  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.
