Zhiye Zhao

Summary: The high‐order approximation with regularly patterned flat‐top partition of unity mesh in one‐ and two‐dimensional cases has been proven linearly independent. However, for problems with stress concentration or stress singularity, local refinement within the regular mesh is necessary to improve the accuracy and efficiency. This paper introduces local refinement of flat‐top partition of unity mesh within the framework of high‐order approximation in one‐ and two‐dimensional spaces, respectively. Based on the traditional PU mesh, the construction of locally refined flat‐top PU mesh is straightforward. With the rank deficiency counting approach, linear independence is proven from element level for the locally refined mesh system. Based on the numerical solution procedure presented, two numerical examples are analyzed to verify the proposed approximation method.