A time dependent approach for removing the cell boundary error in elliptic homogenization problems

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• 作者：Doghonay Arjmand^a ; ^{doghonay.arjmand@it.uu.se" class="auth_mail" title="E-mail the corresponding author} ; Olof Runborg^b ; ^{olofr@kth.se" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Multiscale problems ; Homogenization ; Elliptic PDEs
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：1 June 2016
• 年：2016
• 卷：314
• 期：Complete
• 页码：206-227
• 全文大小：1372 K

文摘

This paper concerns the cell-boundary error present in multiscale algorithms for elliptic homogenization problems. Typical multiscale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. To solve the micro model, boundary conditions are required on the boundary of the microscopic domain. Imposing a naive boundary condition leads to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021999116001601&_mathId=si1.gif&_user=111111111&_pii=S0021999116001601&_rdoc=1&_issn=00219991&md5=d1e08fb1b4aa36dca3b9fce17c20ea2a" title="Click to view the MathML source">O(ε/η)class="mathContainer hidden">class="mathCode">$O(\epsilon /\eta )$ error in the computation, where ε is the size of the microscopic variations in the media and η   is the size of the micro-domain. The removal of this error in modern multiscale algorithms still remains an important open problem. In this paper, we present a time-dependent approach which is general in terms of dimension. We provide a theorem which shows that we have arbitrarily high order convergence rates in terms of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021999116001601&_mathId=si2.gif&_user=111111111&_pii=S0021999116001601&_rdoc=1&_issn=00219991&md5=ab4da10ed818a376935ed5ae946d31c6" title="Click to view the MathML source">ε/ηclass="mathContainer hidden">class="mathCode">$\epsilon /\eta$ in the periodic setting. Additionally, we present numerical evidence showing that the method improves the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021999116001601&_mathId=si1.gif&_user=111111111&_pii=S0021999116001601&_rdoc=1&_issn=00219991&md5=d1e08fb1b4aa36dca3b9fce17c20ea2a" title="Click to view the MathML source">O(ε/η)class="mathContainer hidden">class="mathCode">$O(\epsilon /\eta )$ error to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021999116001601&_mathId=si3.gif&_user=111111111&_pii=S0021999116001601&_rdoc=1&_issn=00219991&md5=6f455e3c920a9637ac350cfb2fcbf1f5" title="Click to view the MathML source">O(ε)class="mathContainer hidden">class="mathCode">$O(\epsilon )$ in general non-periodic media.