schur¶

Module: core

Category: linearsolver

Type string: "schur"

Parameters¶

Name	Description	Default	Units
`print_level`	print_level	0	[]
`schur_block`	schur_block	0	[]
`do_jacobi`	do_jacobi	false	[]
`zero_D_block`	zero_D_block	false	[]
`schur_pc`	schur_pc	0	[]
`A_solver`			[]
`schur_solver`			[]
`schur_A_solver`			[]

Description¶

The Schur solver is a linear solver that takes a 2x2 block structured matrix and solves the linear system by decomposing it in its Schur-complement form. Thus, instead of solving the following system directly,

\[ \left[\begin{array}{cc} A & B\\ C & D \end{array}\right]\left[\begin{array}{c} u\\ v \end{array}\right]=\left[\begin{array}{c} a\\ b \end{array}\right] \]

it first solves for v,

\[ Sv=b-CA^{-1}a \]

where \(S=D-CA^{-1}B\) is the Schur complement of \(A\). Then the Schur solver solves for \(u\),

\[ u=A^{-1}\left(a-Bv\right) \]

In order to use this solver, users need to select how they wish to solve the A-block, and how to solve the Schur complement. For the A-block users can choose any of the linear solvers listed above, either direct or iterative. For the Schur complement users can only choose a Krylov-based iterative solver (FGMRES or CG) because FEBio does not form the Schur complement explicitly.

This solver is often more effective as a preconditioner for FGMRES. In this case, the A-block and Schur complement only need to be solved approximately. See the examples below for some example configurations. This solver requires a block-structured matrix. In order to generate the block structure you need to set the equation_scheme control parameter to 1 in the model input file.