We investigate using the gmres method with the differentiation operator. This operator is unbounded, and thus does not fall into the framework of existing Krylov subspace theory. We establish conditions under which a function can be approximated by its own derivatives in a domain of the complex plane. These conditions are used to determine when gmres converges. This algorithm outperforms tradit...

In this paper we apply simple GMRES bounds to the nearly singular systems that arise in ill-posed problems. Our bounds depend on the eigenvalues of the coefficient matrix, the right-hand side vector and the nonnormality of the system. The bounds show that GMRES residuals initially decrease, as residual components associated with large eigenvalues are reduced, after which semi-convergence can be...

In this work, a two-layer coupled fluid-structure-structure interaction model is considered, which incorporates an anisotropic structure model into the fluid-structure interaction problems. We propose two domain decomposition solvers for such a class of coupled problems: a Robin-Robin preconditioned GMRES solver combined with an inner Dirichlet-Neumann iterative solver, and a Robin-Robin precon...

Standard Krylov subspace methods only allow the user to choose a single preconditioner, although in many situations there may be a number of possibilities. Here we describe an extension of GMRES, multi-preconditioned GMRES, which allows the use of more than one preconditioner. We give some theoretical results, propose a practical algorithm, and present numerical results from problems in domain ...

In this paper, we use the BiCG, BiCGSTAB methods as preconditioned techniques. Also we compare the preconditioned Krylov subspace methods such as GMRES, GMRES(m), QMR, BiCG, CGS, BiCGSTAB for solving linear systems arising from a class of fourth-order approximations for solving the elliptic partial differential equation Auxx + Buyy = f(x,y,u,ux,uy), where A and B are constants. Numerical result...

We show that any admissible cycle-convergence behavior is possible for restarted GMRES at a number of initial cycles, moreover the spectrum of the coefficient matrix alone does not determine this cycleconvergence. The latter can be viewed as an extension of the result of Greenbaum, Pták and Strakoš (SIAM Journal on Matrix Analysis and Applications 1996; 17(3):465–469) to the case of restarted G...

The discrete convection-diffusion equations obtained from streamline diffusion finite element discretization are solved on both uniform meshes and adaptive meshes. The performance of geometric multigrid method (GMG) and algebraic multigrid method (AMG), both as a solver and as a preconditioner of the generalized minimal residual method (GMRES), are evaluated. Our numerical results show that GMR...

Numerical investigation of BiCG and GMRES methods for solving non-symmetric linear equation systems with dominant skew-symmetric part has been presented. Numerical experiments were carried out for the linear system arising from a 5-point central difference approximation of the two dimensional convection-diffusion problem with different velocity coefficients and small parameter at the higher der...

This paper studies convergence properties of the block gmres algorithm when applied to nonsymmetric systems with multiple right-hand sides. A convergence theory is developed based on a representation of the method using matrix-valued polynomials. Relations between the roots of the residual polynomial for block gmres and the matrix "-pseudospectrum are derived, and illustrated with numerical exp...