The RBF-FD (radial-basis-function generated finite differences) method for solving PDEs can be interpreted as a generalisation of the finite difference method. Unlike the latter it is not restricted to grids, but can solve problems on scattered sets of nodes. It belongs to the category of meshless methods, as it does not require that the nodes form a mesh. This is useful, because generating meshes, such as triangulations for FEM, is often computationally difficult. One major advantage of the finite difference method, however, is that the matrix involved in finding the numerical solution of a problem is of a characteristic banded shape. This makes analysing the method easier. The aim of this master’s paper is to explore the properties of analogous matrices, produced by RBF-FD. First, the properties of such matrices are considered when the discretisation forms a regular grid. As this is not particularly usefull in practice, a more general case is considered, when the discretisation is a perturbed grid. Then, a domain discretisation algorithm is presented, which produces more realistic sets of nodes. Finally we consider the properties of the matrices, when RBF-FD is used to solve problems on such sets.
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