  The mathematical theory of a novel variational approximation scheme for
general second and fourth order partial differential equations
\begin{equation}\label{eq: A} \partial_t u -
\nabla\cdot\Big(u\nabla\frac{\delta\phi}{\delta
u}(u)\Big|\nabla\frac{\delta\phi}{\delta u}(u)\Big|^{q-2}\Big) \ = \ 0,
\quad\quad u\geq0, \end{equation} $q\in(1, +\infty)$, is developed.
