Download A posteriori error estimation techniques for finite element by Rüdiger Verfürth PDF
By Rüdiger Verfürth
A posteriori mistakes estimation options are primary to the effective numerical resolution of PDEs bobbing up in actual and technical purposes. This booklet offers a unified method of those concepts and publications graduate scholars, researchers, and practitioners in the direction of figuring out, utilizing and constructing self-adaptive discretization methods.
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4 (p. 9) implies for the left-hand side of this equation E JE (nE · ∇uT )2 ψE ≥ c–2 I,3 JE (nE · ∇uT ) 2 E and for the three terms on its right-hand side ωE K⊂ωE ∇(u – uT ) · ∇wE ≤ ∇(u – uT ) ωE ≤ ∇(u – uT ) ωE K (f K + uT )wE ≤ fK + uT fK + uT cI,4 hE 2 JE (nE · ∇uT ) wE K E , K 1 K⊂ωE K ωE –1 K⊂ωE ≤ K⊂ωE ∇wE (f – f K )wE ≤ f – fK K⊂ωE K · cI,5 hE2 JE (nE · ∇uT ) wE K E , K 1 ≤ f – fK K⊂ωE K cI,5 hE2 JE (nE · ∇uT ) E . This yields c–2 I,3 JE (nE · ∇uT ) –1 E ≤ cI,4 hE 2 ∇(u – uT ) 1 cI,5 hE2 + K⊂ωE ωE 1 fK + uT K cI,5 hE2 f – f K + K⊂ωE K .
W∈HT μ∈M μ∈M w∈HT Hence, we get for all μ ∈ M – 1 ∇(u – uT ) 2 2 ≥ inf L(w, μ) w∈HT 1 2 = inf w∈HT K∈T ∇w · ∇w – ∇uT · ∇w – fw + K K K ∂K γK w + μ∗ (w) – μ(w) . The particular choice μ = μ∗ therefore yields ∇(u – uT ) 2 ≤ –2 inf w∈HT K∈T 1 2 ∇w · ∇w – K ∇uT · ∇w – fw + K K ∂K γK w . 4 Localisation of the Quadratic Functional In order to write the last identity in a more compact form, we denote by HK the restriction of HD1 ( ) to a single element K ∈ T and set for all w ∈ HK and all K ∈ T JK (w) = ∼ We then have HT = K∈T 1 2 ∇w · ∇w – K ∇uT · ∇w – fw + K K ∂K γK w.
1⎠ –1 2 EQUILIBRATED RESIDUALS | 41 its eigenvalues are given by λk (B) = 4 sin2 kπ 2N , 1 ≤ k ≤ N – 1. 2] on the other hand implies that the eigenvalues λk (A) of A are bounded by min μ2i ≤ λk (A) ≤ 2 for all 1 ≤ k ≤ N – 1. 1≤i≤N Hence, we have for all x ∈ RN 1 min μ2 xt Bx ≤ xt Ax, 4 1≤i≤N i π 2N 2 sin2 xt Ax ≤ xt Bx. 60). 32 The indicator ηZ was ﬁrst proposed by J. Z. Zhu and O. C. Zienkiewicz in . The present analysis follows the lines of . Similar indicators based on a local averaging or extrapolation of the gradient are analysed in [13, 16, 128, 137, 138, 150, 169, 281].