Improvement of preconditioned bi-Lanczos-type algorithms with residual norm minimization for the stable solution of systems of linear equations

In this paper, improved algorithms are proposed for preconditioned bi-Lanczos-type methods with residual norm minimization for the stable solution of systems of linear equations. In particular, preconditioned algorithms pertaining to the bi-conjugate gradient stabilized method (BiCGStab) and the generalized product-type method based on the BiCG (GPBiCG) have been improved. These algorithms are more stable compared to conventional alternatives. Further, a stopping criterion changeover is proposed for use with these improved algorithms. This results in higher accuracy (lower true relative error) compared to the case where no changeover is done. Numerical results confirm the improvements with respect to the preconditioned BiCGStab, the preconditioned GPBiCG, and stopping criterion changeover. These improvements could potentially be applied to other preconditioned algorithms based on bi-Lanczos-type methods.

A stabilized coupled method and its optimal error estimates for elliptic interface problems

In this paper, we present a stabilized coupled algorithm for solving elliptic interface problems, mainly by introducing the jump of the solutions along the interface. A framework of theoretical proofs is provided to show the optimal error estimates of this stabilized method. Several numerical experiments are carried out to demonstrate the computational stability and effectiveness of the method.