# Integral equations and function extension techniques for numerical solution of PDEs

**Time: **
Fri 2021-10-01 14.00

**Location: **
F3, KTH eller via Zoom, Lindstedsvägen 26, Stockholm (English)

**Subject area: **
Mathematics

**Doctoral student: **
Fredrik Fryklund
, Numerisk analys, NA

**Opponent: **
PhD Alex Barnett, Flatiron Institute

**Supervisor: **
Professor Anna-Karin Tornberg, Numerisk analys, NA

## Abstract

Today many phenomena from science and engineering can be simulated accurately thanks to computational methods. Still, many challenges remain, one of them being close interface interactions when simulating e.g. the dynamics of a substance concentration in multiphase flows at the micro level. The challenge is to maintain high accuracy and efficiency as drops, vesicles, etc. are very close to each other, which many numerical methods struggle with. Also, the drops' geometries undergo changes over time. Thus far there is no standardized method for solving the equation modeling a concentration on time-dependent geometries efficiently and accurately. Boundary integral methods are powerful in handling moving and complex geometries, and maintaining high accuracy throughout the domain, even for close interactions. However, they are only efficient for a limited class of problems, and thus do not apply to our problem at hand.

The focus of this dissertation is to expand the class of problems boundary integral methods are applicable to, without sacrificing their most attractive properties, plus presenting how the resulting equations time-dependent geometries can be solved on time-dependent geometries. This is achieved by the development of the algorithm partition of unity extension (PUX). It smoothly extends data from its domain of definition, with compact support, which allows for application of established fast methods.

With our PUX method and state-of-the-art computational algorithms new problems could be studied, and to new levels of accuracy. In the process new underlying dynamics that were previously obscured by large errors appeared. These findings spurred a new set of questions, leading to the design of accurate algorithms for a class of problems that appear when applying boundary integral methods in conjunction with discretizing the governing equations first in time.

In sum, we are now closer to a complete solver for the evolution of the substance concentration on time-dependent geometries. In this endeavor, methods have been development and studied that have applications outside our scope, and have already been applied successfully in other researchers' work. The work with PUX has thus been fruitful and will be developed and investigated further in the future, with adaptivity as the goal.