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Convergence Analysis for Virtual Element Discretizations of the Cardiac Bidomain Model

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Abstract

We propose here a convergence analysis for virtual element discretizations of the cardiac Bidomain model, a degenerate system of parabolic reaction-diffusion equations that models the propagation of the electric signal in the cardiac tissue. The virtual element method is a recent numerical technology that generalizes finite elements by considering polytopal computational grids, thus allowing more flexibility and accuracy in approximating complex computational domains. This can be an advantage when modeling for instance damaged cardiac tissues or structural heterogeneities. A previous similar study was performed in Anaya et al. (IMA J Numer Anal 40(2):1544–1576, 2020), where the propagation was modeled by means of a scalar nonlocal FitzHugh-Nagumo reaction-diffusion model. In the present work, we extend this analysis to the full semi-discrete Bidomain system, providing extensive numerical tests that validate the theoretical result on several structured and unstructured meshes.

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Data Availability

The numerical experiments have been performed using an in-house matlab code available upon request to the author.

Notes

  1. The degenerate nature of this parabolic system will be reflected in the choice of the appropriate virtual element spaces and in the definition of the correct projection operators.

  2. Remark: this request is trivial, since for modeling reasons the applied current is always bounded.

  3. These are the discrete counterparts of the continuous norms defined in the Introduction, Sect. 1.

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Acknowledgements

The author would like to thank Simone Scacchi for many helpful discussions and feedbacks.

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  1. Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 5, 27100, Pavia, Italy

    Ngoc Mai Monica Huynh

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  1. Ngoc Mai Monica Huynh
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The author has been supported by grants of Istituto Nazionale di Alta Matematica (INDAM-GNCS) and the European High-Performance Computing Joint Undertaking EuroHPC under Grant Agreement No. 955495 (MICROCARD) co-funded by the Horizon 2020 programme of the European Union (EU), and the Italian ministry of economic development.

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Huynh, N.M.M. Convergence Analysis for Virtual Element Discretizations of the Cardiac Bidomain Model. J Sci Comput 98, 37 (2024). https://doi.org/10.1007/s10915-023-02435-8

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