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A numerical study of conjugate thermogravitational convection in a closed cavity with a local heater of square or triangular shape placed on a heat-conducting substrate using the double distribution function of the lattice Boltzmann method has been carried out. The side walls of the research area are maintained at a constant minimum temperature. The influence of the geometric shape of the heating element, the Rayleigh number, and the material of the heat-removing substrate on the thermohydrodynamic parameters has been studied. As a result of the research, the joint effect of these mentioned parameters on the efficiency of heat removal from the heater surface has been established. It has been found that a rise of the bottom wall thermal conductivity causes an increase in the average Nusselt number at the heater surface.