Point collocation with mollified piecewise polynomial approximants for high-order partial differential equations

TL;DR

The paper addresses solving high-order PDEs using a strong-form point collocation method with mollified piecewise polynomial approximants. Mollified basis functions are constructed by convolving local polynomials with a smooth kernel, preserving polynomial reproduce up to degree $r_p$ while achieving $C^{k-1}$ smoothness determined by the mollifier. The resulting overdetermined linear system, formed by interior and boundary collocation points and solved via least squares, avoids domain integrals required by Galerkin methods and enables direct imposition of Dirichlet BCs, with ghost cells ensuring polynomial reproduction near boundaries. Numerical experiments demonstrate high-order convergence for Poisson, linear elasticity, and biharmonic problems on polytopic Voronoi meshes, and examine the influence of mollifier width, smoothness, and collocation point distributions. The work shows the mollified-collocation approach to be robust to mesh irregularity and promising for efficient, high-accuracy PDE solving on general discretisations, with potential extensions to boundary-mollification, boundary-fitted Voronoi schemes, and adaptive refinement.

Abstract

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.

Figures (21)

• Figure 1: The proposed mollified-collocation method. First the domain (a) is discretised into a set of polytopic cells through Voronoi tessellation (b). Each cell has an associated piecewise linear polynomial and a $C^{k-1}$ smooth kernel is used to obtain smooth high-order basis functions. The resulting basis functions are used to solve a $k$-th order PDE problem giving the solution (c).
• Figure 2: Mollification of piecewise linear functions with $C^1$-smooth quadratic B-spline mollifier. The resulting function $\widehat{f}(x)$ is $C^2$-smooth. Mollification with symmetric and asymmetric kernels is used to exemplify the parallel between the convolution operator and feature maps in CNN.
• Figure 3: Basis functions (left column) over $n_c = 6$ uniform cells for $r_p \in \{ 0, 1, 2, 3 \}$ using a quadratic B-spline mollifier and their corresponding second derivatives (right column). The blue dots situated along the $x$-axis denote the cell boundaries.
• Figure 4: Bivariate constant and linear mollified basis functions with a $C^2$-smooth spline mollifier on a cell $\omega_i \in \mathbb R^2$. The dashed lines in (a) and (c) indicate the boundary of the cell.
• Figure 5: The intersection of the mollifier located at $\vec{x}$ with the cells $\omega_i$ (a) generates the domain for evaluating the convolution integral $\tau_{i, \vec{x}}$ (b).