Schwarz methods by domain truncation

Schwarz methods use a decomposition of the computational domain into subdomains and need to impose boundary conditions on subdomain boundaries. In truncation one restricts unbounded bounded must also put both fields there are vast bodies literature research is very active ongoing. It turns out be fruitful think in as onto subdomains. Seminal precursors this fundamental idea papers by Hagstrom, Tewarson Jazcilevich (1988), Després (1990) Lions (1990). The first truly optimal method that converges finite number steps was proposed Nataf (1993), used precisely transparent transmission between Approximating these for fast convergence led development optimized – name has become common based truncation. Compared classical methods, which simple Dirichlet have been successfully wide range applications, much less well understood, mainly due their more sophisticated conditions. A key application with such turned time-harmonic wave propagation problems, because simply do not work case. past decade given us many new One review from an algorithmic perspective (Gander Zhang 2019) showed equivalence methods. analysis however, lagging behind development. general abstract framework cannot thus open theoretical questions about convergence. Just practical multigrid Fourier instrumental understanding tuning Similar local mode multigrid, two-subdomain case model its simplicity. Many aspects actual situation, e.g. original problem subdomains, were neglected analysis. While gave important insight, phenomena beyond models discovered. This present situation motivation our survey: give comprehensive precise exploration behaviours analysis, taking account conditions, many-subdomain decompositions layered media. We consider operator $-\Delta + \eta $ diffusive $\eta>0$ (screened Laplace equation) or oscillatory $\eta <0$ (Helmholtz equation), order show difference behaviour solvers problems. we study include lowest-order absorbing (Robin), advanced perfectly matched layers (PMLs), developed Our intensive over last two years several results presented here time: Helmholtz equation, see strong influence imposed global factor asymptotic factors small overlap can differ find scaling when size fixed, robust double-sweep free-space problem, either fixed zeroth-order Taylor logarithmically growing PML, PMLs like smoothers converge faster higher frequencies; particular, plane waves (in error) passing through interfaces at right angle slowly. addition main focus Sections 2 3, start Section 1 expository historical introduction 4 brief interpretation recently cross-points viewpoint conclude 5 summary Appendix provide Matlab program block LU form cross-points, B Maple