Parabolic-type problems, involving a variational complementarity formulation, arise in mathematical models of several applications in Engineering, Economy, Biology and different branches of Physics. These kinds of problems present several analytical and numerical difficulties related, for example, to time evolution and a moving boundary. We develop numerical methods that employs a global convergent nonlinear complementarity (or mixed complementarity) algorithms for solving a discretized problem at each time step. Space discretization is implemented using the finite difference implicit scheme and the finite element method.