A finite element method for singular solutions of the Navier-Stokes equations on a non-convex polygon

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• 作者：Hyung Jun Choi ; ¹ ; ^{choihjs@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Jae Ryong Kweon ^{kweon@postech.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：65N12 ; 65N30
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：292
• 期：Complete
• 页码：342-362
• 全文大小：1676 K

文摘

It is shown in Choi and Kweon (2013) that a solution of the Navier–Stokes equations with no-slip boundary condition on a non-convex polygon can be written as $[u,p]={\mathcal{C}}_1[{桅}_1,{蠒}_1]+{\mathcal{C}}_2[{桅}_2,{蠒}_2]+[u_{\text{<small-caps>r</small-caps>}},p_{\text{<small-caps>r</small-caps>}}]$ near each non-convex vertex, where $[u_{\text{<small-caps>r</small-caps>}},p_{\text{<small-caps>r</small-caps>}}]\in H^2\times H^1$, [桅_i,蠒_i]$[{桅}_i,{蠒}_i]$ are corner singularity functions for the Stokes problem with no-slip condition, and 1609026877fad439812" title="Click to view the MathML source">C_i∈R${\mathcal{C}}_i\in \mathbb{R}$ are coefficients which are called the stress intensity factors. We design a finite element method to approximate the coefficients C_i${\mathcal{C}}_i$ and the regular part $[u_{\text{<small-caps>r</small-caps>}},p_{\text{<small-caps>r</small-caps>}}]$, show the unique existence of the approximations, and derive their error estimates. Some numerical examples are given, confirming convergence rates for the approximations.