Improving the speed of convergence of GMRES for certain perturbed tridiagonal systems

摘要

Numerical approximations of partial differential equations often require the employment of spatial adaptation or the utilization of non-uniform grids to resolve fine details of the solution. While the governing continuous linear operator may be symmetric, the discretized version may lose this essential property as a result of adaptation or utilization of non-uniform grids. Commonly, the matrices can be viewed as a perturbation to a known matrix or to a previous iterate's matrix. In either case, a linear solver is deployed to solve the resulting linear system. Iterative methods provide a plausible and affordable way of completing this task and Krylov subspace methods, such as GMRES, are quite popular. Upon updating the matrices as a result of adaptation or multi-grid methodologies, approximate eigenvector information is known stemming from the prior GMRES iterative method. Hence, this information can be utilized to improve the convergence rate of the subsequent iterative method. A one dimensional Poisson problem is examined to illustrate this methodology while showing notable and quantifiable improvements over standard methods, such as GMRES-DR.