Meshless finite difference methods and overlap splines

Meshless finite difference methods are the finite difference methods on irregular nodes, with stencils obtained by optimizing numerical differentiation using local polynomials or radial basis functions. These techniques are inherently meshless and isogeometric, and efficient in applications. I will report on several recent computational results of my group and collaborators. In addition, I will relate meshless finite difference methods to collocation and least squares methods in a pseudo-functional setting, where the approximation tool is "overlap splines" consisting of local patches of either radial basis functions or multivariate polynomials glued together pointwise, in contrast to the standard piecewise polynomial splines, where the patches are connected across common boundaries of the elements of a partition. This approach promises a progress on the theoretical understanding of the meshless finite difference methods, and has already been used to prove convergence and error bounds for a least squares version in a special setting of elliptic differential equations on closed manifolds.