By Rüdiger Verfürth

A posteriori mistakes estimation recommendations are basic to the effective numerical resolution of PDEs coming up in actual and technical purposes. This publication offers a unified method of those concepts and publications graduate scholars, researchers, and practitioners in the direction of knowing, utilising and constructing self-adaptive discretization methods.

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Similar indicators based on a local averaging or extrapolation of the gradient are analysed in [13, 16, 128, 137, 138, 150, 169, 281]. In [49, 93] it is proved that each averaging technique leads to an a posteriori error estimate. 10 Equilibrated Residuals In [172] P. Ladevèze and D. Leguillon proposed a technique for a posteriori error estimation which is based on a dual variational principle. In what follows we will brieﬂy sketch the underlying idea. For a more detailed presentation we refer to [11] and [172].

25) this extra condition may be dropped at the expense of slightly larger perturbation terms close to the boundary. 15 involving g vanish. 31 Assume that the partition T exclusively consists of triangles and that the Neumann boundary N is the empty set. 3) (p. 5). 15 (p. 57). Then there are two constants cZ1 and cZ2 which only depend on the shape parameter of T such that the estimates cZ1 ηR,E ≤ ηZ ≤ cZ2 ηR,E are valid. Proof Since T exclusively consists of triangles, the normal derivatives nE · ∇uT are edge-wise constant.

29) is based on the following auxiliary result. 10 For every vertex z ∈ N there are constants c2 (ωz ) and c2 (σz ) such that, for a suitable choice wz ∈ R with wz λz ∈ S1,0 D (T ), there hold the following Poincaré-type inequalities 1 1 λz2 (w – wz ) 1 2 h⊥ E λz (w – wz ) ωz 1 2 2 ≤ c2 (ωz )hz λz2 ∇w 1 ≤ c2 (σz )hz λz2 ∇w , ωz E E⊂σz , ωz where hz denotes the diameter of ωz and where h⊥ E = λz ωE E λz . 8 (p. 107), cf. 29 (p. 107). The proof is based on weighted trace inequalities, cf. 3 (p. 87), weighted Poincaré inequalities, cf.