The process of heat transfer in a plate of inhomogeneous material is investigated in the article, based on the solution of a two-dimensional problem of heat conduction by the finite element method. Straight-sided finite elements are used to solve the plane problem of heat conduction. Basic relations for linear triangular finite elements are given. On the basis of the developed solution algorithm and software, the numerical solution of a two-dimensional problem of thermal conductivity of a plate made of an inhomogeneous material and a plate containing an isolated square cutout in the center is investigated. The solutions to the test problem correspond to the results obtained by the finite difference method. The temperature field distributions are studied and isotherms of the temperature distribution in the plate are given.