Partial differential equations on products of Sierpinski gaskets

Abstract

We describe a finite element method based on piecewise pluriharmonic or piecewise pluribiharmonic splines to numerically approximate solutions to partial differential equations on the product SG 2 of two copies of the Sierpinski gasket. We use this method to experimentally study both elliptic equations, and a new class of operators that we call quasielliptic, which has no analog in the standard theory of pde's. The existence of these operators is based on the observation that the set of ratios of eigenvalues of the Laplacian on SG has gaps. We explicitly prove that such a gap exists around the value √5.