Implementation and study of boundary integral operators related to PDE:s in the plane
(2023) In Master's Theses in Mathematical Sciences FMNM01 20231Mathematics (Faculty of Engineering)
- Abstract
- The method of solving boundary value problems of partial differential equations numerically by first reformulating the problem as a boundary integral equation has many advantages over other methods, but also some unique difficulties. Some of these difficulties stem from problems in evaluating singular or nearly singular integral operators, and solving these difficulties is an active field of research. Known results are summarized, and an accessible program package is developed, using underlying Gauss--Legendre quadrature and product integration, which can be applied to boundary value problems with smooth boundaries. The program package is available on GitHub at the link https://github.com/erikandersson98/BIE-CELib. Different methods of... (More)
- The method of solving boundary value problems of partial differential equations numerically by first reformulating the problem as a boundary integral equation has many advantages over other methods, but also some unique difficulties. Some of these difficulties stem from problems in evaluating singular or nearly singular integral operators, and solving these difficulties is an active field of research. Known results are summarized, and an accessible program package is developed, using underlying Gauss--Legendre quadrature and product integration, which can be applied to boundary value problems with smooth boundaries. The program package is available on GitHub at the link https://github.com/erikandersson98/BIE-CELib. Different methods of implementing integral operators related to the Laplace and Helmholtz equations are compared with regards to accuracy and convergence rate, both when used in boundary value problems, and when applied to theoretical identities. The methods are based on product integration, as well as on global and local regularization. To conclude, recommended implementations based on the results are given, as well as possible directions to expand the package. (Less)
- Popular Abstract
- Partial differential equations (PDEs) are used to model a plethora of physical phenomena, like the behaviour of electromagnetic or acoustic waves, heat flow, and stress and strain in structural dynamics. In order to solve a given PDE, a domain where the equations hold as well as conditions on the boundaries of the domain and starting conditions need to be specified. Because of the wide applicability of PDEs in modelling the physical world, it is of great interest to be able to solve these PDEs, with some of the most well known methods of solving PDEs being the finite element method and finite difference methods.
This degree project looks at a different method, which consists of reformulating the PDE as a new problem which takes the form... (More) - Partial differential equations (PDEs) are used to model a plethora of physical phenomena, like the behaviour of electromagnetic or acoustic waves, heat flow, and stress and strain in structural dynamics. In order to solve a given PDE, a domain where the equations hold as well as conditions on the boundaries of the domain and starting conditions need to be specified. Because of the wide applicability of PDEs in modelling the physical world, it is of great interest to be able to solve these PDEs, with some of the most well known methods of solving PDEs being the finite element method and finite difference methods.
This degree project looks at a different method, which consists of reformulating the PDE as a new problem which takes the form of an integral equation over the boundary, a boundary integral equation (BIE). Sparing the details, the basic idea is that the solution is thought to be caused by some density on the boundary, so in the context of electrostatics, the charge in the domain is thought to be caused by a charge density on the boundary. Then an integral relation is assumed to hold which automatically fulfills the partial differential equation. All that remains then is to numerically solve for the density on the boundary, after which the charge in the domain can be reconstructed by using the assumed integral relation.
The method is of interest since it allows for high numerical precision and rate of convergence compared to other methods, as well as other positives like reduction of the dimension of the problem to the dimension of the boundary. But some unique problems come up, stemming from singularities in the integrand of the BIE, leading mostly to worse numerical results close to the boundary. These problems can be fixed in a variety of different ways, and in the project the method of product integration close to the boundary is considered. Instead of just integrating numerically close to the boundary, which leads to bad results, a part of the integrand is replaced with a suitable interpolating polynomial. The polynomial terms can be integrated exactly, and the solution contains errors resulting from the polynomial interpolation, but avoids the larger errors that would result from normal evaluation.
In the project, some known theory is summarized regarding the process of dicretizing the integral equation and utilizing product integration close to the boundary and tests of different. Different implementations of commonly occurring integral operators were tested to see which implementation gives the best numerical results. To conduct these tests a program package for MATLAB was developed and made available at GitHub which can be used to solve different problems of the Laplace and Helmholtz equations using the BIE method with product integration close to the boundary. The resulting package can be found at https://github.com/erikandersson98/BIE-CELib. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9133621
- author
- Andersson, Erik LU
- supervisor
- organization
- course
- FMNM01 20231
- year
- 2023
- type
- H2 - Master's Degree (Two Years)
- subject
- keywords
- Boundary integral equation, Singular integral operator, Product integration, Close evaluation, Numerical integration, MATLAB
- publication/series
- Master's Theses in Mathematical Sciences
- report number
- LUTFNA-3052-2023
- ISSN
- 1404-6342
- other publication id
- 2023:E27
- language
- English
- id
- 9133621
- date added to LUP
- 2023-08-28 11:35:18
- date last changed
- 2023-09-07 11:01:01
@misc{9133621,
abstract = {{The method of solving boundary value problems of partial differential equations numerically by first reformulating the problem as a boundary integral equation has many advantages over other methods, but also some unique difficulties. Some of these difficulties stem from problems in evaluating singular or nearly singular integral operators, and solving these difficulties is an active field of research. Known results are summarized, and an accessible program package is developed, using underlying Gauss--Legendre quadrature and product integration, which can be applied to boundary value problems with smooth boundaries. The program package is available on GitHub at the link https://github.com/erikandersson98/BIE-CELib. Different methods of implementing integral operators related to the Laplace and Helmholtz equations are compared with regards to accuracy and convergence rate, both when used in boundary value problems, and when applied to theoretical identities. The methods are based on product integration, as well as on global and local regularization. To conclude, recommended implementations based on the results are given, as well as possible directions to expand the package.}},
author = {{Andersson, Erik}},
issn = {{1404-6342}},
language = {{eng}},
note = {{Student Paper}},
series = {{Master's Theses in Mathematical Sciences}},
title = {{Implementation and study of boundary integral operators related to PDE:s in the plane}},
year = {{2023}},
}