Variational approximation of flux in conforming finite element methods for elliptic partial differential equations : a model problem

Brezzi, Franco ; Hughes, Thomas J. R. ; Süli, Endre

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 12 (2001), p. 159-166 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

We consider the approximation of elliptic boundary value problems by conforming finite element methods. A model problem, the Poisson equation with Dirichlet boundary conditions, is used to examine the convergence behavior of flux defined on an internal boundary which splits the domain in two. A variational definition of flux, designed to satisfy local conservation laws, is shown to lead to improved rates of convergence.

Résumé

Si affronta il problema di approssimare il flusso del gradiente della soluzione di un problema ai limiti per una equazione lineare ellittica del secondo ordine, prendendo come problema modello il problema di Dirichlet per l’operatore di Laplace in due dimensioni spaziali. La linea $\Gamma$ lungo la quale si vuole approssimare il flusso è supposta essere rettilinea. L’approssimazione è costruita con elementi finiti continui e localmente polinomiali di grado $\le k$, con $k$ intero $\ge 1$. Tramite una opportuna definizione variazionale del flusso approsssimato, si ottengono stime dell’errore ottimali in spazi del tipo di $H^{-k-\frac{1}{2}} (\Gamma)$.

Détail

Publié le : 2001-09-01

MR 1898457 | Zbl 1221.65304

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