B-spline collocation method for boundary value problems in complex domains

Abstract

In this paper, an over-determined, global collocation method based upon B-spline basis functions is presented for solving boundary value problems in complex domains. The method was truly meshless approach, hence simple and efficient to programme. In the method, any governing equations were discretised by global B-spline approximation as the B-spline interpolants. As the interpolating B-spline basis functions were chosen, the present method also posed the Kronecker delta property allowing boundary conditions to be incorporated efficiently. The present method showed high accuracy for elliptic partial differential equations in arbitrary domain with Neumann boundary conditions. For coupled Poisson problems with complex Neumann boundary conditions, the boundary collocation approach was adopted and applied in a simple and less costly manner to further improve the accuracy and stability. Applications from elasticity problems were given to demonstrate the efficacy and capability of the present method. In addition, the relation between accuracy and stability for the method was better justified by the new effective condition number given in literature.