In this paper, we propose a numerical method for solving the heat equations with
interfaces. This method uses the non-traditional finite element method together
with finite difference method to get solutions with second-order accuracy. It is
capable of dealing with matrix coefficient involving time, and the interfaces
under consideration are sharp-edged interfaces instead of smooth interfaces.
Modified Euler Method is employed to ensure the accuracy in time. More than
1.5th order accuracy is observed for solution with singularity (second
derivative blows up) on the sharp-edged interface corner. Extensive numerical
experiments illustrate the feasibility of the method.