A non-intrusive extension to the standard $p$-version of the finite element method is proposed. Meshes with hanging nodes are handled by adapting the reference elements so that the resulting discretisation is always conforming. The shape functions on these adaptive reference elements are not polynomials, but either harmonic extensions of the boundary restrictions of the standard shape functions or solutions to a local Poisson problem. The numerical experiments are taken from computational function theory and the efficiency of the proposed extension resulting in exponential convergence in the quantities of interest is demonstrated.

中文翻译：

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通过谐波扩展和相关内部模式的自适应参考元件

对标准的非侵入式扩展 $p$提出了有限元方法的版本化方法。带有悬挂节点的网格通过调整参考元素进行处理，以使最终的离散化始终符合要求。这些自适应参考元素上的形状函数不是多项式，而是标准形状函数的边界限制的谐波扩展或局部泊松问题的解决方案。数值实验取自计算函数理论，并证明了所提出的扩展的效率，该扩展导致感兴趣量的指数收敛。