The approximation properties of the finite element method can often be substantially improved by choosing smooth high-order basis functions. It is extremely difficult to devise such basis functions for partitions consisting of arbitrarily shaped polytopes. We propose the mollified basis functions of arbitrary order and smoothness for partitions consisting of convex polytopes. On each polytope an independent local polynomial approximant of arbitrary order is assumed. The basis functions are defined as the convolutions of the local approximants with a mollifier. The mollifier is chosen to be smooth, to have a compact support and a unit volume. The approximation properties of the obtained basis functions are governed by the local polynomial approximation order and mollifier smoothness. The convolution integrals are evaluated numerically first by computing the boolean intersection between the mollifier and the polytope and then applying the divergence theorem to reduce the dimension of the integrals. The support of a basis function is given as the Minkowski sum of the respective polytope and the mollifier. The breakpoints of the basis functions, i.e. locations with non-infinite smoothness, are not necessarily aligned with polytope boundaries. Furthermore, the basis functions are not boundary interpolating so that we apply boundary conditions with the non-symmetric Nitsche method as in immersed/embedded finite elements. The presented numerical examples confirm the optimal convergence of the proposed approximation scheme for Poisson and elasticity problems.

翻译：限制元素方法的近似特性通常可以通过选择平滑的高阶基函数得到大幅改进。 很难为由任意形状的多面形组成的分区设计这样的基础函数。 我们提议由 convex 多元面组成的分区的任意秩序和平滑性等离子的软化基本功能。 假设每个多面体中, 独立的本地多点相邻的任意秩序。 基函数的定义是带有一个摩托机的当地相邻者的演进。 摩托者被选择为平滑的, 拥有一个紧凑的支持和单位体积。 获得的基础函数的近似特性由本地多面形近似顺序和摩托光性平滑性来调控。 计算每个多面体和多面体多面体相之间的布尔性交叉点, 然后应用偏差的调调调调来减少组合体的维度。 基础函数的支持以相应的 Minkowski 和软体积体积支持。 获得的基础函数的断点, 由本地多面线/ 边框中, 无法使用平面的平整的边框度函数 。 。 。