Spline element methods allowing multiple level hanging nodes

In this paper, a systematic algorithm of spline element method allowing multiple level hanging nodes on triangular mesh is proposed. The classical smoothness condition of the piecewise polynomials on triangular mesh is generalized to the case of meshes with hanging nodes, and is treated as the linear constraints of the system equations. We then derive a concise formulation for such linear constraints according to the special hierarchical configurations of the hanging nodes. Moreover, an efficient linear equation solver is proposed when applied to solve the elliptic partial differential equations. Numerical examples are illustrated to show the accuracy of the proposed methods, and the comparisons are made between different levels of hanging nodes.