Issues with Accurate Boundary Representation in Oversampled Meshless Methods

A premier example of meshless methods is the Radial Basis Function-generated Finite Difference (RBF-FD) method, which approximates a linear operator based on values in discrete computational points. Compared to mesh based methods this approach offers several advantages for solving partial differential equations (PDEs), including a simpler spatial discretisation that eliminates the need for mesh generation. However, any constraints we impose only hold in computational points i.e. we solve the equation in a strong formulation. Sometimes, it is beneficial to consider the solution in a more continuous sense. One way to achieve this is with oversampling i.e. evaluating the approximation in additional points. Unfortunately, this means that the system of equations is now overdetermined and can no longer be solved exactly. This can result in large errors due to parts of the system that are important for the solution i.e. the relatively small number of nodes on the boundary, having almost no effect on the residual norm we are minimising. One possible solution to this problem is increasing the weighting for important parts of the system. In this paper, we observe how the changes in boundary weights affect the accuracy of the considered PDE solutions.