Mathematical analysis of boundary integral equations and domain decomposition methods with applications in polarisable electrostatics
Hassan, Muhammad; Cancès, Eric (Thesis advisor); Hiptmair, Ralf (Thesis advisor); Stamm, Benjamin (Thesis advisor)
Dissertation / PhD Thesis
This dissertation is primarily concerned with the analysis of mathematical models that arise in the study of polarisable electrostatics, either in the context of dielectric particles undergoing mutual polarisation or implicit solvation modelling in theoretical chemistry. The main objective of our work is to understand how the accuracy and computational cost of the numerical methods used to solve the governing equations in both cases scale with the number of objects in the problem. To aid the reader, we have divided the dissertation into three broadly independent parts: In the first part of this dissertation (Chapters 2-5), we analyse a boundary integral equation that models the electrostatic interaction between N dielectric spherical particles undergoing mutual polarisation. Our main result is to prove that under appropriate geometrical assumptions, the three key quantities of interest in this problem, namely, the induced surface charge on each particle, the total electrostatic energy of system, and the electrostatic forces acting on each particle can be computed with linear scaling in accuracy, i.e., it requires only O(N) operations to compute approximations of these quantities with a given and fixed relative error. In the second part of this dissertation (Chapters 6-8), we perform a scalability analysis of the parallel Schwarz method which is used to solve the COSMO implicit solvation model for the reaction potential generated by a solute molecule with a given charge distribution embedded in a polarisable medium. Our main result is to construct a new, systematic framework that allows us to characterise the norm of the Schwarz operator for domains representing solute molecules in which multiple atoms have a common overlap and atoms can be completely buried in the interior of the molecule with no access to the external medium. The third part of this dissertation (Chapter 9) is a departure from the themes of polarisable electrostatics and scalability analysis that relate Parts I and II but is connected to both integral operators and domain decomposition in a general sense. Our main result here is to establish a fundamental continuity property for Riesz potentials on polygonal boundaries in two spatial dimensions. As weakly singular integral operators, Riesz potentials can be used to construct exponentially convergent, non-overlapping domain decomposition methods for the Helmholtz equation, and our result fills a gap in the analysis of such methods.