Proofs of the stability and convergence of a weakened weak method using PIM shape functions

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• 作者：J.H. Yue^a ; ^{woyuejunhong@163.com" class="auth_mail" title="E-mail the corresponding author} ; M. Li^a ; ^{liming01@tyut.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; G.R. Liu^b ; ^c ; ^{liugr@ucmail.uc.edu" class="auth_mail" title="E-mail the corresponding author} ; R.P. Niu^a ; ^{niu_ruiping2007@126.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：GG space ; Smoothed point interpolation method ; Weakened weak form ; Stability ; Convergence
• 刊名：Computers & Mathematics with Applications
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：72
• 期：4
• 页码：933-951
• 全文大小：1025 K

文摘

Recently, the smoothed point interpolation method (S-PIM) regarded as a weakened weak (class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116303340&_mathId=si22.gif&_user=111111111&_pii=S0898122116303340&_rdoc=1&_issn=08981221&md5=e8900f7c0332fac17d97ed6c3a9a2546">class="imgLazyJSB inlineImage" height="14" width="23" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0898122116303340-si22.gif">class="mathContainer hidden">class="mathCode">$W^2$) formulation method has been developed for solving engineering mechanics problems. It works well with distorted meshes. The class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116303340&_mathId=si23.gif&_user=111111111&_pii=S0898122116303340&_rdoc=1&_issn=08981221&md5=9eb7d0175f4290ea9d08500515f6ee0f" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$ space theory offers the theoretical base for all the class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116303340&_mathId=si22.gif&_user=111111111&_pii=S0898122116303340&_rdoc=1&_issn=08981221&md5=e8900f7c0332fac17d97ed6c3a9a2546">class="imgLazyJSB inlineImage" height="14" width="23" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0898122116303340-si22.gif">class="mathContainer hidden">class="mathCode">$W^2$ methods that use smoothing operations. In this paper, we first prove mathematically that if a function is of Lipschitz continuity and its interpolated function is established using PIM shape functions, then the interpolated function belongs to a class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116303340&_mathId=si25.gif&_user=111111111&_pii=S0898122116303340&_rdoc=1&_issn=08981221&md5=40de56bd8c7da6e46b0b927c3bfa614a">class="imgLazyJSB inlineImage" height="19" width="27" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0898122116303340-si25.gif">class="mathContainer hidden">class="mathCode">$G_{h,0}^s$ space. Our proofs work for smoothing operations that are the node-based, cell-based and a mixture of both smoothing domains. In addition, when mesh is refined under a given regularity condition, a sufficiently smooth target function can be approximated by its interpolated function with arbitrary accuracy, meaning that the interpolation error norm approaches to zero. Therefore, the stability and convergence of a class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116303340&_mathId=si22.gif&_user=111111111&_pii=S0898122116303340&_rdoc=1&_issn=08981221&md5=e8900f7c0332fac17d97ed6c3a9a2546">class="imgLazyJSB inlineImage" height="14" width="23" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0898122116303340-si22.gif">class="mathContainer hidden">class="mathCode">$W^2$ method using PIM shape functions and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116303340&_mathId=si23.gif&_user=111111111&_pii=S0898122116303340&_rdoc=1&_issn=08981221&md5=9eb7d0175f4290ea9d08500515f6ee0f" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$ space theory can be ensured.