Domain Decomposition with Local Impedance Conditions for the Helmholtz Equation with Absorption

Analysis of the SORAS Domain Decomposition Preconditioner for Non-self-adjoint or Indefinite Problems

“…The following theorem, which generalizes the theory for the Helmholtz equation developed in [17], identifies assumptions that are sufficient to obtain the two bounds. In particular, the proof was inspired by the one of [17,Theorem 3.11] and by the analysis in subsection [17, §3.2].…”

“…Assumptions (3.2) and (3.3) are classical inequalities in the domain decomposition framework. Inequality (3.2) is dubbed in [17] 'a kind of converse to the stable splitting result', and it can be viewed as a continuity property of the reconstruction operator {W j } N j=1 → N j=1 R T j W j . In [17,Lemma 3.6] the inequality is proved at the continuous level for the Helmholtz energy norm (see [17,…”

“…In the literature, GMRES convergence estimates are based for instance on the field of values [13,12,4] or on the pseudo-spectrum (see [31] and references therein) of the preconditioned operator. For example, field of values bounds were derived for overlapping domain decomposition preconditioners for non-symmetric parabolic problems which are small perturbations of SPD operators [9], and later for the high-frequency Helmholtz [16,17,15] and time-harmonic Maxwell [5] equations.…”

“…Here, by generalizing the work of [17], we analyze for generic problems the convergence of the preconditioned GMRES method in its weighted version [14]. We identify a list of assumptions and estimates that are sufficient to obtain an upper bound on the norm of the preconditioned matrix, and a lower bound on the distance of its field of values from the origin.…”

“…It was later studied in ( [21], 2015) for generic symmetric positive definite problems and viewed as a symmetric variant of the ORAS preconditioner ( [30], 2007), hence called SORAS (Symmetrized Optimized Restricted Additive Schwarz ). Note that in [24] several one-level and two-level versions, with a coarse space based on plane waves, were tested numerically, and more than ten years later the one-level OBDD-H version was rigorously analyzed in [17], for the Helmholtz equation. In [21] a two-level version, with a spectral coarse space, was rigorously analyzed for generic SPD problems.…”