Lagrange Multiplier Based Substructuring: FETI Method

In this chapter, we describe the FETI method (the Finite Element Tearing and Interconnecting method) [FA2, FA16, FA15, MA25, FA14, KL8]. It is a Lagrange multiplier based iterative substructuring method for solving a finite element discretization of a self adjoint and coercive elliptic equation, based on a non-overlapping decomposition of its domain. In traditional substructuring, each subdomain solution is parameterized by its Dirichlet value on the boundary of the subdomain. The global solution is sought by solving a reduced Schur complement system for determining the unknown Dirichlet boundary values of each subdomain solution. By contrast, in Lagrange multiplier substructuring, each subdomain solution is parameterized by a Lagrange multiplier flux variable which represents the Neumann data of each subdomain solution on the subdomain boundary. The global solution is then sought by determining the unknown Lagrange multiplier flux variable, by solving a saddle point problem, resulting in a highly parallel algorithm with Neumann subproblems. Applications include elasticity, shell and plate problems [FA2, FA16, FA15].