On a generalized \(L_2\) projection and some related stability estimates in Sobolev spaces
type: Item
instance of: scholarly article
title: On a generalized \(L_2\) projection and some related stability estimates in Sobolev spaces
zbMATH Open document ID: 0997.65120
published in: Numerische Mathematik
publication date: 2002-05-21T00:00:00Z
review text: The paper deals with the stability estimate needed in hybrid finite and boundary element methods, especially in hybrid coupled domain decomposition methods including mortar finite elements. This stability estimate is equivalent to the stability of a generalized \(L_2\) projection in certain Sobolev spaces. The construction of the dual mesh is described. The stability estimate \[ {1\over c_s} \|u_h\|_{H^s(\Gamma)}\leq \sup_{0\neq W_h\in W_h} {\langle u_h,w_h\rangle\over \|w_h\|_{\widetilde H^{-s}(\Gamma)}}\text{ for all }u_h\in V_h\subset H^s(\Gamma),\;c_s> 0, \] is crucial for the stability analysis of hybrid coupled domain decomposition methods. (Here, \(\Gamma\) is an open or closed Lipschitz' manifold in \(\mathbb{R}^n\), \(n= 2,3\) and \(V_h\subset H^s(\Gamma)\) with \(s\in (0,1)\) is given finite element space, \(W_h\subset (H^s(\Gamma))^*\).) Main result: Using piecewise linear trial spaces and appropriate piecewise constant test spaces, the stability of the generalized \(L_2\) projection is proved assuming some mesh conditions locally.
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65N38
Mathematics Subject Classification ID: 65N55
zbMATH DE Number: 1742797
zbMATH Keywords: stability estimate
zbMATH Keywords: hybrid finite element method
zbMATH Keywords: boundary element method
zbMATH Keywords: domain decomposition method
zbMATH Keywords: mortar finite elements
zbMATH Keywords: generalized \(L_2\) projection
zbMATH Keywords: Sobolev spaces
author: O. Steinbach
reviewed by: Jan Lovíšek
MaRDI profile type: Publication
full work available at URL: s002110100329
OpenAlex ID: W2040746334
DOI: 10.1007/S002110100329
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description: scientific article; zbMATH DE number 1742797