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Convergence Analysis of the Dirichlet‐Neumann Iteration for Finite Element Discretizations
Author(s) -
Monge Azahar,
Birken Philipp
Publication year - 2016
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201610355
Subject(s) - spectral radius , discretization , mathematics , convergence (economics) , dirichlet distribution , mathematical analysis , rate of convergence , matrix (chemical analysis) , context (archaeology) , finite element method , preconditioner , coupling (piping) , limit (mathematics) , eigenvalues and eigenvectors , physics , boundary value problem , linear system , computer science , thermodynamics , materials science , computer network , channel (broadcasting) , paleontology , quantum mechanics , metallurgy , economics , composite material , biology , economic growth
We analyze the convergence rate of the Dirichlet‐Neumann iteration for the fully discretized unsteady transmission problem. Specifically, we consider the coupling of two linear heat equations on two identical non overlapping domains with jumps in the material coefficients across these. In this context, we derive the iteration matrix of the coupled problem. In the 1D case, the spectral radius of the iteration matrix tends to the ratio of heat conductivities in the semidiscrete spatial limit, but to the ratio of the products of density and specific heat capacity in the semidiscrete temporal one. This explains the fast convergence previously observed for cases with strong jumps in the material coefficients. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)