Johan Helsing: On the polarizability and capacitance of the cube

An electrostatic solver is constructed for problems on domains with cuboidal inclusions. A particular characteristic of the solver is that it takes advantage of sharp edges and corners, rather than being a victim of them. In this way we circumvent the need to round sharp geometric features slightly -- a common practice which leads to complications as new length-scales are introduced and, in extreme cases, to radically different solutions.

Our solver can be used to compute the polarizability of a dielectric cube in a dielectric background medium at virtually every permittivity ratio for which it exists. For example, polarizabilities accurate to between five and ten digits are obtained (as complex limits) for negative permittivity ratios in minutes on a standard workstation. In passing, the capacitance of the unit cube is determined with unprecedented accuracy.

Some aspects of polarizabilities and their representing measures are also clarified, including limiting behavior both when approaching the support of the measure and when deforming smooth domains into a non-smooth domain.

Finally, stepping down to two dimensions, we show a high-resolution animation illustrating how the effective permittivity of an array of dielectric squares evolves as the geometry approaches that of a dielectric checkerboard.

The talk is based on joint work with Graeme Milton, Karl-Mikael Perfekt, and Ross McPhedran.