Numerical instabilities are often encountered in FE solution of continuum topology optimization. The essence of the numerical instabilities is given from the inverse partial differential equation (PDE) point of view. On the basis of the strict mathematical theory, a novel method, named as window filter and multi-grid method, which solves the numerical instabilities, is proposed. Convergent analyses and a numerical example are presented. Numerical instabilities are often encountered in FE solution of continuum topology optimization. The essence of the numerical instabilities is given from the inverse partial differential equation (PDE) point of view. On the basis of the strict mathematical theory, a novel method, named as window filter and multi-grid method, which solves the numerical instabilities, is proposed. Convergent analyzes and a numerical example are presented.