Isogeometric Schwarz preconditioners for linear elasticity systems

Hybrid and Multiplicative Overlapping Schwarz Algorithms with Standard Coarse Spaces for Mixed Linear Elasticity and Stokes Problems

“…We do not have a very precise explanation for this difference, which could be due to the use of different coarse problems and it might not hold for the more general 3D problems considered in [11]. Nevertheless, in our particular configuration, it seems that it could be possible to improve the bounds (14) and (17) to κ(P OAS ) ≤ C(H/δ) 3 (1 + log(H/δ)) and κ(P OHS ) ≤ C(H/δ) 3 (1 + log(H/δ)), indicating that the two-level overlapping Schwarz algorithms with standard coarse spaces developed in this work are actually optimal, i.e., in the generous overlap case H/δ = Constant, these bounds become independent of H/h. Moreover, we also tested our preconditioners for the checkerboard test (see the coefficient distribution in Section 5.3) in which neither Assumption 1 nor Assumption 2 are satisfied.…”

“…Theorem 3.2 Assuming that the same assumptions of Theorem 3.1 hold, we have κ(P OHS ) ≤ C(H/δ) 3 (1 + log(H/δ))(1 + log(H/h)).…”

“…Earlier works on overlapping Schwarz methods for linear elasticity have focused on the compressible case in which the Poisson ratio ν is bounded away from 1/2. The related recent developments are as follows: some nonoverlapping domain decomposition algorithms for mixed elasticity and Stokes systems have studied Wirebasket and Balancing Neumann-Neumann methods, see [1,19,34,35], and FETI-DP and BDDC methods for the incompressible limit, see [20,22,25,26,36,38]; more recent applications include fluid-structure interaction [2], computational fluid dynamics [15,20], and isogeometric analysis [3]. We will consider two different approaches to solve the resulting saddle point systems: a) a preconditioned conjugate gradient (PCG) method applied to the symmetric positive definite reformulation of the almost incompressible linear elasticity system obtained by eliminating the pressure unknowns element by element; b) a GMRES method with indefinite overlapping Schwarz preconditioner applied directly to the saddle point formulation of both the elasticity and Stokes systems.…”

Parallelization of continuous and discontinuous Galerkin dual–primal isogeometric tearing and interconnecting methods

“…Depending on the considered formulation, one needs to use the right space V X h , X ∈ {cG, dG}. The IgA schemes (4) and (6) are equivalent to the system of linear IgA equations…”

Parallel Sum Primal Spaces for Isogeometric Deluxe BDDC Preconditioners

“…For earlier work on the iterative solution of isogeometric approximations, see Beirão da Veiga et al (2013b), Collier et al (2013), Gahalaut et al (2013), Kleiss et al (2012).…”