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 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 (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 O(h²)$O\left(h^2\right)$ for original variable u$u$ in H¹$H^1$ norm and flux $\overrightarrow{q}$ in 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 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 h$h$ is the subdivision parameter and τ$\tau$ is the time step.