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 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.

翻译：研究表明，基于变分或弱公式的简支边界条件板弯曲问题的离散化，在使用多边形域近似且施加的边界条件与某些正则函数限制到近似域上的节点插值相容时，不会导致收敛失败。进一步证明，此结果在以下意义下是最优的：对于协调方法，完全实现边界条件会导致收敛失败。这些抽象条件意味着标准非协调和间断Galerkin方法能正确收敛，而协调方法需要适当放松边界条件。数值实验验证了该结论。