Applications of Symmetric and Nonsymmetric MSSOR Preconditioners to Large-Scale Biot's Consolidation Problems with Nonassociated Plasticity

Table 1

Block preconditioners recently proposed for Biot’s linear system.

Authors	Block preconditioner
	Type	𝐊	𝐒
Phoon et al. [3]	Diagonal	Diag(𝐾)	diag[𝐶+𝐵^𝑇diag(𝐾)^−1𝐵]
		Direct inverse	Direct inverse
Simoncini [6]	Triangular	𝐾	𝐶+𝐵^𝑇𝐵
		Incomplete Cholesky	Incomplete Cholesky
Toh et al. [10]	All three types	diag(𝐾)	𝐶+𝐵^𝑇𝐾^−1𝐵
		Direct inverse	Incomplete or sparse Cholesky
Bergamaschi et al. [11]	Constraint	𝐿K𝐿K^𝑇&(𝑍𝑍^𝑇)^−1	𝐶+𝐵^𝑇𝐾^−1𝐵
		Incomplete Cholesky and approximate inverse	Incomplete Cholesky
Haga et al. [13]	Constraint	diag(𝐾) & 𝐾	diag[𝐶+𝐵^𝑇diag(𝐾)^−1𝐵] & 𝐶+𝐵^𝑇diag(𝐾)^−1𝐵
		AMG	Direct inverse & AMG
Chaudhary [15]	Diagonal	Partitioned block diagonal	diag[𝐶+𝐵^𝑇diag(𝐾)^−1𝐵]
		Cholesky & direct inverse	Direct inverse