B-spline Collocation with Domain Decomposition Method

A global B-spline collocation method has been previously developed and successfully implemented by the present authors for solving elliptic partial differential equations in arbitrary complex domains. However, the global B-spline approximation, which is simply reduced to Bezier approximation of any 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 in the global method would be more prominent in domains of large dimension. For the increased collocation points, it may also lead to the ill-conditioning problem. In this study, overlapping domain decomposition of multiplicative Schwarz algorithm is combined with the global method. Our objective is two-fold that improving the accuracy with the combination technique, and also investigating influence of the combination technique to the employed B-spline basis orders with respect to the obtained accuracy. It was shown that the combination method produced higher accuracy with the B-spline basis of much lower order than that needed in implementation of the initial method. Hence, the approximation stability of the B-spline collocation method was also increased.