Local Polynomial Regression and Filtering for a Versatile Mesh‐Free PDE Solver

A high-order, mesh-free finite difference method for solving partial differential equations is presented. The spatial domain is sampled by an irregular point cloud, and at each point, a local polynomial approximation is formulated. This polynomial is used to approximate both the function and its derivatives. The function approximation is used as a filter to regularise the solution and ensure stability of the method. The local polynomial is obtained by straightforward least squares minimisation, which is solved by using the singular value decomposition method, with truncation. A non-parametric regression with robust M-estimator functions is also investigated and exhibits additional robustness and accuracy. A straightforward semi-discrete form is obtained by adopting an explicit time discretisation, and combined with the local polynomial regression, a modular assembly of complete solver algorithms is possible. The resulting numerical method is accurate, simple and versatile. Challenging numerical benchmark tests are investigated, namely the solution to inviscid Burgers' equation, linearised Euler equations and incompressible Navier-Stokes equations (with the artificial compressibility and vorticity transport formulations) in Eulerian and Lagrangian reference frames.