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

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Abstract

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(\(\ell \)), 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.

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Notes

  1. Galerkin approximation for (18) is inaccurate if the mesh is not fine enough to resolve the layers in the solution and these inaccuracies may also propagate and pollute the approximated solution in regions where the exact solution is well behaved. An alternative way to handle boundary layers is by using Shishkin grids; see [19].

  2. Note that for BICGSTAB2, \(k_1\), \(k_2\) are the iteration counts for the built in stopping criterion in the IFISS implementation of BICGSTAB2 to tolerances of 1e−6, 1e−9 respectively. The iteration counts \(k^*\) for BICGSTAB2 is obtained by incorporating the weak stopping criterion in this IFISS implementation of BICGSTAB2.

  3. The average cputimes were computed for each h by averaging over cputimes generated from 10 independent runs (of eigs and the IFISS script for computing \(\eta _h\)) for each h.

  4. This observation is clear from \(\Lambda _h\) values in Table 5, however a rigorous mathematical proof is still under research.

  5. \(\mathbf{X}^h_{{E}}\) is not a vector space unless its elements (which are functions) are zero on the boundary.

  6. This is not a rigorous mathematical statement. A proof for this statement is an ongoing research.

  7. This \(k^*\) will in general be different for different l.

  8. This a posteriori approximation error estimate is for the linearized part \((\delta \overrightarrow{u}_h^{(l_k)}, \delta {p}_h^{(l_k)})\).

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Authors and Affiliations

  1. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, USA

    Pranjal

  2. School of Mathematics, University of Manchester, Manchester, UK

    David Silvester

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Pranjal, Silvester, D. Balanced Iterative Solvers for Linear Nonsymmetric Systems and Nonlinear Systems with PDE Origins: Efficient Black-Box Stopping Criteria. J Sci Comput 81, 271–290 (2019). https://doi.org/10.1007/s10915-019-01018-w

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