TY - JOUR
T1 - Domain Decomposition Methods for Diffusion Problems with Discontinuous Coefficients Revisited
AU - Na , Xuyang
AU - Xu , Xuejun
JO - Communications in Computational Physics
VL - 1
SP - 212
EP - 238
PY - 2024
DA - 2024/01
SN - 35
DO - http://doi.org/10.4208/cicp.OA-2023-0184
UR - https://global-sci.org/intro/article_detail/cicp/22901.html
KW - Diffusion problem, discontinuous coefficients, finite elements, domain decomposition.
AB - <p style="text-align: justify;">In this paper, we revisit some nonoverlapping domain decomposition methods for solving diffusion problems with discontinuous coefficients. We discover some
interesting phenomena, that is, the Dirichlet-Neumann algorithm and Robin-Robin algorithms may make full use of the ratio of coefficients in some special cases. Detailedly,
in the case of two subdomains, we find that their convergence rates are $\mathcal{O}(ν_1/ν_2)$ if $ν_1 &lt; ν_2,$ where $ν_1, \ ν_2$ are coefficients of two subdomains. Moreover, in the case of
many subdomains with red-black partition, the condition number bounds of Dirichlet-Neumann algorithm and Robin-Robin algorithm are $1+\epsilon(1+{\rm log}(H/h))^2$ and $C+\epsilon(1+ {\rm log}(H/h))^2,$ respectively, where $\epsilon$ equals ${\rm min}\{ν_R/ν_B,ν_B/ν_R\}$ and $ν_R,ν_B$ are the coefficients of red and black domains. By contrast, Neumann-Neumann algorithm and
Dirichlet-Dirichlet algorithm could not obtain such good convergence results in these
cases. Finally, numerical experiments are preformed to confirm our findings.</p>