Extending the local radial basis function collocation methods for solving semi-linear partial differential equations

This work addresses local radial basis function (RBF) collocation methods for solving a major class of non-linear boundary value problems, i.e., Lu = f(x, u) being f a non-linear function of u. This class of problems has been largely analyzed in the BEM community. To our knowledge, few works are reported where the local RBF collocation methods (LRBFCM) based on the generalized Hermite RBF interpolation (double collocation) have been extended successfully to solve semi-linear problems even when extending to more complex nonlinear cases are not reported yet. The studied schemes are based on a strong-form approach of the PDE and an overlapping multi-domain procedure combining with standard iterative schemes. At each sub-domain, a locally meshless approximation solution by a standard or Hermite RBF expansion can be constructed.We studied also the performance respect to the shape parameter of RBF. It is confirmed that the local RBF double collocation can improve greatly the accuracy order. Some 2D benchmark problems with mixed boundary conditions showing the accuracy, convergence property and implementation issues of LRBFCM are presented.