It is shown that discretizations based on variational or weak formulations of the plate bending problem with simple support boundary conditions do not lead to the failure of convergence when polygonal domain approximations are used and the imposed boundary conditions are compatible with the nodal interpolation of the restriction of certain regular functions to approximating domains. It is further shown that this is optimal in the sense that a full realization of the boundary conditions leads to failure of convergence for conforming methods. The abstract conditions imply that standard nonconforming and discontinuous Galerkin methods converge correctly while conforming methods require a suitable relaxation of the boundary condition. The results are confirmed by numerical experiments.

中文翻译：

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单纯网格上避免 Babuška 悖论的充要条件


结果表明，当使用多边形域近似并且施加的边界条件与限制的节点插值兼容时，基于​​具有简单支撑边界条件的板弯曲问题的变分或弱公式的离散化不会导致收敛失败。某些正则函数来逼近域。进一步表明，从边界条件的完全实现导致一致方法收敛失败的意义上来说，这是最优的。抽象条件意味着标准的非相容和不连续伽辽金方法正确收敛，而相容方法需要适当放宽边界条件。结果通过数值实验得到证实。