Fast-multipole accelerated regularized meshless method for large-scale isotropic heat conduction problems

The regularized meshless method (RMM) belongs to the family of meshless boundary collocation methods and can be viewed as one kind of modified method of fundamental solutions (MFS). This method circumvents the fictitious boundary issue associated with the traditional MFS while remaining the merits of the later of being truly meshless, integration-free, and easy-to-program. However, like the traditional MFS, the conventional RMM also produces dense and unsymmetrical coefficient matrices which require O(N2) memory and O(N3) operations to compute the system of equations using direct solvers, with N being the total number of unknowns. This paper documents the first attempt to apply the fast-multipole method (FMM) to accelerate the solutions of the RMM for large-scale problems. Combining the FMM and RMM can potentially reduce the operations in formation and solution of the RMM system, as well as the memory requirement, all to O(N). It is shown that the new RMM proposed here can now solve large-scale problems with several million unknowns on a desktop computer.