Inconel 625 alloy is widely used in the aerospace industry because of its high strength, especially at high temperatures, corrosion resistance and excellent weldability. The study presents modeling of the TIG welding process using a Gaussian heat source. The calculations were performed using our in-house software Mod_FEM_met, designed for modeling, among others, welding processes using finite element method and employed as a service of PL-Grid+ infrastructure. The program has a modular structure, with the modules for solving the Navier-Stokes and heat transport equations together with a coupling super-module used in welding simulations. To increase computing speed and accuracy in areas with large error of approximation, adaptive meshes were used. Calculations were performed for plate made of Inconel 625. In the calculations the temperature dependent properties of Inconel 625 alloy, as well as thermal phenomena at the edges and inside the weld pool were taken into consideration. The results of the calculations include the dimensions and shape of the weld pool, as well the velocity and temperature fields. The results indicate how the efficiency of heat source can be used as a parameter to optimize the fitting of calculations to the experimental data.