The Birth and Evolution of Fuzzy Logic by Zadeh L.A.

By Zadeh L.A.

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Osserviamo che la matrice di rigidezza nel caso di elementi finiti di grado 2 sarà pentadiagonale. 5). Inoltre, come ben noto dalla teoria dell’interpolazione polinomiale, l’uso di gradi elevati combinato con distribuzioni equispaziate dei nodi, conduce ad approssimazioni sempre meno stabili, a discapito dell’aumento teorico di accuratezza. Un rimedio brillante è fornito dall’approssimazione agli elementi spettrali che, facendo uso di nodi opportuni (quelli delle formule di quadratura di Gauss), consente di generare approssimazioni di accuratezza arbitrariamente elevata.

R Indicato con Π K l’operatore di interpolazione locale che associa ad una funzione v r il polinomio Π K v ∈ Pr (K), interpolante v negli r + 1 nodi dell’elemento K ∈ T h , definiamo Πhr v come Πhr v ∈ Xhr : Πhr v K r = ΠK (v K ∀K ∈ Th . 18). Vale allora la seguente stima dell’errore di interpolazione: |v − Πhr v|Hk (I) ≤ Ck,r hr+1−k |v|Hr+1 (I) per k = 0, 1. 19) Le costanti Ck,r sono indipendenti da v ed h. Ricordiamo che H 0 (I)=L2 (I) e che | · |H0 (I) = · L2 (I) . Dimostrazione. 19) per il caso r = 1, rimandando a [QV94, Cap.

Problemi ellittici per operatori del quarto ordine sono proposti negli Esercizi 4 e 6, mentre un problema ellittico derivante dalla teoria dell’elasticità lineare è affrontato nell’Esercizio 7. 1 (di Lax-Milgram) Sia V uno spazio di Hilbert, a(·, ·) : V ×V → R una forma bilineare continua e coerciva. Sia F (·) : V → R un funzionale lineare e continuo. Allora esiste unica la soluzione del problema: trovare u ∈ V : a(u, v) = F (v) ∀v ∈ V. 32) Dimostrazione. Si basa su due classici risultati di Analisi Funzionale: il teorema di rappresentazione di Riesz, ed il teorema dell’immagine chiusa di Banach.

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