The finite-element equations for coupled nonlinear heat conduction with phase change and steady current flow in the case where the electric field moves with constant speed are derived. The resulting matrix equations for quasi-static problems are nonsymmetric for the heat conduction part and symmetric for the electric field. The formulation is applied to several classical problems in heat conduction and compared to the more usual transient analysis formulation. It is shown that the quasi-static formulation is many times faster than the equivalent transient analysis. The formulation is then applied to a coupled problem in which the thermal and electrical properties are temperature dependent.