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Partial diff. equations and functional analysis: the by Koelink E., van Neerven J., et al. (eds.)

By Koelink E., van Neerven J., et al. (eds.)

Shooting the cutting-edge of the interaction among partial differential equations, practical research, maximal regularity, and likelihood idea, this quantity was once initiated on the Delft convention at the party of the retirement of Philippe Clément. will probably be of curiosity to researchers in PDEs and useful research.

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Example text

May enter v). 3. Residual representation formula Without stating examples in explicit form, it is stressed that linear and nonlinear boundary value problems of second order elliptic PDEs typically lead (after some partial integration) to some explicit representation of the residual R := b−a(uh , ·). Let T denote the underlying regular triangulation of the domain and let E denote a set of edges for n = 2. 3) E holds for all v ∈ V . The goal of a posteriori error control is to establish guaranteed lower and upper bounds of the (energy norm of the) error and hence of the (dual norm of the) residual.

Set f := z − zh . Based on the Galerkin orthogonality a(e, zh ) = 0 one infers (e) = a(e, z) = a(e, z − zh ) = a(e, f ). 10) Cauchy inequalities lead to the a posteriori estimate | (e)| ≤ e a f a ≤ ηu ηz . Indeed, utilizing the primal and dual residual Ru and Rz in V ∗ , defined by Ru := b − a(uh , ·) and Rz := − a(·, zh ), computable upper error bounds for e V ≤ ηu and f V ≤ ηz can be found by the arguments of the energy error estimators [1, 3]. Indeed, the parallelogram rule shows 2 (e) = 2 a(e, f ) = e + f 2 a − e 2 a − f 2 a.

Bartels, S. and Klose, R. (2001). An experimental survey of a posteriori Courant finite element error control for the Poisson equation. Adv. Comput. , 15, 1-4, 79–106. [14] Carstensen, C. A. Constants in Cl´ ement-interpolation error and residual based a posteriori estimates in finite element methods. East-West J. Numer. Math. 8(3) (2000) 153–175. [15] Carstensen, C. A. Fully reliable localised error control in the FEM. SIAM J. Sci. Comp. 21(4) (2000) 1465–1484. [16] Carstensen, C. and Verf¨ urth, R.

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