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On Efficient Numerical Solution of Linear Algebraic Systems Arising in Goal-Oriented Error Estimates

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Abstract

We deal with the numerical solution of linear partial differential equations (PDEs) with focus on the goal-oriented error estimates including algebraic errors arising by an inaccurate solution of the corresponding algebraic systems. The goal-oriented error estimates require the solution of the primal as well as dual algebraic systems. We solve both systems simultaneously using the bi-conjugate gradient method which allows to control the algebraic errors of both systems. We develop a stopping criterion which is cheap to evaluate and guarantees that the estimation of the algebraic error is smaller than the estimation of the discretization error. Using this criterion and an adaptive mesh refinement technique, we obtain an efficient and robust method for the numerical solution of PDEs, which is demonstrated by several numerical experiments.

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Notes

  1. After the restart, the machine accuracy is achieved in few steps and the error estimators give vanishing values, therefore we do not show them.

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Acknowledgements

The authors are thankful to their colleagues from the Charles University, namely M.Kubínová, T. Gergelits and F. Roskovec for a fruitful discussion.

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  1. Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83, 186 75, Prague, Czech Republic

    Vít Dolejší & Petr Tichý

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  1. Vít Dolejší
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Correspondence to Vít Dolejší.

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This work was supported by Grants Nos. 17-04150J and 20-01074S of the Czech Science Foundation.

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Dolejší, V., Tichý, P. On Efficient Numerical Solution of Linear Algebraic Systems Arising in Goal-Oriented Error Estimates. J Sci Comput 83, 5 (2020). https://doi.org/10.1007/s10915-020-01188-y

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