Convergence Rate of the Dirichlet-Neumann Algorithm for Coupled Poisson Equations

Abstract: This thesis presents and tests the convergence rate of the Dirichlet-Neumann algorithm for two Poisson equations coupled by transmission boundary conditions. Three second order discretisation methods are used when analyzing the convergence: standard equidistant finite difference, standard adaptive linear finite element, and standard adaptive finite volume discretisation of Poisson's equation. The convergence rate of the Dirichlet-Neumann algorithm, when using each of the discretisations for both sub problems, is presented and proved. Using elements of the proofs for the intermediate results leads to a theorem when combining the discretisations. The theorem states that the Dirichlet-Neumann algorithm's convergence rate is entirely independent of the grid used for any combination of the discretisations analyzed. Inspired by these results a general theorem of the convergence rate is presented. Using semi-discrete analysis it is possible to generalize the results to a large subset of discretisations in the asymptotic case. It is possible to remove the asymptotic argument if the discretisations approximate the homogenous solution exactly. All theoretical results were numerically confirmed. The numerical results aligned with the theoretical conclusions.