To ensure the sustainable development of the surrounding environment and the sustainable operation of landfills, detecting landfill leakage is of great significance. In landfills lacking a leakage monitoring system, the inability to detect and locate damaged High-Density Polyethylene (HDPE) membranes can lead to the contamination of soil and groundwater by landfill leachate. To address this issue, this study proposes a resistivity tomography inversion model based on the external-electrode power supply mode. Utilizing the resistivity difference between the leakage zone and the surrounding soil, electrodes are arranged symmetrically for both power supply and measurement. Upon applying direct current (DC) excitation, potential data are collected, with the finite volume method employed for inversion and the Gauss-Newton method integrated with an adaptive particle swarm optimization algorithm for parameter fitting. Experimental results show that the combined algorithm provides better clarity in edge recognition of low-resistance models compared with single algorithms. The maximum deviation between inferred leakage coordinates and the actual location is 10.1 cm, while the minimum deviation is 6.4 cm, satisfying engineering requirements. This method can effectively locate point sources and line sources, providing an accurate solution for subsequent leakage point filling and improving repair efficiency.