On the ill-conditioning in the new higher order multipoint method

The new higher order approximation multipoint technique in the meshless FDM is considered. The method, following the original Collatz multipoint concept, is based on arbitrary irregular meshes, moving weighted least squares approximation and the global, local or global-local formulations of boundary value problems. Two basic versions of the multipoint meshless FD approach are developed: the general and the specific ones. Unfortunately, ill-conditioning phenomenon may be encountered, when the general multipoint approach is considered. Therefore, several modifications of the multipoint algorithm have been proposed and tested to avoid this problem. A variety of tests done show that the proposed concepts work well. However, efficiency problem in some versions which involves additional calculations is the main issue now. Alternatively, the simplest solution in this situation is to use a regularization term in the weighted error functional. Numerical results demonstrate the effectiveness of the proposed versions in situation, when ill-conditioning or singularity could be encountered.