Local RBF Algorithms for Elliptic Boundary Value Problems in Annular Domains (Communications in Computational Physics, 2018)
A local radial basis function method (LRBF) is applied for the solution ofboundary value problems in annular domains governed by the Poisson equation, theinhomogeneous biharmonic equation and the inhomogeneous Cauchy-Navier equationsof elasticity. By appropriately choosing the collocation points we obtain linearsystems in which the coefficient matrices possess block sparse circulant structures andwhich can be solved efficiently using matrix decomposition algorithms (MDAs) andfast Fourier transforms (FFTs). The MDAs used are appropriately modified to takeinto account the sparsity of the arrays involved in the discretization. The leave-one-outcross validation (LOOCV) algorithm is employed to obtain a suitable value for theshape parameter in the radial basis functions (RBFs) used. The selection of the nearestcentres for each local influence domain is carried out using a modification of the kd-treealgorithm. In several numerical experiments, it is demonstrated that the proposed algorithmis both accurate and capable of solving large scale problems.
Keywords: Radial basis functions; Kansa method; Poisson equation; biharmonic equation; Cauchy-Navier equations of elasticity; matrix decomposition algorithms; fast Fourier transforms.