The Study Of Quasi Wavelets Method And Meshless Method Based On Natural Boundary Element Method

The boundary element method(BEM) is a numerical method of solving differential equations that attributes the boundary-value problem of differential equations in a domain of the boundary of the domain and discretizes on the boundary. This method has an advantage of the dimensionality. It is very convenient to solve a series of problems that has characteristic of the high-dimensionality because of the advantage. Differing from the general BEM that is attributed from the general boundary integral equation(BIE), The natural boundary element method attributed from the natural BIE has not only the advantage of the dimensionality, but is confirmed and gain the only NBIE from the original problem. Its results are existent and unique. A meshless method need only nodal data on the boundary and avoid the generating of meshes in the FEM in the numerical implementation of the method, and the computation is simple and stable. The quasi wavelets method introduce regularizer and make the scale function regularity and localization attributes. A meshless method and a quasi wavelets method based on the natural boundary integral equation in this paper are presented. Corresponding formulas are developed for the implementation of the present method. This paper proposes the combination of MLS interpolants with NBEM in order to retain the meshless attribute of the former and the dimensionality and computation advantage of latter and the combination of the quasi wavelets bases with NBEM in order to retain the distinctive smoothness, regularity and localization attributes of the former and the advantage of latter. The above presented methods are applied to solve the 2D harmonic equation. Numerical examples show satisfactory results.