Regularized meshless method

In numerical mathematics, the regularized meshless method (RMM), also known as the singular meshless method or desingularized meshless method, is a meshless boundary collocation method designed to solve certain partial differential equations whose fundamental solution is explicitly known. The RMM is a strong-form collocation method with merits being meshless, integration-free, easy-to-implement, and high stability. Until now this method has been successfully applied to some typical problems, such as potential, acoustics, water wave, and inverse problems of bounded and unbounded domains. In numerical mathematics, the regularized meshless method (RMM), also known as the singular meshless method or desingularized meshless method, is a meshless boundary collocation method designed to solve certain partial differential equations whose fundamental solution is explicitly known. The RMM is a strong-form collocation method with merits being meshless, integration-free, easy-to-implement, and high stability. Until now this method has been successfully applied to some typical problems, such as potential, acoustics, water wave, and inverse problems of bounded and unbounded domains. The RMM employs the double layer potentials from the potential theory as its basis/kernel functions. Like the method of fundamental solutions (MFS), the numerical solution is approximated by a linear combination of double layer kernel functions with respect to different source points. Unlike the MFS, the collocation and source points of the RMM, however, are coincident and placed on the physical boundary without the need of a fictitious boundary in the MFS. Thus, the RMM overcomes the major bottleneck in the MFS applications to the real world problems. Upon the coincidence of the collocation and source points, the double layer kernel functions will present various orders of singularity. Thus, a subtracting and adding-back regularizing technique is introduced and, hence, removes or cancels such singularities. These days the finite element method (FEM), finite difference method (FDM), finite volume method (FVM), and boundary element method (BEM) are dominant numerical techniques in numerical modelings of many fields of engineering and sciences. Mesh generation is tedious and even very challenging problems in their solution of high-dimensional moving or complex-shaped boundary problems and is computationally costly and often mathematically troublesome.