The simplest nonconforming mixed finite element method for linear elasticity in the symmetric formulation on -rectangular grids

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• 作者：Jun Hu^a ; ^b ; ^{hujun@math.pku.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Hongying Man^c ; ^{manhy@bit.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Jianye Wang^a ; ^{jianguoyeluo@163.com" class="auth_mail" title="E-mail the corresponding author} ; Shangyou Zhang^d ; ^{szhang@udel.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：First order system ; Symmetric stress field ; Nonconforming mixed finite element ; Inf&ndash ; sup condition
• 刊名：Computers & Mathematics with Applications
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：71
• 期：7
• 页码：1317-1336
• 全文大小：472 K

文摘

A family of mixed finite elements is proposed for solving the first order system of linear elasticity equations in any space dimension, where the stress field is approximated by symmetric finite element tensors. This family of elements has a perfect matching between the stress and the displacement. The discrete spaces for the normal stress class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300190&_mathId=si42.gif&_user=111111111&_pii=S0898122116300190&_rdoc=1&_issn=08981221&md5=39164ff97ef886729a39253c75067a3a" title="Click to view the MathML source">τ_iiclass="mathContainer hidden">class="mathCode">${\tau }_{ii}$, the shear stress class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300190&_mathId=si43.gif&_user=111111111&_pii=S0898122116300190&_rdoc=1&_issn=08981221&md5=776d178f3bee0afbfaed8b286fd4570c" title="Click to view the MathML source">τ_ijclass="mathContainer hidden">class="mathCode">${\tau }_{ij}$ and the displacement class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300190&_mathId=si44.gif&_user=111111111&_pii=S0898122116300190&_rdoc=1&_issn=08981221&md5=1c257a777128c3302639b5bcdab158c9" title="Click to view the MathML source">u_iclass="mathContainer hidden">class="mathCode">$u_i$ are class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300190&_mathId=si45.gif&_user=111111111&_pii=S0898122116300190&_rdoc=1&_issn=08981221&md5=51e0cfb0695523e46a1c7e4bd7847a15" title="Click to view the MathML source">span{1,x_i}class="mathContainer hidden">class="mathCode">$span\{1,x_i\}$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300190&_mathId=si46.gif&_user=111111111&_pii=S0898122116300190&_rdoc=1&_issn=08981221&md5=d6171dcf971f80773d20927fbcedd4b1" title="Click to view the MathML source">span{1,x_i,x_j}class="mathContainer hidden">class="mathCode">$span\{1,x_i,x_j\}$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300190&_mathId=si47.gif&_user=111111111&_pii=S0898122116300190&_rdoc=1&_issn=08981221&md5=e2adb2bce7a1b3c8cfcc63d895f9ea2c" title="Click to view the MathML source">span{1}class="mathContainer hidden">class="mathCode">$span\left\{1\right\}$, respectively, on rectangular grids. In particular, the definition remains the same for all space dimensions. As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. In 1D, the element is nothing else but the 1D Raviart–Thomas element, which is the only conforming element in this family. In 2D and higher dimensions, they are new elements but of the minimal degrees of freedom. The total degrees of freedom per element are class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300190&_mathId=si48.gif&_user=111111111&_pii=S0898122116300190&_rdoc=1&_issn=08981221&md5=cf11269ff309de940cfc0e26eabc3fb3" title="Click to view the MathML source">2class="mathContainer hidden">class="mathCode">$2$ plus class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300190&_mathId=si49.gif&_user=111111111&_pii=S0898122116300190&_rdoc=1&_issn=08981221&md5=1092d59951fd7064274b266fade43d95" title="Click to view the MathML source">1class="mathContainer hidden">class="mathCode">$1$ in 1D, 7 plus 2 in 2D, and 15 plus 3 in 3D. These elements are the simplest element for any space dimension.

The well-posedness condition and the optimal a priori error estimate of the family of finite elements are proved. Numerical tests in 2D and 3D are presented to show a superiority of the new elements over others, as a superconvergence is exhibited and proved.