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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.