Abstract We consider the Lame system posed in a domain with the reentrant corner of 2 π as a mathematical model for the crack problem. We construct a version of the weighted finite-element method (FEM) on the base of a novel definition of the R ν -generalized solution. This allows us to suppress the influence of the singularity caused by the presence of the reentrant corner on the accuracy of computation of the approximate solution. Comparative numerical analysis of the presented approach with classical FEM and the method with mesh refinement has shown its advantages in the computational accuracy and stability as well as in the use of high-dimensional meshes. The results of investigation of the accuracy of solution to the model problem are presented in the Sobolev and energy norms. An absolute error in the mesh nodes is also analyzed.

中文翻译：

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带裂纹弹性问题的加权有限元法

摘要 我们考虑在重入角为 2 π 的域中提出的 Lame 系统作为裂纹问题的数学模型。我们在 R ν -广义解的新定义的基础上构建了加权有限元方法 (FEM) 的一个版本。这使我们能够抑制由于重入角的存在而引起的奇异性对近似解计算精度的影响。所提出的经典有限元方法和网格细化方法的比较数值分析显示了其在计算精度和稳定性以及使用高维网格方面的优势。模型问题解的准确性的调查结果在 Sobolev 和能量范数中给出。还分析了网格节点中的绝对误差。