Abstract
The paper deals with the solution of large non-symmetric two-by-two block linear systems with a singular leading submatrix. Our algorithm consists of two levels. The outer level combines the Schur complement reduction with the orthogonal projectors that leads to the linear equation on subspaces. To solve this equation, we use a Krylov-type method representing the inner level of the algorithm. We propose a general technique how to get from the standard Krylov methods their projected variants generating iterations on subspaces. Then we derive the projected GMRES. The efficiency of our approach is illustrated by examples arising from the combination of the fictitious domain and FETI method.
Highlights
We consider two-by-two block linear systems A u λ = f g (1) where A= A B1 B2 −C ∈ R(n+m)×(n+m)Element Tearing and Interconnecting) domain decomposition method [8], [6], [20] is used for the numerical solution of elliptic PDEs, we get a saddle-point linear system, i.e., Eq (1) with A being symmetric, positive semidefinite, B1 = B2, and C = 0
Instead of Eq (33), we propose to solve the following fictitious domain (FD) formulation of Eq (32) in Ω: where σ(u) is the stress tensor in ω corresponding to u and ν stands for the unit outward normal vector to γ
Notice that λΓ plays the role of the control variable defined on Γ enforcing the c 2014 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING
Summary
The outer level combines the Schur complement reduction requiring a generalized inverse to A with the null-space method performed by orthogonal projectors It results in the linear equation with the singular matrix that is, the symmetric and positive definite operator in a subspace V ⊂ Rm. It results in the linear equation with the singular matrix that is, the symmetric and positive definite operator in a subspace V ⊂ Rm This equation can be solved in V by the projected CGM representing the inner level of the FETI algorithm. The outer level is based on the same ideas as in the FETI algorithm, we arrive at the linear equation whose matrix is the invertible operator between two different subspaces V1 and V2 in Rm. the projected iterative method for non-symmetric, indefinite operators is needed in the inner level of the PSCM.
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