Meshless Methods for Singular and Nonlinear Elliptic PDEs

Asymmetric RBF collocation (where RBF stands for radial basis function) is a relatively new alternative to finite elements and volumes for the numerical solution of PDEs in irregular domains. It is a meshless method -requires no connectivity between the discretization nodes-, which approximates the strong form of the PDE in an interpolation space made up of shifted radial functions. These features confer appealing properties on it, such as geometric flexibility, exponential convergence, and great ease of implementation. There are some practical drawbacks as well, most notably, ill-conditioning and the lack of sound theoretical foundations. In this talk, the RBF setting will be first introduced. Then, the performance of this meshless scheme in two non-standard applications will be discussed: 1) elliptic problems with boundary singularities, and 2) quasilinear elliptic PDEs. It will be shown that, with minor modifications, the properties of the method carry over to these kind of problems.