Restricted overlapping balancing domain decomposition methods and restricted coarse problems for the Helmholtz problem

High performance domain decomposition methods on massively parallel architectures with freefem++

“…The approach to build the partition of unity is the same as in most domain decomposition methods, see for example [7]. Let χ i be a continuous piecewise linear function of Ω l i defined as such:…”

Scalable domain decomposition preconditioners for heterogeneous elliptic problems

“…In our experiments, we use a pretty simple partition of unity, already used for example in [19]. Let χi be a continuous piecewise linear function (hence mesh elements vertices and degrees of freedom coincide) defined on Ω δ i as such:…”

Optimized Schwarz Methods with Overlap for the Helmholtz Equation

“…Després' method with overlap, c.f. [3,12]) denoted by TO0, because it corresponds to a Taylor expansion of zero order of the symbol of the DtN operator, and the second-order low frequency absorbing condition, which is denoted by TO2. Since the Schwarz methods can be used as a stationary iterative solver, or as a preconditioner for GMRES, both cases are tested, except for the classical Schwarz stationary iteration, which can not converge.…”

“…For the Helmholtz equation, simple absorbing conditions of the form ∂ n − iω were proposed as transmission condition (TC) in Schwarz methods first without overlap in [4], and later also with overlap, see [3,12]. More advanced TCs can also be used, see e.g.…”