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

Sammanfattning: 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.