  In [KP16] (arXiv:1605.07880) the authors introduced a second-order
variational problem in $L^{\infty}$. The associated equation, coined the
$\infty$-Bilaplacian, is a \emph{third order} fully nonlinear PDE given by
$\Delta^2_\infty u\, := (\Delta u)^3 | D (\Delta u) |^2 = 0.$ In this work we
build a numerical method aimed at quantifying the nature of solutions to this
problem which we call $\infty$-Biharmonic functions. For fixed $p$ we design a
mixed finite element scheme for the pre-limiting equation, the $p$-Bilaplacian
$\Delta^2_p u\, := \Delta(| \Delta u |^{p-2} \Delta u) = 0.$ We prove
convergence of the numerical solution to the weak solution of $\Delta^2_p u =
0$ and show that we are able to pass to the limit $p\to\infty$. We perform
various tests aimed at understanding the nature of solutions of
$\Delta^2_\infty u$ and in 1-$d$ we prove convergence of our discretisation to
an appropriate weak solution concept of this problem, that of $\mathcal
D$-solutions.
