Matrix time-extrapolation algorithm for solving semilinear parabolic problems

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• 作者：Xiangqi Wang ^{xiangqi.wang@forxmail.com" class="auth_mail" title="E-mail the corresponding author} ; Chuanmiao Chen ; ^{cmchen@hunnu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Crank&ndash ; Nicolson ; Finite element ; Matrix time-extrapolation ; Semilinear parabolic
• 刊名：Applied Mathematics Letters
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：64
• 期：Complete
• 页码：162-169
• 全文大小：1088 K

文摘

In this letter, we propose a fast matrix time-extrapolation algorithm to solve semilinear parabolic problems of Crank–Nicolson-based finite element scheme, which employs exact matrix values computed by integral at time levels class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302762&_mathId=si19.gif&_user=111111111&_pii=S0893965916302762&_rdoc=1&_issn=08939659&md5=4e7fc6c0332dc5ab13a15139c51eae51" title="Click to view the MathML source">mclass="mathContainer hidden">class="mathCode">$m$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302762&_mathId=si20.gif&_user=111111111&_pii=S0893965916302762&_rdoc=1&_issn=08939659&md5=6995d5ed613324b5c0b0b8d13427522a" title="Click to view the MathML source">m+pclass="mathContainer hidden">class="mathCode">$m+p$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302762&_mathId=si21.gif&_user=111111111&_pii=S0893965916302762&_rdoc=1&_issn=08939659&md5=68179437cace877c1c0a94f2d01a7a96" title="Click to view the MathML source">m+2pclass="mathContainer hidden">class="mathCode">$m+2p$ to construct quadratic interpolation so that we can estimate matrix values at levels class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302762&_mathId=si22.gif&_user=111111111&_pii=S0893965916302762&_rdoc=1&_issn=08939659&md5=3c64f83b14c86bb61f2131ca8096c0c1" title="Click to view the MathML source">m+2p+1,m+2p+2,…,m+3p−1class="mathContainer hidden">class="mathCode">$m+2p+1,m+2p+2,\dots ,m+3p-1$, then the matrix value is recalculated at the level class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302762&_mathId=si23.gif&_user=111111111&_pii=S0893965916302762&_rdoc=1&_issn=08939659&md5=7a9f614be85b946bae1cc7b88476ee11" title="Click to view the MathML source">m+3pclass="mathContainer hidden">class="mathCode">$m+3p$. This process is performed iteratively, and finally, the calculation for matrices decreases to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302762&_mathId=si24.gif&_user=111111111&_pii=S0893965916302762&_rdoc=1&_issn=08939659&md5=13d6da0b6ceff25c118663b54e100aac" title="Click to view the MathML source">1∕pclass="mathContainer hidden">class="mathCode">$1∕p$. The error estimate of this algorithm is proven, and numerical examples are established to support this theory.