The theoretical guarantees on the convergence of the truncated preconditioned conjugate gradient (TPCG) method for the singular linear system in the multipatch setting are less established compared to the non-singular case. However, several insights can be drawn from existing literature on Krylov subspace methods and their behavior with singular systems.
It is known that Krylov solvers, including the conjugate gradient method, can perform well with singular systems provided that the right-hand side lies within the range of the system matrix. In the context of the multipatch method, the low-rank approximations and the overlapping Schwarz preconditioner can help maintain the convergence properties of the TPCG method. The use of low-rank Tucker approximations allows for a more compact representation of the system, which can mitigate the effects of singularity.
While the convergence of the TPCG method is not guaranteed in the singular case, empirical results from numerical experiments suggest that the method exhibits robust performance, similar to that observed in non-singular cases. The convergence behavior can be further analyzed by studying the spectral properties of the preconditioned system and ensuring that the preconditioner effectively reduces the condition number of the system matrix. Future work could focus on establishing more rigorous theoretical frameworks to analyze the convergence of TPCG in the context of singular linear systems.