In this paper, we address the nonrigid shape matching with outliers by a novel and effective pointwise map refinement method, termed Locality Preserving Refinement. For accurate pointwise conversion from a given functional map, our method formulates a two-step procedure. Firstly, starting with noisy point-to-point correspondences, we identify inliers by leveraging the neighborhood support, which yields a closedform solution with linear time complexity. After obtained the reliable correspondences of inliers, we refine the pointwise correspondences for outliers using local linear embedding, which operates in an adaptive spectral similarity space to further eliminate the ambiguities that are difficult to handle in the functional space. By refining pointwise correspondences with local consistency thus embedding geometric constraints into functional spaces, our method achieves considerable improvement in accuracy with linearithmic time and space cost. Extensive experiments on public benchmarks demonstrate the superiority of our method over the state-of-the-art methods. Our code is publicly available at https://github.com/XiaYifan1999/LOPR.