Balanced Iterative Solvers for Linear Nonsymmetric Systems and Nonlinear Systems with PDE Origins: Efficient Black-Box Stopping Criteria

This paper discusses the design of efficient algorithms for solving linear nonsymmetric systems and nonlinear systems associated with FEM approximation of elliptic PDEs. The novel feature of the designed linear solvers like GMRES, BICGSTAB( ℓ ), TFQMR, and nonlinear solvers like Newton and Picard, is the incorporation of error control in the ‘natural norm’ in combination with an effective a posteriori estimator for the PDE approximation error. This leads to robust black-box stopping criteria in the sense that the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error. Such a solver is called ‘balanced’ in this paper since the stopping criteria are obtained by balancing (comparing) the algebraic error and the approximation error.