Superconvergence analysis of an -Galerkin mixed finite element method for Sobolev equations

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• 作者：Dongyang Shi ; ^{shi_dy@zzu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Junjun Wang ^{wjunjun8888@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Sobolev equations ; H1H1-Galerkin MFEM ; Semi-discrete scheme ; Fully-discrete scheme ; Superconvergence results ; Bramble&ndash ; Hilbert lemma
• 刊名：Computers & Mathematics with Applications
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：72
• 期：6
• 页码：1590-1602
• 全文大小：447 K

文摘

An method=retrieve&_eid=1-s2.0-S0898122116304254&_mathId=si56.gif&_user=111111111&_pii=S0898122116304254&_rdoc=1&_issn=08981221&md5=4b462d525944bc613c8ccb1f1b2ebd97" title="Click to view the MathML source">H¹$H^1$-Galerkin mixed finite element method (MFEM) is discussed for the Sobolev equations with the bilinear element and zero order Raviart–Thomas element method=retrieve&_eid=1-s2.0-S0898122116304254&_mathId=si58.gif&_user=111111111&_pii=S0898122116304254&_rdoc=1&_issn=08981221&md5=2bd36e1a8b32087c3157bc3632feb5d6" title="Click to view the MathML source">(Q₁₁+Q₁₀×Q₀₁)$(Q_{11}+Q_{10}\times Q_{01})$. The existence and uniqueness of the solutions about the approximation scheme are proved. Two new important lemmas are given by using the properties of the integral identity and the Bramble–Hilbert lemma, which lead to the superclose results of order method=retrieve&_eid=1-s2.0-S0898122116304254&_mathId=si59.gif&_user=111111111&_pii=S0898122116304254&_rdoc=1&_issn=08981221&md5=4b15d77528c79c61b5a69f978d0f89bb" title="Click to view the MathML source">O(h²)$O\left(h^2\right)$ for original variable method=retrieve&_eid=1-s2.0-S0898122116304254&_mathId=si60.gif&_user=111111111&_pii=S0898122116304254&_rdoc=1&_issn=08981221&md5=e16216ccc485bdccfa14c26de77deefe" title="Click to view the MathML source">u$u$ in method=retrieve&_eid=1-s2.0-S0898122116304254&_mathId=si56.gif&_user=111111111&_pii=S0898122116304254&_rdoc=1&_issn=08981221&md5=4b462d525944bc613c8ccb1f1b2ebd97" title="Click to view the MathML source">H¹$H^1$ norm and flux the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116304254&_mathId=si62.gif&_user=111111111&_pii=S0898122116304254&_rdoc=1&_issn=08981221&md5=1cd0bc57bdbaea9311b00f68a8ed3d6c">the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0898122116304254-si62.gif"> in method=retrieve&_eid=1-s2.0-S0898122116304254&_mathId=si63.gif&_user=111111111&_pii=S0898122116304254&_rdoc=1&_issn=08981221&md5=c8e4ec76359e5c859f07c76056af923c" title="Click to view the MathML source">H(div;Ω)$H(div;\Omega )$ norm under semi-discrete scheme. Furthermore, two new interpolated postprocessing operators are put forward and the corresponding global superconvergence results are obtained. On the other hand, a second order fully-discrete scheme with superclose property method=retrieve&_eid=1-s2.0-S0898122116304254&_mathId=si64.gif&_user=111111111&_pii=S0898122116304254&_rdoc=1&_issn=08981221&md5=9395b01369038b4c918feb90c12d9657" title="Click to view the MathML source">O(h²+τ²)$O(h^2+{\tau }^2)$ is also proposed. At last, numerical experiment is included to illustrate the feasibility of the proposed method. Here method=retrieve&_eid=1-s2.0-S0898122116304254&_mathId=si65.gif&_user=111111111&_pii=S0898122116304254&_rdoc=1&_issn=08981221&md5=4fbe7d49811f9058f22c4a1e7e046e8e" title="Click to view the MathML source">h$h$ is the subdivision parameter and method=retrieve&_eid=1-s2.0-S0898122116304254&_mathId=si66.gif&_user=111111111&_pii=S0898122116304254&_rdoc=1&_issn=08981221&md5=7bc9eddc77253fb21bd31253665d5424" title="Click to view the MathML source">τ$\tau$ is the time step.