Superconvergent isogeometric collocation method with Greville points

Abstract

A superconvergent isogeometric collocation method that employs the conventional Greville abscissae as collocation points is proposed. Firstly, a basis transformation is presented for isogeometric basis functions, which preserves the exact geometry representation characterized by isogeometric analysis. More specifically, the geometry is exactly represented by the transformed isogeometric basis functions and the Greville points that function as control points in the current formulation. Meanwhile, the proposed basis transformation enables a very straightforward formulation of the consistency conditions. It also turns out that the transformed isogeometric basis functions span the same solution space as the standard isogeometric basis functions. Subsequently, with the aid of these consistency conditions, an accuracy analysis is presented for multi-dimensional isogeometric collocation formulation and the basis degree discrepancy issue is theoretically identified. In order to resolve this issue, a recursive gradient formulation is further introduced to effectively construct the gradients of transformed isogeometric basis functions. It is shown that these recursive gradients meet additional higher order consistency conditions. Consequently, a superconvergent isogeometric collocation method is finally established, which utilizes the recursive gradients to replace the conventional gradients of isogeometric basis functions and directly collocates at Greville points. The superconvergence of the proposed method regarding odd degree basis functions is then proved, which is also well manifested by numerical results.