Abstract: This paper focuses on the problems for computing correspondences between non-isometric shapes with not fully automatic and have a low accuracy rate. The novel approach we propose in this paper is based on the Localized Manifold Harmonics basis and shape alignment in intrinsic and extrinsic space. Firstly, we use the Localized Manifold Harmonics basis as the intrinsic information of the shape, and combine it with extrinsic information such as Cartesian coordinates by embedding the input shapes into an intrinsic-extrinsic product space to align the internal features with the external information. Secondly, we integrate the optimization problem with the Coherent Point Drift method to improve the stability and accuracy of the results. Finally, we use an alternating scheme based on the Manifold Alternating Direction Method of Multipliers method to solve the optimization problem and get the final result. The experimental results have shown that compared with the existing algorithms, this algorithm has the lowest geodesic error and the highest accuracy of global correspondence on SMAL, SHERC’19, TOSCA and SHERC’16 Topology datasets. Meanwhile, our method can deal with the topological noise and symmetric ambiguity problems.