Mimetic finite difference approximation of quasilinear elliptic problems
type: Item
instance of: scholarly article
title: Mimetic finite difference approximation of quasilinear elliptic problems
zbMATH Open document ID: 1316.65092
author: Nadia Bigoni
author: Paola F. Antonietti
author: M. Verani
published in: Calcolo
publication date: 2015-03-23T00:00:00Z
review text: The authors consider the following quasilinear elliptic problem \[ -\mathrm{div} (\kappa (|\nabla u|^2)\nabla u)=f \quad \text{in}\, \Omega, \qquad u=0\quad \text{on}\, \partial \Omega , \tag{1} \] where \(f\) is a given function and \(\Omega\) is an open, bounded set of \(\mathbb{R}^2\) with boundary \(\partial \Omega\). For the approximation of the problem (1), the mimetic finite difference (MFD) method is used. It is shown that the MFD-solution converges to the solution of (1) in a suitable discrete energy norm. The nonlinear discrete problem is solved iteratively via linearization by applying the Kačanov method. The convergence of this method is proved in the discrete mimetic framework. In numerical examples the function \(\kappa(t)\) is chosen equal to the Carreau law: \(\kappa(t)=\eta_\infty+(\eta_0-\eta_\infty)(1+\lambda t)^{\frac{p-2}{p}}\), \(t\geq 0\), with \(\eta_0=3\), \(\eta_\infty=1\) and \(p=1.7\) and with a known solution.
reviewed by: Vladimir L. Makarov
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 35J62
zbMATH DE Number: 6418685
zbMATH Keywords: Kačanov method
zbMATH Keywords: quasilinear elliptic problems
zbMATH Keywords: mimetic finite difference method
zbMATH Keywords: convergence
zbMATH Keywords: numerical example
MaRDI profile type: Publication
full work available at URL: s10092-014-0107-y
OpenAlex ID: W2122838361
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DOI: 10.1007/S10092-014-0107-Y
description: scientific article; zbMATH DE number 6418685