  By introducing height dependency in the surface energy density, we propose a
novel regularized variational model to simulate wetting/dewetting problems. The
regularized model leads to the appearance of a precursor layer which covers the
bare substrate, with the precursor height depending on the regularization
parameter $\varepsilon$. The new model enjoys lots of advantages in analysis
and imulations. With the help of the precursor layer, the regularized model is
naturally extended to a larger domain than that of the classical
sharp-interface model, and thus can be solved in a fixed domain. There is no
need to explicitly track the contact line motion, and difficulties arising from
free boundary problems can be avoided. In addition, topological change events
can be automatically captured. Under some mild and physically meaningful
conditions, we show the positivity-preserving property of the minimizers of the
new model. By using asymptotic analysis and $\Gamma$-convergence, we
investigate the convergence relations between the new regularized model and the
classical sharp-interface model. Finally, numerical results are provided to
validate our theoretical analysis, as well as the accuracy and efficiency of
the new regularized model.
