A local radial basis function method (LRBF) is applied for the solution of boundary value problems in annular domains governed by the Poisson equation, the inhomogeneous biharmonic equation and the inhomogeneous Cauchy-Navier equations of elasticity. By appropriately choosing the collocation points we obtain linear systems in which the coefficient matrices possess block sparse circulant structures and which can be solved efficiently using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). The MDAs used are appropriately modified to take into account the sparsity of the arrays involved in the discretization. The leave-one-out cross validation (LOOCV) algorithm is employed to obtain a suitable value for the shape parameter in the radial basis functions (RBFs) used. The selection of the nearest centres for each local influence domain is carried out using a modification of the kd-tree algorithm. In several numerical experiments, it is demonstrated that the proposed algorithm is both accurate and capable of solving large scale problems.