# Conformal Mapping Improvement of the Boundary Element Method Solution for Underground Water Flow in a Domain with a Very Singular Boundary

August 2018

Thesis

## Degree Name

Master of Science (MS)

## Department

Mathematical Sciences

Angel Muleshkov

Zhonghai Ding

## Third Committee Member

Michelle Robinette

Stephen Lepp

85

## Abstract

Numerical solutions using a Boundary Element Method (BEM) for a confined flow in a very singular finite domain are developed. Typically, in scientific journal publications, authors avoid domains with many and more malignant singularities due to the extremely big and difficult to estimate errors in the numerical calculations. Using exact Conformal Mapping solutions for simplified domains with the same singularity as in the original domain, this problem can be solved numerically with improvements introduced by Conformal Mapping solutions. Firstly, to experiment with improving the BEM solution by Conformal Mapping, a domain inside a rectangle is considered. The exact solution inside a rectangle is found by Conformal Mapping. Then solution is obtained from BEM. The singularities in number and malignancy will lead to failure of the Boundary Element solution near the singular points and even much further from them. Then, the BEM solution is improved by Conformal Mapping. For every singularity, the domain will be extended in an easy enough shape, which allows for using Conformal Mapping method to find the exact solution near the singularity for the extended domain. The function that describes the behavior of the solution close to the singularity is imposed to the elements that are adjacent to the corresponding singularities in the BEM calculations, which leads to the BEM solution improved by Conformal Mapping. The different methods for finding the solution of the problem inside a rectangle are compared to show the need for improvement of the BEM solution. Then, a more complicated realistic finite domain for which it is impossible to find the exact solution is considered. The problem is solved by BEM and by improving BEM by Conformal Mapping as described above. The calculations are expressed in the form of tables and figures. Additional analysis of the effect of the singularities is given in the conclusions.

## Keywords

2D Laplace PDE; Boundary Element Method (BEM); Conformal Mapping; Exact Solution; Singularities; Underground Water Flow

## Disciplines

Applied Mathematics | Engineering | Physics

pdf

English