Abstract Two novel finite element schemes were earlier proposed to reduce the meshing effort needed for practical finite element analysis and their promising performance was demonstrated in the AMORE (AMORE stands for Automatic Meshing with Overlapping and Regular Elements) framework. In the first scheme “overlapping finite elements” are established that combine advantages of meshless and traditional finite element methods. A key step is to use polynomial interpolations for the rational shape functions in the meshless method. The scheme enables effective, accurate, and element distortion insensitive numerical solutions. In the second scheme, individual meshes are allowed to overlap quite freely. In our earlier papers we gave illustrative examples and also brief discussions on the convergence of the schemes when used in AMORE. We now focus on presenting deeper insights into the convergence properties through theory and novel illustrative solutions.

中文翻译：

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关于重叠单元和重叠网格的收敛

摘要 早先提出了两种新颖的有限元方案，以减少实际有限元分析所需的网格划分工作，并且在 AMORE（AMORE 代表具有重叠和规则元素的自动网格划分）框架中证明了它们的良好性能。在第一个方案中，“重叠有限元”结合了无网格和传统有限元方法的优点。一个关键步骤是在无网格方法中对有理形函数使用多项式插值。该方案可实现有效、准确且对元素失真不敏感的数值解。在第二种方案中，允许单独的网格非常自由地重叠。在我们之前的论文中，我们给出了说明性的例子，并简要讨论了在 AMORE 中使用时方案的收敛性。