<p>This paper aims to numerically solve the blow-up solution to the semi-linear Caputo–Hadamard fractional evolution equation. Firstly, the moving mesh finite element scheme is established to solve the linear Caputo–Hadamard fractional evolution equation which has a sharp solution, where the L1 formula is used for time fractional derivative. The numerical stability, convergence, and error estimates for the fully discrete scheme are analyzed. And the numerical scheme is successfully used to solve the sharp solution to the linear problem. Then, based on the method and technique, the moving mesh finite element method is built up for the semi-linear problem. The numerical simulations show that the derived fully discrete scheme is efficient and effective for solving the finite time blow-up solution to the semi-linear Caputo–Hadamard fractional evolution equation. And the numerical experiments disclose the fact that blow-up time increases when the fractional derivative order increases.</p>