TY - JOUR
T1 - Local RBF Algorithms for Elliptic Boundary Value Problems in Annular Domains
AU - C. S. Chen & Andreas Karageorghis
JO - Communications in Computational Physics
VL - 1
SP - 41
EP - 67
PY - 2018
DA - 2018/09
SN - 25
DO - http://doi.org/10.4208/cicp.OA-2018-0031
UR - https://global-sci.org/intro/article_detail/cicp/12662.html
KW - Radial basis functions, Kansa method, Poisson equation, biharmonic equation, Cauchy-Navier
equations of elasticity, matrix decomposition algorithms, fast Fourier transforms.
AB - <p style="text-align: justify;">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.</p>