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High order trapezoidal rule-based quadratures for boundary integral methods on non-parametrized surfaces

Time: Wed 2022-11-02 10.00

Location: F3, Lindstedtsvägen 26 & 28, Stockholm

Language: English

Subject area: Mathematics

Doctoral student: Federico Izzo , Numerisk analys, NA

Opponent: Professor Robert Krasny, University of Michigan,US

Supervisor: Olof Runborg, Numerisk analys, NA

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This thesis is concerned with computational methods for solving boundary integral equations (BIE) on surfaces defined without explicit parametrization, called Implicit Boundary Integral Methods (IBIM). Using implicit methods for describing surfaces, such as the level-set method, can be advantageous for complex geometries and problems where the surface evolves over time. 

In the IBIM setting, the surface integrals appearing in the BIE are written as volume integrals over domains surrounding the surface using the signed distance function. The singular integrands defined on the surface become functions singular along a straight line in the volume. Accurately integrating such functions is challenging as the special quadrature rules previously developed for BIE only deal with point singularities aligned with the grid in R2, and not line singularities in R3.

In this thesis we focus on developing a framework for integrating three-dimensional functions singular along a line using the trapezoidal rule. We first split the three-dimensional problem in a composition of two-dimensional problems, where the singularity is only in a point unaligned with the grid. We then develop corrected trapezoidal rules to deal with these two-dimensional singular integrands with point singularities unaligned with the grid. Moreover we develop generalizations to such rules to Rn for a wide class of functions which can reach arbitrarily high order. Then we develop expressions and approximations of the singular layer kernels from IBIM in a way that can be used with the corrected trapezoidal rules. The expressions are related to the approximation of the surface in the target points, and the higher the order of approximation of the surface the more accurate the expressions of the kernels.

We adapt and apply the quadrature methods to the computation of the electrostatic potential of macromolecules immersed in aqueous solvent. For this application, the surface represents the solute-solvent interface where the molecule and the solvent particles interact. The potential solves the linearized Poisson-Boltzmann equation, but can be written as the solution of a coupled system of BIE. 

The corrected trapezoidal rules developed aim to showcase IBIM as a valid and robust alternative to standard techniques for BIE for computationally intensive applications.