# LUP Student Papers

## LUND UNIVERSITY LIBRARIES

### Solving the steady-state heat equation using overdetermined non-overlapping domain decomposition methods

(2018) In Master's Theses in Mathematical Sciences FMN820 20172
Mathematics (Faculty of Engineering)
Abstract
Domain decomposition methods can be used to numerically solve partial differential equations for certain problems, for example in cases where the domain has an irregular shape, or if there are differences in material constants. By splitting the domain into subdomains, these problems can be solved using domain decomposition methods. In this thesis, the topic is solving the steady-state heat equation using more than one boundary condition for each subdomain, causing the domain decomposition method to be overdetermined. The least squares method is used to handle this, and so it is explored if, by modifying the method to use parts of the mathematical formulation as constraints, the method will find an adequate approximation to the steady-state... (More)
Domain decomposition methods can be used to numerically solve partial differential equations for certain problems, for example in cases where the domain has an irregular shape, or if there are differences in material constants. By splitting the domain into subdomains, these problems can be solved using domain decomposition methods. In this thesis, the topic is solving the steady-state heat equation using more than one boundary condition for each subdomain, causing the domain decomposition method to be overdetermined. The least squares method is used to handle this, and so it is explored if, by modifying the method to use parts of the mathematical formulation as constraints, the method will find an adequate approximation to the steady-state heat equation. It was found that overdetermined domain decomposition methods can indeed find a good approximation of the temperature distribution, and that using a constrained least squares method with different types of relaxation, can decrease the number of iterations to reach termination. This paves way for more work in relation to the use of overdetermined domain decomposition methods. (Less)
author
supervisor
organization
course
FMN820 20172
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Overdetermined non-overlapping domain decomposition methods, constrained least squares methods, steady-state heat equation
publication/series
Master's Theses in Mathematical Sciences
report number
LUFTNA-3043-2018
ISSN
1404-6342
other publication id
2018:E22
language
English
id
8946627
2018-06-11 15:53:37
date last changed
2018-06-11 15:53:37
```@misc{8946627,
abstract     = {Domain decomposition methods can be used to numerically solve partial differential equations for certain problems, for example in cases where the domain has an irregular shape, or if there are differences in material constants. By splitting the domain into subdomains, these problems can be solved using domain decomposition methods. In this thesis, the topic is solving the steady-state heat equation using more than one boundary condition for each subdomain, causing the domain decomposition method to be overdetermined. The least squares method is used to handle this, and so it is explored if, by modifying the method to use parts of the mathematical formulation as constraints, the method will find an adequate approximation to the steady-state heat equation. It was found that overdetermined domain decomposition methods can indeed find a good approximation of the temperature distribution, and that using a constrained least squares method with different types of relaxation, can decrease the number of iterations to reach termination. This paves way for more work in relation to the use of overdetermined domain decomposition methods.},
author       = {Thor, Filip},
issn         = {1404-6342},
keyword      = {Overdetermined non-overlapping domain decomposition methods,constrained least squares methods,steady-state heat equation},
language     = {eng},
note         = {Student Paper},
series       = {Master's Theses in Mathematical Sciences},
title        = {Solving the steady-state heat equation using overdetermined non-overlapping domain decomposition methods},
year         = {2018},
}

```