Partition of Unity Refinement for Local Approximation

In this article, we propose a Partition ofUnity Refinement (PUR)method to improve the local approximations of elliptic boundary value problems in regions of interest. The PUR method only needs to refine the local meshes and hanging nodes generate no difficulty. The mesh qualities such as uniformity or quasi-uniformity are kept. The advantages of the PUR include its effectiveness and relatively easy implementation. In this article, we present the basic ideas and implementation of the PUR method on triangular meshes. Numerical results for elliptic Dirichlet boundary value problem on an L-shaped domain are shown to demonstrate the effectiveness of the proposed method. The extensions of the PUR method to multilevel and higher dimensio