Block FETI–DP/BDDC preconditioners for mixed isogeometric discretizations of three-dimensional almost incompressible elasticity

Abstract

A block FETI-DP/BDDC preconditioner for mixed formulations of almost incompressible elasticity is constructed and analyzed; FETI-DP (dual primal finite element tearing and interconnection) and BDDC (balancing domain decomposition by constraints) are two very successful domain decomposition algorithms for a variety of elliptic problems. The saddle point problems of the mixed problems are discretized with mixed isogeometric analysis with continuous pressure fields. As in previous work by Tu and Li (2015), for finite element discretizations of the incompressible Stokes system, the proposed preconditioner is applied to a reduced positive definite system involving only the pressure interface variable and the Lagrange multipliers of the FETI-DP algorithm. The novelty of this preconditioner consists in using BDDC with deluxe scaling for the interface pressure block as well as deluxe scaling for the FETI-DP preconditioner for the Lagrange multiplier block. A convergence rate analysis is presented with a condition number bound for the preconditioned operator which depends on the inf-sup parameter of the fully assembled problem and the condition number of a closely related BDDC algorithm for compressible elasticity. This bound is scalable in the number of subdomains, poly-logarithmic in the ratio of subdomain and element sizes, and robust with respect to material incompressibility and presence of discontinuities of the Lamé parameters across subdomain interfaces. Parallel numerical experiments validate the theory and indicate how the rate of convergence varies with respect to the spline polynomial degree and regularity and the deformation of the domain. Of particular interest is the development of variants of the algorithm with a coarse component of small dimension.

Dual-Primal Isogeometric Tearing and Interconnecting methods for the Stokes problem

“…In [9], a solver has also been proposed for the elasticity problem for quasi incompressible materials. These results have recently been extended in [14].…”

Recent advances in domain decomposition methods for large-scale saddle point problems

“…As for a convergence rate analysis for a discretization with a continuous pressure, a recent article [44] generalizes the theory developed in [45] to the case of non-zero pressure block but under the assumption that the discontinuities are resolved by the subdomains.…”

Robust and scalable adaptive BDDC preconditioners for virtual element discretizations of elliptic partial differential equations in mixed form

“…In what follows, we will present a concise description of their theoretical aspects by focusing on the VEM discretization; for further algorithmic details, see [61]. For solvers designed for continuous discretizations of the pressure space, the interested reader is referred to [57] and the references therein.…”