Approximation of Degenerate Parabolic Problems Using Numerical Integration

A class of multidimensional degenerate parabolic equations is considered: the two-phase Stefan problem and the porous medium equation are analyzed as models of singular parabolic equations; nonstationary filtration (with gravity) is also treated as a model of elliptic-parabolic equations. A fully discrete scheme combined with a smoothing procedure (when needed) is proposed. The discretization is given by $C^0 $ piecewise linear finite elements in space and a semi-implicit scheme in time. The effects of numerical integration and domain change are taken into account; thus the proposed method is easy to implement. Several error estimates for the solutions and the free boundaries are proven. The results of some numerical tests are presented in order to show the accuracy of our scheme to approximate both the solution and the interface.