An efficient alternating-direction finite-element method is applied to the solution of nonlinear heat transfer problems. The technique is comparable in accuracy to standard finite-element methods while offering a substantial reduction in computer costs in terms of both processing time and core storage. The alternating-direction formulation is presented together with a discussion of the resulting matrix system and treatment of the boundary conditions. A heat conduction problem involving phase change is solved as a demonstration of the efficiency of the algorithm. The latent heat effect is accounted for by rapid variations in the heat capacity term.