The authors propose two new fully discrete interior penalty discontinuous Galerkin methods for the 2D and 3D Allen-Cahn equation \[ u_t-\Delta u+\frac{1}{\epsilon^2}f(u)=0, \] where \(f(u)=u(u^2-1)\). In this setting, the authors discuss a fully implicit and energy-splitting time stepping scheme. Sharp error bounds depending on \(\epsilon\) are deduced. Further, convergence of the zero level sets for the fully discrete interior penalty discontinuous Galerkin solutions for the classical and generalized mean curvature flow is investigated. Numerical experiments are also presented to support the theoretical findings.