This paper presents a planar curve matching framework based on computing similarities between shape parts. We propose an elastic similarity measure issued from shape geodesics in the shape space. As the transition from global matching to partial matching leads to additional difficulties, we bypass them with a shape decomposition process based on the discrete curve evolution (DCE). This decomposition aims to obtain significant parts to match and it leads to a robust and efficient 2D shape matching algorithm. The comparison of the proposed method with the state of the art demonstrates its ability to handle elastic deformations leading to an overall optimal partial correspondence between shapes.