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A Robust and Mass Conservative Virtual Element Method for Linear Three-field Poroelasticity

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Abstract

We present and analyze a robust and mass conservative virtual element method for the three-field formulation of Biot’s consolidation problem in poroelasticity. The displacement and fluid flux are respectively approximated by enriched \({\varvec{H}}({\text {div}})\) virtual elements and \({\varvec{H}}({\text {div}})\) virtual elements, while the pressure is discretized by piecewise polynomial functions. Optimal a priori error estimates are obtained, including the semi-discrete scheme and the fully-discrete scheme with the implicit Euler approximation in time. Moreover, our method achieves robustness with respect to the constants hidden in the error estimates, even for the Lamé coefficient tending to infinity and the arbitrarily small constrained specific storage coefficient, and therefore it is free of both volumetric (Poisson) locking and nonphysical pressure oscillations. Meanwhile, it also conserves pointwise mass conservation for Biot’s consolidation problem on the discrete level.

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Acknowledgements

The authors are very grateful to Editors and anonymous reviewers for their valuable suggestions on an earlier version.

Funding

Research supported by the National Natural Science Foundation of China (No.11971337) and the Scientific Research Foundation of Chengdu University of Information Technology (KYTZ202184).

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Authors and Affiliations

  1. College of Mathematics, Sichuan University, Chengdu, 610064, People’s Republic of China

    Jun Guo & Minfu Feng

  2. College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, 610225, People’s Republic of China

    Jun Guo

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  1. Jun Guo
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Correspondence to Minfu Feng.

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Guo, J., Feng, M. A Robust and Mass Conservative Virtual Element Method for Linear Three-field Poroelasticity. J Sci Comput 92, 95 (2022). https://doi.org/10.1007/s10915-022-01960-2

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