FETI

struct DirichletPreconditioner <: Preconditioner

Dirichlet preconditioner that uses a Schur-complement to precondition the interface problem. This is the most accurate but also most expensive preconditioner.

The exact definition for the preconditioner $D_{I}^{-1}$ for $N_s$ subdomains is given below for interior dofs $i$, interface dofs $b$, subdomain $s$, interface $I$, and connectivity operator $B$

$D_{I}^{-1} = \sum_{s=1}^{N_s} B^{s} \left[\begin{array}{c} 0 & 0 \\ 0 & S_{bb}^{s} \end{array}\right] B^{s^{T}},$

where the Schur complement $S_{bb}^{s} = K_{bb}^{s} - K_{ib}^{s^T}K_{ii}^{s^{-1}}K_{ib}^{s}$.

References

[1] : Equation (38) and (43) from Farhat, C., Mandel, J., and Roux, F.X. Optimal convergence properties of the FETI domain decomposition method. Computer Methods in Applied Mechanics and Engineering. vol 115, 365-385. 1994.

struct LumpedPreconditioner <: Preconditioner

Preconditions the dual-interface problem with an economical preconditioner only involving the the stiffness matrix $K_{j}^{I}$ for a subdomain $j$ at the interface $I$.

The exact definition for the preconditioner $L_{I}^{-1}$for $N_s$ subdomains is given below for interface dofs $b$, subdomain $s$, and connectivity operator $B$

$L_{I}^{-1} = \sum_{s=1}^{N_s} B^{s} \left[\begin{array}{c} 0 & 0 \\ 0 & K_{bb}^{s} \end{array}\right] B^{s^{T}}.$

References

[1] : Equation (44) from Farhat, C., Mandel, J., and Roux, F.X. Optimal convergence properties of the FETI domain decomposition method. Computer Methods in Applied Mechanics and Engineering. vol 115, 365-385. 1994.

struct TracePreconditioner <: Preconditioner

Simple preconditioner for the interface problem. Uses the sum of the trace $tr(\cdot)$ of local stiffness matrices ${K}_{bb}^{s}$ on the interface $I$ with interface dofs denoted by $b$ and subdomains by $s$ for $N_s$ subdomains.

$P_{I}^{-1} = \sum_{s=1}^{N_s} tr(K_{bb}^{s}).$

References

[1] : Equation (24) from Farhat, C. A Method of Finite Element Tearing and Interconnecting and its Parallel Solution Algorithm. International Journal for Numerical Methods in Engineering. vol 32, 1205-1227. 1991.