B-Spline Collocation with Domain Decomposition Method and Its Application for Singularly Perturbed Convection-Diffusion Problems

Abstract

Global collocation method using B-spline basis functions is shown to be capable for solving elliptic partial differential equations in arbitrary complex domains. The global B-spline collocation approach is effectively alleviating difficulties commonly associated to B-spline based methods in handling such domains. Nonetheless, the global method, which is simply reduced to Bezier approximation of degree p with C 0 continuity, has led to the use of B-spline basis of high order in order to achieve high accuracy. The need for B-spline bases of high order is also more prominent in domains of large dimension. In addition, it may also lead to the ill-conditioning problem for combination of the use of B-spline bases of high order and increasing number of collocation points. In this chapter, global B-spline collocation scheme with domain decomposition techniques is introduced for solving Poisson equations in arbitrary complex domains. Overlapping Schwarz multiplicative and additive domain decomposition techniques are examined in this study. It is shown that the combination method produces higher accuracy with the B-spline basis of much lower order than that needed in implementation of the global method. The B-spline collocation with domain decomposition method hence improves the approximation stability of the global B-spline collocation method. Numerical simulations of singularly perturbed convection-diffusion problems are presented to further show the method efficacy and capability.