We derive and analyze adaptive solvers for boundary value problems in which the differential operator depends affinely on a sequence of parameters. These methods converge uniformly in the parameters and provide an upper bound for the maximal error. Numerical computations indicate that they are more efficient than similar methods that control the error in a mean square sense.

We derive and analyze adaptive solvers for boundary value problems in which the differential operator depends affinely on a sequence of parameters. These methods converge uniformly in the parameters and provide an upper bound for the maximal error. Numerical computations indicate that they are more efficient than similar methods that control the error in a mean square sense.

In this paper, our attention is concentrated on the GMRES method for the solution of the system $(I-T)x=b$ of linear algebraic equations with a nonsymmetric matrix. We perform $m$ pre-iterations $y_{l+1}=T{y}_l+b$ before starting GMRES and put $y_m$ for the initial approximation in GMRES. We derive an upper estimate for the norm of the error vector in dependence on the $m$th powers of eigenvalues of the matrix $T$. Further we study under what eigenvalues lay-out this upper estimate is the best one. The estimate shows and numerical...