Adaptive Coarse Spaces for FETI-DP and BDDC Methods

Iterative substructuring methods are well suited for the parallel iterative solution of elliptic partial differential equations. These methods are based on subdividing the computational domain into smaller nonoverlapping subdomains and solving smaller problems on these subdomains. The solutions are then joined to a global solution in an iterative process. In case of a scalar diffusion equation or the equations of linear elasticity with a diffusion coefficient or Young modulus, respectively, constant on each subdomain, the numerical scalability of iterative substructuring methods can be proven. However, the convergence rate deteriorates significantly if the coefficient in the underlying partial differential equation (PDE) has a high contrast across and along the interface of the substructures. Even sophisticated scalings often do not lead to a good convergence rate. One possibility to enhance the convergence rate is to choose appropriate primal constraints. In the present work three different adaptive approaches to compute suitable primal constraints are discussed. First, we discuss an adaptive approach introduced by Dohrmann and Pechstein that draws on the operator P_D which is an important ingredient in the analysis of iterative substructuring methods like the dual-primal Finite Element Tearing and Interconnecting (FETI-DP) method and the closely related Balancing Domain Decomposition by Constraints (BDDC) method. We will also discuss variations of the method by Dohrmann and Pechstein introduced by Klawonn, Radtke, and Rheinbach. Secondly, we describe an adaptive method introduced by Mandel and Sousedík which is also based on the P_D-operator. Recently, a proof for the condition number bound in this method was provided by Klawonn, Radtke, and Rheinbach. Thirdly, we discuss an adaptive approach introduced by Klawonn, Radtke, and Rheinbach that enforces a Poincaré- or Korn-like inequality and an extension theorem. In all approaches generalized eigenvalue problems are used to compute a coarse space that leads to an upper bound of the condition number which is independent of the jumps in the coefficient and depend on an a priori prescribed tolerance. Proofs and numerical tests for all approaches are given in two dimensions. Finally, all approaches are compared