2016/03/13 by Martin J. Gander, Gander, Martin J., Soheil Hajian +1
Engineering · Mathematics · #65F08 #65F10 #65H10 #65N22 #65N55 #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1603.04073
openalex publication_date 2016/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Schwarz methods are attractive parallel solution techniques for solving\nlarge-scale linear systems obtained from discretizations of partial\ndifferential equations (PDEs). Due to the iterative nature of Schwarz methods,\nconvergence rates are an important criterion to quantify their performance.\nOptimized Schwarz methods (OSM) form a class of Schwarz methods that are\ndesigned to achieve faster convergence rates by employing optimized\ntransmission conditions between subdomains. It has been shown recently that for\na two-subdomain case, OSM is a natural solver for hybridizable discontinuous\nGalerkin (HDG) discretizations of elliptic PDEs. In this paper, we generalize\nthe preceding result to the many-subdomain case and obtain sharp convergence\nrates with respect to the mesh size and polynomial degree, the subdomain\ndiameter, and the zeroth-order term of the underlying PDE, which allows us for\nthe first time to give precise convergence estimates for OSM used to solve\nparabolic problems by implicit time stepping. We illustrate our theoretical\nresults with numerical experiments.\n