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：利用光滑基函数可显著改善偏微分方程（PDE）的求解逼近效果。近期提出的光滑化基函数通过将单元定义的分段多项式与特定性质的光滑化函数进行光滑化（即卷积）运算而构建。此类基函数的性质由分段多项式的阶次及光滑化函数的光滑度共同决定。本研究利用光滑化基函数的高阶次与高光滑度特性，通过点配置法求解偏微分方程。基函数在计算域内的一组配置点处进行取值计算。此外，在域边界上分布的边界配置点集处施加边界条件。为保证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。所得线性方程组为超定系统，采用最小二乘法进行求解。数值算例验证了所提逼近格式在泊松问题、线性弹性问题及双调和问题中的收敛性。我们重点研究了光滑化函数与配置点空间分布的影响规律。