This work aims to build a bridge between probability methods and finite element methods. It starts with considering probability distributions supported in an interval $[a,b]$, which incorporate the traditional probability distributions defined on the whole real space as limit cases on the one hand, lead to a type of spherical probability models with wide potential applications on the other hand. This type of probability has scaling and symmetry feature, and sufficient conditions under which a density function can be generated through discrete polynomial spectrum are given in this work followed by concrete examples. The density function $\rho$ obtained in this way has the advantage of being positive definite. Computer based numerical simulation shows that the theoretically verified criteria for probability distribution are almost optimal with respect to our testing examples. After the establishment of an approximation theorem in $L^1$ space, we propose a probabilistic Galerkin scheme that can be either continuous or discontinuous, which is potentially useful to asymptotically solve some PDEs on the sphere locally and globally.

这项工作旨在在概率方法和有限元方法之间架起一座桥梁。首先考虑间隔中支持的概率分布$[\mathrm{一种}，b]$一方面结合了在整个现实空间中定义的传统概率分布作为极限情况，另一方面导致了一种具有广泛潜在应用范围的球形概率模型。这种类型的概率具有缩放和对称性特征，并且在此工作中给出了可以通过离散多项式谱生成密度函数的充分条件，随后给出了具体示例。密度函数$\rho$以这种方式获得的优点是具有正定的优点。基于计算机的数值模拟表明，相对于我们的测试示例，概率分布的理论验证标准几乎是最佳的。在建立一个近似定理之后${\mathrm{大号}}^{\mathrm{1个}}$ 在空间方面，我们提出了一种概率Galerkin方案，该方案可以是连续的也可以是不连续的，这对于局部和全局渐近求解该球体上的某些PDE有潜在的帮助。