Continuum solvation models are widely used to model quantum effects of molecules in liquid solutions. Our research focuses on the polarizable continuum model (PCM). In this model, the molecule under study (the solute) is located inside a cavity, surrounded by a homogeneous dielectric (the solvent). The solute-solvent interactions between the charge distributions which compose the solute and the dielectric are reduced to those of electrostatic origin.

According to classical electrostatics, the charge distribution of the solute, inside the molecule, polarizes the dielectric continuum, which in turn polarizes the solute's charge distribution. This interaction might be expressed in terms of an apparant surface charge (ASC) which resides on the molecule's surface. This is known as the ASC approach. The underlying electrostatic problem is described by a transmission problem for the Laplacian in the whole space ℝ3. Therefore, the integral equation formalism (IEF) offers a favourable approach to compute the electrostatic solute-solvent interaction.

Boundary integral equations are generally solved by the boundary element method (BEM). BEM is a well established tool in PCM. However, traditional discretizations lead to densely populated and possibly ill-conditioned system matrices. Both features pose serious obstructions to the efficient numerical treatment of such problems. Modern methods for the rapid BEM solution reduce the complexity to almost or even optimal rates. We apply the wavelet Galerkin scheme which produces the approximate solution with discretization error accuracy in linear complexity.