A Nitsche-type method for Helmholtz equation with an embedded acoustically permeable interface

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• 作者：Esubalewe Lakie Yedeg^a ; ^{yedegl@cs.umu.se" class="auth_mail" title="E-mail the corresponding author} ; Eddie Wadbro^a ; ^{eddiew@cs.umu.se" class="auth_mail" title="E-mail the corresponding author} ; Peter Hansbo^b ; ^{Peter.Hansbo@ju.se" class="auth_mail" title="E-mail the corresponding author} ; Mats G. Larson^c ; ^{mats.larson@math.umu.se" class="auth_mail" title="E-mail the corresponding author} ; Martin Berggren^a ; ^{martin.berggren@cs.umu.se" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Helmholtz equation ; Finite element method ; Nitsche&rsquo ; s method ; Acoustic impedance ; Surface wave ; Gå ; rding inequality
• 刊名：Computer Methods in Applied Mechanics and Engineering
• 出版年：2016
• 出版时间：1 June 2016
• 年：2016
• 卷：304
• 期：Complete
• 页码：479-500
• 全文大小：1261 K

文摘

We propose a new finite element method for Helmholtz equation in the situation where an acoustically permeable interface is embedded in the computational domain. A variant of Nitsche’s method, different from the standard one, weakly enforces the impedance conditions for transmission through the interface. As opposed to a standard finite-element discretization of the problem, our method seamlessly handles a complex-valued impedance function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0045782516300627&_mathId=si29.gif&_user=111111111&_pii=S0045782516300627&_rdoc=1&_issn=00457825&md5=54f40e5810539f96448bd783ff90b905" title="Click to view the MathML source">Zclass="mathContainer hidden">class="mathCode">$Z$ that is allowed to vanish. In the case of a vanishing impedance, the proposed method reduces to the classic Nitsche method to weakly enforce continuity over the interface. We show stability of the method, in terms of a discrete Gårding inequality, for a quite general class of surface impedance functions, provided that possible surface waves are sufficiently resolved by the mesh. Moreover, we prove an a priori error estimate under the assumption that the absolute value of the impedance is bounded away from zero almost everywhere. Numerical experiments illustrate the performance of the method for a number of test cases in 2D and 3D with different interface conditions.