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boundary knot method
In numerical mathematics, the boundary knot method (BKM) is proposed as an alternative boundary-type meshfree distance function collocation scheme.
Recent decades have witnessed a research boom on the meshfree numerical PDE techniques since the construction of a mesh in the standard finite element method and boundary element method is not trivial especially for moving boundary, and higher-dimensional problems. The boundary knot method is different from the other methods based on the fundamental solutions, such as boundary element method, method of fundamental solutions and singular boundary method in that the former does not require special techniques to cure the singularity. The BKM is truly meshfree, spectral convergent (numerical observations), symmetric (self-adjoint PDEs), integration-free, and easy to learn and implement. The method has successfully been tested to the Helmholtz, diffusion, convection-diffusion, and Possion equations with very irregular 2D and 3D domains.
== Description ==
The BKM is basically a combination of the distance function, non-singular general solution, and dual reciprocity method (DRM). The distance function is employed in the BKM to approximate the inhomogeneous terms via the DRM, whereas the non-singular general solution of the partial differential equation leads to a boundary-only formulation for the homogeneous solution. Without the singular fundamental solution, the BKM removes the controversial artificial boundary in the method of fundamental solutions. Some preliminary numerical experiments show that the BKM can produce excellent results with relatively a small number of nodes for various linear and nonlinear problems.
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