The Dirichlet-Neumann iteration – Three-field case: Methods and analyses

Sammanfattning: We construct and analyze Dirichlet-Neumann iterations for the 1D Poisson equation. Specifically, we wish to gain insight into how the convergence depends on material coefficients when solving coupled linear heat equations on three non-overlapping domains. We first consider the two-domain case and then extend the method to three domains. A finite element method is used to discretize the Laplacian. Exact formulae are provided for the spectral radii of the iteration matrices for all methods considered. Their validity as predictors for the convergence rates is verified through numerical tests. We show that the different methods for the three-field case have distinct and complementary convergence properties and give an overview of problems, specifying which method is the most suitable.