may be found in [10]. When applied to the corresponding particular examples, our error estimates are similar to those obtained in =-=[7]-=-, [8]. However, only upper bounds on the error are established there. Moreover, the techniques and, most important, the discretizations considerably differ from ours. The discontinuous Galerkin method... |

133 |
and Quasilinear Parabolic Problems
- Amann, Linear
- 1995
(Show Context)
Citation Context ... (0, T ; W 1,ρ 0 (Ω), W −1,π (Ω)), and � T 0 u(·, 0) = u0 in W −1,π (Ω) (2.1) 〈 ∂u � T � (·, t), v(·, t)〉dt + {a(x, u, ∇u)∇v + b(x, u, ∇u)v} dxdt = 0 ∂t 0 Ω ∀v ∈ L p′ (0, T ; W 1,π′ 0 (Ω)) (2.2) (cf. =-=[2]-=-). Note that W 1,ρ 0 (Ω) is continuously embedded in W −1,π (Ω) for all ρ and π since Ω ⊂ IR 2 . 4s3. Finite element discretization For the discretization we choose an integer N ≥ 1 and intermediate t... |

115 |
Adaptive Finite Element Methods for Differential Equations
- Bangerth, Rannacher
- 2003
(Show Context)
Citation Context ... have to pay for making computable this dual norm. This extra work is comparable to the one required by the now popular estimators that are based on the solution of suitable discrete adjoint problems =-=[3]-=-. We prove that the error estimator yields upper and lower bounds for the error measured in a suitable L r (0, T ; W 1,ρ 0 (Ω))-norm (cf. Section 2 for a definition of these spaces and their norms). T... |

92 | Toward a universal h-p adaptive finite element strategy. Part 2, A posteriori error estimates - Oden, Demkowicz, et al. - 1989 |

90 |
Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems
- Amann
- 1993
(Show Context)
Citation Context ...T;W 1,σ 0 (Ω)), + 1 p ′ =1, where ρ ≤ σ<∞. In order to better understand the flavour of problem (1.1) and definition (4.2), we recall the notions of weak and very weak solutions of problem (4.1) (cf. =-=[2]-=-, Sections 11 and 13). A function u ∈ W r (0,T;W 1,ρ 0 (Ω),W−1,π(Ω)) is called a weak solution of problem (4.1) if u(., 0) = u0 in W −1,π (Ω) and ∫ T 0 ∫ T ∫ 〈∂tu(., t),ϕ(., t)〉 W 1,π ′ + {a(x, u, ∇u)... |

53 | An Adaptive Multilevel Approach to Parabolic Problems - Bornemann - 1991 |

49 | An adaptive finite element method for linear elliptic problems - Eriksson, Johnson - 1988 |

48 |
Some optimal error estimates for piecewise linear element approximations
- Rannacher, Scott
- 1982
(Show Context)
Citation Context ...denote by RT : W 1,1 0 (Ω) → S1 (T ) the Ritz projection which is defined by � Ω � ∇(RT v)∇wT = Ω ∇v∇wT ∀v ∈ W 1,1 0 (Ω), wT ∈ S 1 (T ). Consider first the case q ∈ [1, 2]. Then we have q ′ ≥ 2. From =-=[7]-=- and [4, Chap. 7] we know that RT is stable in the W 1,q′ -norm, i.e., there is a constant cq ′ > 0 such that 1,q′ �∇(RT w)�0,q ′ ≤ cq ′�∇w�0,q ′ ∀w ∈ W0 22 (Ω).sThe constant cq ′ only depends on q′ ,... |

38 |
The finite element method for elliptic problems
- G
- 1978
(Show Context)
Citation Context ...on in space. Let Th be an admissible and shape regular partition of Ω into n-simplices. We denote by Ih : L1 (Ω) → S 1,0 h,0 the quasi-interpolation operator of Clément (cf. [5] and Exercise 3.2.3 in =-=[4]-=-). Lemma 3.2. The operator Ih satisfies the following error estimates for all K ∈ Th,E ∈Eh,and1≤p<∞: ‖u−Ihu‖ W k,p (K) ≼ h l−k K ‖u‖ W l,p (U(K)) ∀0 ≤ k ≤ l ≤ 2,u∈W l,p (U(K)), l− 1 p ‖u − Ihu‖Lp (E) ... |

32 | Time discretization of parabolic problems by the discontinuous Galerkin method - Eriksson, Johnson, et al. - 1985 |

31 | An a posteriori error estimate for finite element approximations of the Navier-Stokes equations, Comput. Methods - Oden, Wu, et al. - 1994 |

24 | A posteriori error estimates for the Stokes problem - Bank, Welfert - 1991 |

21 |
Sobolev spaces
- A, Adams
- 1975
(Show Context)
Citation Context ...ed open subset ω of Ω with Lipschitz boundary γ we denote by W k,p (ω), k ∈ IN, 1 ≤ p ≤ ∞, L p (ω) = W 0,p (ω), and L p (γ) the usual Sobolev and Lebesgue spaces equipped with the standard norms (cf. =-=[1]-=-, [6, Vol. 3, Chap. IV ]): and ⎧ ⎨ � � �u�k,p;ω = ⎩ |α|≤k |D ω α u(x)| p dx �u�k,∞;ω = max |α|≤k ess.sup |D α u(x)| �� �u�p;γ = x∈ω �u�∞;γ = ess.sup |u(x)|. x∈γ γ ⎫ ⎬ ⎭ 1,p , p < ∞, |u(x)| p �1,p ds(x... |

19 | The finite element method for parabolic equations. II. A posteriori error estimation and adaptive approach - Bieterman, Babuška - 1982 |

18 | Some remark on the Zienkiewicz{Zhu estimator - Rodriguez - 1994 |

14 | Rheinboldt Error estimates for adaptive finite element computations - Babu˜ka, C - 1978 |

13 | Basic principles of feedback and adaptive approaches in the finite element method - Babuˇska, Gui - 1986 |

13 | On the asymptotic exactness of error estimators for linear triangular finite elements - Durán, Muschietti, et al. - 1991 |

13 | An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem - Johnson, Nie, et al. - 1990 |

12 | Amri. Estimateurs a posteriori d’erreurs pour le calcul adaptatif d’écoulements quasi-newtoniens - Baranger, El - 1991 |

10 | On the asymptotic exactness of Bank-Weiser’s estimator - Duran, Rodriguez - 1992 |

9 | A local refinement finite element method for two-dimensional parabolic systems - Adjerid, Flaherty - 1988 |

8 |
The Dirichlet problem for the Laplacian in bounded and unbounded domains
- Simader, Sohr
- 1996
(Show Context)
Citation Context ...sed on the W 1,q -stability of the Laplacian both in its analytical and discrete form. We start with the analytical case. The following result is well-known for domains with smooth C 1 -boundary (cf. =-=[8]-=-). For polygonal domains, however, we are not aware of a proof. Recall that for q ∈ [1, ∞] the dual Lebesgue exponent is denoted by q ′ ∈ [1, ∞] and is defined by 1 q + 1 q ′ = 1. 8.1 Lemma. For every... |

8 | An adaptive local mesh refinement method for time dependent partial differential equations - Arney, Flaherty - 1989 |

8 | A posteriori error estimation with finite element semi- and fully discrete methods for nonlinear parabolic equations in one space dimension - Moore - 1994 |

7 | a posteriori estimates in finite elements: aims, theory, and experience - Feedback, adaptivity - 1986 |

6 | Navier-Stokes Equations 3rd Edition - Temam - 1984 |

2 | Thomée Galerkin finite element methods for parabolic problems - unknown authors - 1997 |

1 | Alt: Lineare Funktionalanalysis - W - 1985 |

1 | Scriven: An adaptive finite element method for steady and transient problems - Benner, Davis, et al. - 1987 |

1 | Improved accuracy by adapted mesh-refinement in the finite element method - Eriksson - 1985 |

1 | Rheinboldt: On a theory of mesh-refinement processes - C - 1980 |