Optimized Schwarz Methods with Robin Conditions for the Advection-Diffusion Equation

Abstract

Summary. We study optimized Schwarz methods for the stationary advectiondiffusion equation in two dimensions. We look at simple Robin transmission conditions, with one free parameter. In the nonoverlapping case, we solve exactly the associated min-max problem to get a direct formula for the optimized parameter. In the overlapping situation, we solve only an approximate min-max problem. The asymptotic performance of the resulting methods, for small mesh sizes, is derived. Numerical experiments illustrate the improved convergence compared to other Robin conditions.

Optimized Schwarz methods for unsymmetric layered problems with strongly discontinuous and anisotropic coefficients

“…Differently from what is done in [12], we do not have an explicit formula linking the real part and the imaginary part of the eigenvalues of the matrices Λ j D −1 , j = 1, 2, and we cannot solve the optimization problem (5.1) analytically. Let…”

Optimized Schwarz methods for the Stokes–Darcy coupling

“…Since Optimized Schwarz Methods do not require overlap to converge, they have become quite popular in the last decade, and are a natural framework to deal with a spatial decomposition of the domain driven by a multi-physics problem (see Gerardo-Giorda et al (2010)). Although in general Optimized Schwarz methods based on one-sided interface conditions (α f = α p ) have been extensively used along the years (see, e.g., Lions (1990); Gander (2006); Japhet et al (2001); Collino et al (1997)), the use of two-sided interface condition (α f = α p ) has recently become increasingly popular due to better convergence properties of the associated algorithms, see Alonso-Rodriguez & Gerardo-Giorda (2006); Dolean et al (2009);Dubois (2007); Gander et al (2007);Gerardo-Giorda & Perego (2013). Since such parameters are in general obtained by suitable approximations of the symbols in the Fourier space of the Steklov-Poincaré operator (or Dirichlet-to-Neumann mapping) associated to the problem within the subdomain (Gander (2006)), the two-sided interface conditions are a natural choice in the presence of multi-physics problems where different problems have to be solved in different regions of the computational domain (Gerardo-Giorda et al (2010).…”

Domain decomposition methods to model heat exchanges between a well and a rock mass

“…Let us consider the Robin Robin Schwarz algorithm [13,12,11] with Robin coefficients β rock ≥ 0 and β ff ≥ 0, β rock β ff = 0, which updates the temperatures (T n−1 ff , T n−1 wall , T n−1 rock ) at iteration n − 1 ≥ 0 by the temperatures (T n ff , T n wall , T n rock ). They are defined by the solution T n rock of the following rock subproblem with Robin condition at the well boundary:…”

“…The convergence rate of domain decomposition methods depends crucially on the boundary conditions applied at the interface between both subdomains. In this work, we investigate Robin-Robin optimized Schwarz domain decomposition methods based on Robin boundary conditions for both subproblems [20,13,12,11]. It will be compared with the more usual Dirichlet Neumann domain decomposition algorithm based on a Dirichlet boundary condition on the rock mass subproblem combined with a Neumann boundary condition on the well subproblem.…”