The solution approximation for partial differential equations (PDEs) can be substantially improved using smooth basis functions. The recently introduced mollified basis functions are constructed through mollification, or convolution, of cell-wise defined piecewise polynomials with a smooth mollifier of certain characteristics. The properties of the mollified basis functions are governed by the order of the piecewise functions and the smoothness of the mollifier. In this work, we exploit the high-order and high-smoothness properties of the mollified basis functions for solving PDEs through the point collocation method. The basis functions are evaluated at a set of collocation points in the domain. In addition, boundary conditions are imposed at a set of boundary collocation points distributed over the domain boundaries. To ensure the stability of the resulting linear system of equations, the number of collocation points is set larger than the total number of basis functions. The resulting linear system is overdetermined and is solved using the least square technique. The presented numerical examples confirm the convergence of the proposed approximation scheme for Poisson, linear elasticity, and biharmonic problems. We study in particular the influence of the mollifier and the spatial distribution of the collocation points.

翻译：偏微分方程的解近似可以通过使用光滑基函数得到显著改善。近期引入的光滑化基函数是通过对单元定义的分段多项式与具有特定性质的光滑磨光器进行卷积（即磨光化处理）构建的。光滑化基函数的性质由分段函数的阶数和磨光器的光滑性共同决定。本文利用光滑化基函数的高阶与高光滑性，通过点配置法求解偏微分方程。基函数在区域内一组配置点上进行评估。此外，边界条件通过分布在域边界上的边界配置点集施加。为确保所得线性方程组的稳定性，配置点数量设置为大于基函数总数。生成的超定线性方程组采用最小二乘法求解。数值算例验证了所提近似格式在泊松问题、线弹性问题及双调和问题中的收敛性。我们特别研究了磨光器与配置点空间分布对求解的影响。