Accuracy improvement of collocation method by using the over-range collocation points for 2-D and 3-D problems

Abstract

It has been shown that the significance of the positivity conditions in the collocation methods (CM), and the violation of the positivity conditions can significantly result in a large error in the numerical solution. For boundary points, however, the positivity conditions cannot be satisfied, obviously. To overcome the demerit of the CM, the over-range collocation method (ORCM) has been proposed. In the ORCM, some over-range collocation points are introduced which are located at the outside of the domain of an analyzed body, and at the over-range collocation points no satisfaction of any governing partial differential equation or boundary condition is needed. In this paper, it is shown that the positivity conditions of boundary points in the ORCM are satisfied by calculated results on the positivity conditions, while the positivity conditions of boundary points in the CM are not satisfied. The boundary value problems on the 2-D and 3-D Poisson’s equations and the 3-D Helmholtz’s equation are analyzed by using the ORCM and the CM. The numerical solutions by using both the ORCM and the CM are compared with the exact solutions. The relative errors by using the ORCM are smaller than those by using the CM, for both the unknown variables and their derivatives of 2-D problems and for the unknown variables of 3-D problems, and the relative errors of the unknown's derivatives of 3-D problems by using the ORCM are about same as those by using the CM. Convergence studies in the numerical examples show that the ORCM possesses good convergence for both the unknown variables and their derivatives. Because the ORCM does not demand any specific type of partial differential equations, it is concluded that the ORCM promises the wide engineering applications.

Keywords

• meshless method
• collocation method
• positivity conditions
• over-range points
• kronecker-delta