Computing with spherical geometries: from implicit solvation models to stochastic homogenisation

In this talk, a framework to solve elliptic interface problems on geometries involving spherical shapes is presented. Applications range from implicit solvation models in computational chemistry to electrostatic interaction of charged dielectric spheres to stochastic homogenisation with spherical inclusions. One of the main issues of elliptic problems is that the Green’s function is decaying very slowly. Therefore, the strategy is in all cases the same: we first transform the problem into an integral equation allowing for capturing the right asymptotic behaviour of the solution towards infinity. The integral equation is subsequently discretised by means of spherical harmonics on each sphere. In particular in the case of stochastic homogenisation, this approach allows for a new strategy of the corrector problem as one can solve problems on the unbounded domain where the stochastic medium is embedded in an infinite continuous medium. Numerical illustrations will underline the methodological developments.