In inverse design, the design and background areas can be represented by different spatial resolutions; thus, adaptive meshes are more efficient than structured meshes. In this study, a second-order interpolation scheme is introduced to realize an inverse design process on an adaptive mesh. Experiment results show that the proposed scheme yields a 1.79-fold acceleration over that achieved using a structured mesh, aiding design time reduction or design area expansion. As the design area can be divided into multiple areas with different spatial resolutions, in future work, adaptive meshes can be combined with machine learning algorithms to further improve the inverse-design-process efficiency.