  Finite time singularity formation in a fourth order nonlinear parabolic
partial differential equation (PDE) is analyzed. The PDE is a variant of a
ubiquitous model found in the field of Micro-Electro Mechanical Systems (MEMS)
and is studied on a one-dimensional (1D) strip and the unit disc. The solution
itself remains continuous at the point of singularity while its higher
derivatives diverge, a phenomenon known as quenching. For certain parameter
regimes it is shown numerically that the singularity will form at multiple
isolated points in the 1D strip case and along a ring of points in the radially
symmetric 2D case. The location of these touchdown points is accurately
predicted by means of asymptotic expansions. The solution itself is shown to
converge to a stable self-similar profile at the singularity point. Analytical
calculations are verified by use of adaptive numerical methods which take
advantage of symmetries exhibited by the underlying PDE to accurately resolve
solutions very close to the singularity.
