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Mesh-Dependent \(L^2\)-Like Norm a Posteriori Error Estimates for Elliptic Problems with Non-essential Boundary Conditions

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Abstract

This work is concerned with the proof of \(L^2\)-like norm residual-type a posteriori error estimates for finite element methods for elliptic problems with non-essential boundary conditions, such as Neumann or Robin type. To ensure the proof of lower bounds (efficiency), a non-standard mesh-dependent \(L^2\)-like norm is used for the error. The proof of lower bounds requires a carefully constructed \(C^1\)-conforming ’bubble’-function. A series of numerical experiments is presented, showcasing the good performance of the estimators.

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No external data sets have been used in the manuscript. All plots are generated from purpose-built computer code. The code is available upon reasonable request to the corresponding author.

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Acknowledgements

This research work was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under 1) the “First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant" (Project Number: 3270, support for KC and EHG) and 2) the 3rd Call for H.F.R.I. PhD Fellowships (Fellowship Number: 6532, support for VDP). Also, EHG wishes to acknowledge the financial support of EPSRC (grant number EP/W005840/2) and of The Leverhulme Trust (grant number RPG-2021-238). We would like to thank Dr Gabriel Barrenechea (University of Strathclyde) and the two anonymous referees for their insightful comments which, we feel, have resulted in an improved revised manuscript.

Funding

Hellenic Foundation for Research and Innovation (3270, 6532), Engineering and Physical Sciences Research Council (EP/W005840/2), Leverhulme Trust (RPG-2021-238).

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Authors and Affiliations

  1. Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, 15780, Zografou, Greece

    Konstantinos Chrysafinos, Emmanuil H. Georgoulis & Vassilis D. Papadopoulos

  2. The Maxwell Institute for Mathematical Sciences & Department of Mathematics, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Emmanuil H. Georgoulis

  3. Institute of Applied and Computational Mathematics-FORTH, Heraklion, Crete, Greece

    Konstantinos Chrysafinos & Emmanuil H. Georgoulis

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  1. Konstantinos Chrysafinos
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  2. Emmanuil H. Georgoulis
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  3. Vassilis D. Papadopoulos
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Correspondence to Emmanuil H. Georgoulis.

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Chrysafinos, K., Georgoulis, E.H. & Papadopoulos, V.D. Mesh-Dependent \(L^2\)-Like Norm a Posteriori Error Estimates for Elliptic Problems with Non-essential Boundary Conditions. J Sci Comput 100, 8 (2024). https://doi.org/10.1007/s10915-024-02559-5

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  • DOI: https://doi.org/10.1007/s10915-024-02559-5

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