Adaptive radial basis function collocation method for elliptic interface problems

In this paper, we propose an adaptive residual sub-sampling algorithm for solving elliptic interface problems with discontinuous coefficients and singular source terms. The method employs the radial basis function (RBF) collocation technique and dynamically refines the computational grid by adding or removing nodes based on residuals evaluated at finer points. To mitigate the growth of the interpolation matrix’s condition number, we adjust the RBF shape parameters according to node distances. This adaptive procedure identifies regions of the domain requiring refinement and selectively adjusts point distribution. We solve the resulting ill-posed coefficient matrix using numerical linear algebra techniques such as Tikhonov regularization and singular value decomposition (SVD). Numerical results validate the proposed approach and highlight the algorithm’s effectiveness.